An engineer decides to set up a solar pond, which stores thermal energy by making a high salinity

layer at the bottom of the pond that is sufficiently dense that it will not float to the top, minimizing

convection and thus sequestering heat in the lower layer of the pond. She looks to accomplish this

by first covering the bottom of a reservoir with a solid layer of salt and then adding fresh water to a

depth $L=1\mathrm{.}0m$ at time $t=0.$ Diffusion of the salt into the fresh water will then generate the

necessary salinity gradient.

The salt is NaCl, which has a diffusivity in water of $D=1\mathrm{.}2*{10}^{-9}\frac{m^2}{s}$ at $30C$ and saturation

density in water ${}_{\text{N}\text{a}\text{C}\text{l}\text{,}\text{s}\text{a}\text{t}}=400k\frac{g}{m^3}.$

As a first approximation, it can be assumed that the diffusion coefficient and the total mass density

in the solution, ${}_{\text{t}\text{o}\text{t}},$ are constant. Note that the assumption of constant total density implies that a

certain mass of salt is replaced by an equal mass of water during dissolution.

A$)$ Starting from Fick$$s rate equation and the general differential equation for mass transfer,

derive the differential equation for mass density of NaCl in the solar pond, Dnacl Make

sure to clearly state your assumptions.

B$)$ Provide the initial and$/$or boundary conditions needed to solve the differential equation for

the early stages of the salt dissolution process.

C$)$ According to laboratory experiments, the solar pond will start working once the salt

concentration at $20$ cm from the bottom reaches $25\%$ of saturation. How long will it take

before the pond is operational?

D$)$ If the density of salt is DNaCl $=2000$ kg$/$m$3,$ calculate by how much the thickness of the salt

layer at the bottom has changed when the pond becomes operational.

E$)$ Under operating conditions, the water temperature varies from $30\backslash$deg C at the water surface to

$90\backslash$deg C near the bottom of the pond. Give a quantitative assessment of the initial assumption

that D $=1\mathrm{.}210-9$ m$2/$s is constant in the solar pond by calculating how much D can be

expected to vary. Hint: look at the key parameters potentially changing in this situation

based on$,$ for example, Stokes$-$Einstein and Wilke and Chang equations.

F$)$ How does the answer in part E affect the time you calculated in part C$...$ meaning, is the

answer in C accurate, and if not then in what direction is it potentially inaccurate?