GATE Civil Engineering (CE) 2022 Shift 2 Solved Paper

The concentration s(x, t) of pollutants in a one-dimensional reservoir at position x and time t satisfies the diffusion equation
$\frac{\partial \mathit{s}\left(\mathit{x},\mathit{t}\right)}{\partial \mathit{t}}=\mathit{D}\frac{{\partial }^2\mathit{s}\left(\mathit{x},\mathit{t}\right)}{\partial {\mathit{x}}^2}$
on the domain 0 ≤ x ≤ L, where D is the diffusion coefficient of the pollutants. The initial condition s(x, 0) is defined by the step-function shown in the figure.

The boundary conditions of the problem are given by $\frac{\partial \mathit{s}\left(\mathit{x},\mathit{t}\right)}{\partial \mathit{t}}=0$ at the boundary points x = 0 and x = L at all times. Consider D = 0.1 m²/s, s₀ = 5 μmol/m, and L = 10 m.
The steady state concentration $\overline{\mathit{s}}\left(\frac{\mathit{L}}{2}\right)=\mathit{s}\left(\frac{\mathit{L}}{2},\mathit{\infty }\right)$ at the center $\mathit{x}=\frac{\mathit{L}}{2}$ of the reservoir (in μmol/m) is _______. (in integer)