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 $H_A=0\mathrm{.}87\frac{m\frac{g}{c}m3\text{g}\text{e}\text{l}}{m\frac{g}{c}m3\text{f}\text{l}\text{u}\text{i}\text{d}}.$

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.