Problem $3(35$ Points$)$

The first step in making caramel is to dissolve sugar $($sucrose$,C_{12}{H}_{22}{O}_{11},$ MW $342\frac{g}{m}$ole,

density $1600\frac{g}{?}$ liter$)$ in water. The aqueous solubility of sucrose is $C_{\text{s}\text{a}\text{t}}=5$mole$\frac{s}{?}$ liter. Suppose

you start with $50g$ of dry sugar, in the form of solid spheres with an initial diameter $d_p=1e-3m.$

You add $0\mathrm{.}2$ liter of water and stir. Because of the stirring, the average sugar crystal has liquid

flowing by it at $0\mathrm{.}005\frac{m}{s}.$ The dissolution process is mass$-$transfer limited $-$ this is why you

have to stir.

Because it is external mass transfer limited$,$ you need a mass$-$transfer coefficient $k_c.$ The solid

spheres of sugar move around more or less independently so it is not unreasonable to use the

mass$-$transfer coefficient for flow past an isolated sphere $($the Froessling correlation $Sh=2+0\mathrm{.}6$

$R{e}^{\frac{1}{2}}S{c}^{\frac{1}{3}},$ where the first term is negligible compared to the second$).$ In this concentration range,

the diffusion coefficient of sucrose in water $D=2e-10\frac{m^2}{s}.$ Using $v=1e-6\frac{m^2}{s}$ as the kinematic

viscosity of the liquid, and neglecting the change in the volume and the viscosity of the liquid as

the sucrose concentration increases,

$k_c=Sh\frac{D}{d_p}=(1\mathrm{.}4e-7m^{\frac{3}{2}}{s}^{-1})d_p(t{)}^{-\frac{1}{2}}$

Approximately how long does it take the sugar to dissolve?

Hint: The rate of dissolution changes with time, both because the sugar crystals are getting

smaller, changing surface area and $k_c,$ and because the concentration of sucrose in the liquid is

increasing.