GATE CE 2022 Set 2 — Question 48

The given diffusion equation is:$$\frac{\partial S(x, t)}{\partial t} = D \frac{\partial^2 S(x, t)}{\partial x^2}$$For steady-state conditions, the concentration does not change with time, so $\frac{\partial S(x, t)}{\partial t} = 0$.
Therefore, the equation becomes:$$D \frac{\partial^2 S(x, t)}{\partial x^2} = 0$$Since $D \ne 0$, we have:$$\frac{\partial^2 S(x, t)}{\partial x^2} = 0$$Integrating this equation twice with respect to $x$ gives the steady-state concentration profile:$$S(x) = Ax + B$$where $A$ and $B$ are constants.The boundary conditions are given as no-flux conditions:$$\frac{\partial S(x, t)}{\partial x} = 0 \quad \text{at } x=0 \text{ and } x=L$$From $S(x) = Ax + B$, the first derivative with respect to $x$ is $\frac{\partial S}{\partial x} = A$.
Applying the boundary conditions:
At $x=0$, $A = 0$.
At $x=L$, $A = 0$.
Both conditions imply that $A=0$. Therefore, the steady-state concentration is a constant:$$S(x) = B$$To find the value of $B$, we use the principle of conservation of mass. The total mass of the pollutant in the reservoir must remain constant over time.Initial total mass ($M_{initial}$):
The initial condition $s(x, 0)$ is a step function:
$s(x, 0) = S_0$ for $0 < x < 0.4L$
$s(x, 0) = 0$ for $0.4L < x < L$$$M_{initial} = \int_0^L S(x, 0) dx = \int_0^{0.4L} S_0 dx + \int_{0.4L}^L 0 dx$$$$M_{initial} = S_0 [x]_0^{0.4L} = S_0 (0.4L - 0) = 0.4L S_0$$Final total mass ($M_{final}$) at steady state:$$M_{final} = \int_0^L S(x) dx = \int_0^L B dx = B [x]_0^L = BL$$By conservation of mass, $M_{initial} = M_{final}$:$$0.4L S_0 = BL$$Since $L \ne 0$, we can divide by $L$:$$B = 0.4 S_0$$Given values:
$S_0 = 5 \text{ µmol/m}$Substitute $S_0$ into the equation for $B$:$$B = 0.4 \times 5 = 2 \text{ µmol/m}$$The steady-state concentration is constant throughout the reservoir, so at $x = L/2$, the concentration $S(L/2, \infty)$ is $B$.$$S(L/2, \infty) = 2 \text{ µmol/m}$$The question asks for the numerical value in integer.The final answer is $\boxed{2}$