Question 27.3: Recall the drug capsule described in Example 2. The present ...

This problem is readily solved using the charts given in Appendix F.

(a) Spherical capsule: First, calculate the relative time \left(X_{D}\right), relative position (n), and relative resistance (m) based on the spherical coordinate system

\begin{gathered} X_{D}=\frac{D_{A B} t}{R^{2}}=\frac{\left(3 \times 10^{-7}  \frac{\mathrm{cm}^{2}}{\mathrm{~s}}\right)\left(48 \mathrm{~h} \frac{3600 \mathrm{~s}}{1 \mathrm{~h}}\right)}{(0.326 \mathrm{~cm})^{2}}=0.488 \\ n=\frac{r}{R}=\frac{0 \mathrm{~cm}}{0.326 \mathrm{~cm}}=0(\text { center of sphere }) \\ m=\frac{D_{A B}}{k_{c} R} \approx 0 \end{gathered}

From Figure F.1 or Figure 18.3, the value for Y, which in this case is the unaccomplished concentration change at the center of the spherical bead, is about 0.018. We can now calculate c_{A}

Y=0.018=\frac{c_{A s}-c_{A}}{c_{A s}-c_{A o}}=\frac{0-c_{A}}{0-68.9  \mathrm{mg} / \mathrm{cm}^{3}}

The residual Dramamine concentration at the center of the bead after 48 \mathrm{~h}\left(c_{A}\right) is 1.24  \mathrm{mg} / \mathrm{cm}^{3}.

(b) For the cube-shaped capsule the distance from the midpoint of the cube to any of the six faces is 0.652 \mathrm{~cm} / 2. The relative time X_{D} is now defined as

X_{D}=\frac{D_{A B} t}{x_{1}^{2}}=\frac{\left(3 \times 10^{-7}  \frac{\mathrm{cm}^{2}}{\mathrm{~s}}\right)\left(48 \mathrm{~h} \frac{3600 \mathrm{~s}}{1 \mathrm{~h}}\right)}{(0.326 \mathrm{~cm})^{2}}=0.488

Values for n and m are unchanged, with n=0 and m=0. As all of the faces of the cube are of equal dimension, let

Y=Y_{a} Y_{b} Y_{c}=Y_{a}^{3}

From the appendix Figure F.4, given X_{D}=0.488, \mathrm{~m}=0 and n=0, Y_{a} is 0.4 for a flat plate of semi-thickness x_{1}=a=0.326 \mathrm{~cm}. Extending this value to a three-dimensional cube using the above relationship, we have

Y=Y_{a}^{3}=(0.4)^{3}=0.064

Finally,

Y=0.064=\frac{c_{A s}-c_{A}}{c_{A s}-c_{A o}}=\frac{0-c_{A}}{0-68.9  \mathrm{mg} / \mathrm{cm}^{3}}

with c_{A}=4.41  \mathrm{mg} / \mathrm{cm}^{3} after 48 \mathrm{~h}.

(c) For a cylindrical capsule with exposed ends, R=0.652 \mathrm{~cm} / 2 for the radial coordinate, and x_{1}=a=0.3 \mathrm{~cm} / 2 for the axial coordinate. The relative times are

X_{D}=\frac{D_{A B} t}{R^{2}}=\frac{\left(3.0 \times 10^{-7}  \frac{\mathrm{cm}^{2}}{\mathrm{~s}}\right)\left(48 \mathrm{~h} \frac{3600 \mathrm{~s}}{1 \mathrm{~h}}\right)}{(0.326 \mathrm{~cm})^{2}}=0.488

for the cylindrical dimension and

X_{D}=\frac{D_{A B} t}{x_{1}^{2}}=\frac{\left(3.0 \times 10^{-7}  \frac{\mathrm{cm}^{2}}{\mathrm{~s}}\right)\left(48 \mathrm{~h} \frac{3600 \mathrm{~s}}{1 \mathrm{~h}}\right)}{(0.15 \mathrm{~cm})^{2}}=2.30

for the axial dimension. Values for n and m are unchanged, with n=0 and m=0. From Figures F.1 and F.2, respectively, Y_{\text {cylinder }}=0.1 for the cylindrical dimension and Y_{a}= 0.006 for the axial dimension. Therefore,

Y=Y_{\text {cylinder }} Y_{a}=(0.1)(0.006)=0.0006

and finally

Y=0.006=\frac{c_{A s}-c_{A}}{c_{A s}-c_{A o}}=\frac{0-c_{A}}{0-68.9  \mathrm{mg} / \mathrm{cm}^{3}}

with c_{A}=0.413  \mathrm{mg} / \mathrm{cm}^{3} after 48 \mathrm{~h}. As Y_{a} \ll Y_{\text {cylinder }}, the flux directed out of the exposed ends of the cylindrical tablet along the axial dimension dominates.

The above calculations assume that convective mass-transfer resistances associated with external fluid flow over the surface of the capsule are negligible. Problems in Chapter 30 will reconsider the drug release for unsteady-state diffusion and convection in series.