Question $2$: A farmer uses a special type of fertilizer, called GrowFast, to boost the growth of his crops. GrowFast is a chemical agent aimed at enhancing plant productivity. GrowFast has been continuously diffusing into a stagnant water body located at the edge of the farm at $x=0$ meters and $y=0$ meters. The concentration profile of GrowFast in the water body is illustrated in Figure $1.$ The water body is characterized by a perfect mixing along its depth $($z$-$axis$).$ It's given that the diffusion coefficient of GrowFast in the water along the $x-$direction is $5{10}^{-4}\frac{m^2}{s}.$

Figure $1.$ Top View of Concentration Contours $(m\frac{g}{L}$ for Gaussian Plume originating at $x=0,y=0m$

a$)$ Draw concentration of GrowFast profile on $x-$axis at $y=0m$ as depicted in Figure $1($Do not put numbers into the graph and do not calculate any concentration at that step, only draw your expectation about shape of the graph$).$

b$)$ Write $1-$D mass balance equation on $x-$axis at $y=0m$ for GrowFast that explains spatial and temporal variations in the stagnant water body. Please do not forget to write initial and boundary conditions.

c$)$ Write the solution of the mass balance equation for concentration of GrowFast.

d$)$ Determine the time at which concentration distribution for GrowFast at $x=55m$ and $y=0m$ in Figure $1$ is obtained.

e$)$ At $x=55m$ and $y=0m,$

i$)$ Calculate GrowFast concentration for $t=10,50,100,200$ and $350$ days.

ii$)$ Plot $\frac{C}{C}0$ vs $t$ graph for $t=10,50,100,200$ and $350$ days. Comment on your expectations for the concentration when time approaches infinity.

f$)$ If there was an instantaneous spill instead of continuous leakage to the stagnant water body at $x=75m,y=0m,$ how would the concentration profile in Figure $1$ change at $t$ in part d$?$ Without making any calculations, draw expected concentration contours like Figure $1$ and comment on it$.$

g$)$ If there was an instantaneous spill instead of continuous leakage to a tube with in the experiment, what would be the challenge of writing the boundary condition and stating your assumptions to define the boundary conditions? Explain it$.($Hint: Please check $1D$ transient diffusion of nonreactive chemicals from an instantaneous local source $($pulse input$)$ in your lecture notes$).$