Mammalian tissue is to be grown in the annular space between a tube of $($poly$)$ dimethyl

siloxane $($PDMS$)$ and an impermeable outer shell $($see figure$).$ Oxygen will flow through

the interior of the tubing. In order to meet the metabolic demands of the tissue, the

oxygen will diffuse through the tubing wall and then into the tissue. For this problem, it

may be assumed that any resistance to mass transport associated with the tubing may be

neglected. Thus, the appropriate boundary condition at $r=R_1$ is

$C_{O_2}=C_{O_2}$@$r=R_1$

It is assumed that the $O_2$ reacts with the tissue in accordance with

$N_{o_2}=$ constant $=-7{10}^{-5}mo\frac{l}{m^3}-s$

Furthermore, the diffusivity of $O_2$ in tissue is known to be about $2{10}^{-5}c\frac{m^2}{s}.$ In

addition, the value of $C_{O_2}$ is about $1\mathrm{.}4mo\frac{l}{m^3}.$ Convective transport of $O_2$ may be

neglected.

i$.$ What is the expression for $C_{O_2}(r)?$

ii$.$ There is a critical value of $R_2$ at which $C_{O_2}=0$ at $R_2.$ What is the

expression for this critical radius? For the parameters given above, show

that $R_{\text{e}\text{r}\text{i}\text{t}}<mn\; id="EditorElement\_1494611">4</mn><msub\; id="EditorElement\_1494620"><mrow><mi\; id="EditorElement\_1494613">R</mi></mrow><mrow\; id="EditorElement\_1494619"><mn\; id="EditorElement\_1494616">1</mn></mrow></msub>$ if $R_1=0\mathrm{.}25cm$

DATA

In cylindrical coordinates

grad$*F_{-}=\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r{F}_r)+\frac{1}{r}\frac{del{F}_{}}{\text{d}\text{e}\text{l}}+\frac{del{F}_z}{\text{d}\text{e}\text{l}\text{z}}$

$gra{d}^{2}f=\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{r}})+\frac{1}{r^2}\frac{de{l}^{2}f}{\text{d}\text{e}\text{l}{}^2}+\frac{de{l}^{2}f}{del{z}^2}$