There is an artificial pond located near a great lake. The pond is connected to the

main lake through a small channel, which is used for water intake. In winter, the pond is used

as a storage reservoir for the flow from the lake. In summer, the pond is used for irrigation

purposes. The water levels between the lake and the pond remain the same after wintertime, so

the velocity in the channel becomes zero. However, before the irrigation season, a new industry

started to discharge its wastewater into the lake. This caused the boron concentration in the lake

to rise to $3$ ppm$,$ which is hazardous if used in high concentrations for agricultural irrigation. It

is important to note that boron is an inert substance. The connecting channel is $10$ meters long,

$4$ meters wide, and $2$ meters deep. Assume the lake and the pond are completely mixed and the

diffusion constant for boron is $10-9$ m$2/$s$.$

a$)$ Please draw a conceptual model of the system.

b$)$ Derive steady$-$state $1-$D mass balance equation equation in the channel.

c$)$ Please solve the mass balance equation you developed in part b$.$ Do not forget to indicate

the IC and$/$or BCs$.$ Please do not put numerical values in this step.

d$)$ Derive steady diffusive mass flux using the information you developed in part c$.$ Please

do not put numerical values in this step.

e$)$ What is the characteristic diffusion time of boron in the channel?

f$)$ In summertime, farmers want to treat boron in the irrigation pond using chemical

precipitation techniques up to a safe level, which is $0\mathrm{.}5ppm.$ Calculate the amount of

boron in the precipitate over $1$ month before the irrigation season.