PROBLEM STATEMENT A spherical gel particle $0\mathrm{.}5cm$ in diameter is infused with a drug solution to a uniform concentration of $30\mathrm{.}6m\frac{g}{c}{m}^3$ throughout the sphere $(2mg$ total of drug in the sphere$).$ When the sphere reaches the small intestine, the drug starts diffusing out. The solution in the intestine surrounding the gel quickly reaches a constant drug concentration of $0\mathrm{.}23m\frac{g}{c}{m}^3.$

a$)$ For the in vitro experiment, the sphere is immersed in a flask containing water stirred with a bulk velocity of $0\mathrm{.}5c\frac{m}{s}($resulting in a Reynolds number of $38$ for this flow$).$ Using the most appropriate correlation in WRF Chapter $30,$ calculate the mass transfer coefficient for convection of the drug away from the surface of the sphere. Be sure to note the reason for selecting the correlation you use.

b$)$ Calculate the Biot number $B{i}_m$ for the drug release from the gel during the in vitro experiment $($defined as $B{i}_m=\frac{1}{m}$ in Appendix $F$ Table $F9).$ Does the magntiude of the Biot number suggest that the solution is being stirred rapidly enough to ensure that the data obtained is controlled by diffusion in the gel and not be affected the rate of convection in the solution?

c$)$ In the small intestine, a typical Biot number for mass transfer $B{i}_m$ from a spherical drug delivery device is $Bi=5.$ Using this value, estimate the time required for the concentration of the drug in the center to fall to $0\mathrm{.}26m\frac{g}{c}{m}^3.$ At this point, the gel sphere is effectively depleted of the drug $($i$.$e$.,C_A$ approaches zero everywhere in the gel$).$

d$)(1$ pt extra credit$)$ Contrast the Biot number found for the in vitro experiment with the value used in the in vivo experiment. Do their relative values make sense considering the differences in the experiments $($ release in the flask vs$.$ release in the intestine$)?$ Explain briefly.

Physical properties:

Diffusion coefficient of the drug in the gel:

$4\mathrm{.}0{10}^{-6}c\frac{m^2}{s},$

Diffusion coefficient of the drug in the intestinal fluid: $,1\mathrm{.}0{10}^{-5}c\frac{m^2}{s}.$

Partition coefficient of the drug between the gel and solution: $0\mathrm{.}87,$ defined as $H_A=0\mathrm{.}87\frac{m\frac{g}{c}m3\text{g}\text{e}\text{l}}{m\frac{g}{c}m\text{f}\text{l}\text{u}\text{i}\text{d}}.$

Gel density: $1\mathrm{.}0\frac{g}{c}{m}^3$ and

All fluid properties are equal to those of water at $37C(310K)$ and $1$atm.