$(10$ pts; $5$ pts for each$)$ Tubular membranes of silicone rubber can be used for "bubbleless"

aeration of water. A cross section of the tube is shown in the figure. Pure oxygen $(O_2)$ gas is

pressurized to $2\mathrm{.}0$atm inside a section of silicone rubber tubing of inner diameter $12\mathrm{.}7mm$ and

wall thickness $3\mathrm{.}2mm.$ The tubing is immersed in a large volume of an aqueous solution.

Silicone rubber is permeable to oxygen $(O_2)$ gas but is also hydrophobic so that water does not

seep through the rubber. The $O_2$ gas "dissolves" into the silicone rubber with a concentration $C_A$

$($ mole of $\frac{A}{m_3}$ silicone rubber$)$ diffuses through the wall of the tubing, and then re$-$dissolves into

the water of concentration $C_{\text{A}\text{c}\text{d}\text{o}\text{t}\text{s}}.$ As you can see, this is steady state.

a$.$ Develop an equation, in final integrated form, to predict the $O_2$ flux across the tube wall from

$r=R_1$ to $r=R_0,$ using $C_A^{\text{'}}$ to describe the concentration of $O_2$ dissolved in the tube wall

material itself. State all assumptions. You may neglect convective mass$-$transfer resistances

associated with the liquid boundary layer surrounding the tube.

b$.$ At the conditions given above, determine the oxygen transfer rate $(mo\frac{l}{s})$ to the water.

$$ Use these properties $>$

The solubility of $O_2$ gas in silicone rubber in contact with $2\mathrm{.}0atm{O}_2$ gas is

$C_A^{\text{'}}=6\mathrm{.}30$ mole $\frac{O_2}{m^3}.$

The solubility of $O_2$ gas in silicone rubber in contact with the water is $C_A^{\text{'}}=0\mathrm{.}0012$mole$\frac{O_2}{m^3}.$

The molecular diffusivity of $O_2$ in silicone rubber is $0\mathrm{.}1c\frac{m^2}{s}$ in this condition.