PROBLEM $1(50$ POINTS$)$

A proposed device for the oxygenation of blood involves a series of vertical tubes within an oxygen$-$rich gas environment. The tube walls are made of a polymeric membrane that is highly permeable to oxygen. Blood enters from the top and as it flows through the tubes, it becomes oxygenated by contact with an oxygen$-$rich environment.

The device is made of $5$ identical tubes of inner diameter $D_1=8mm$ and outer diameter $D_0=12ommm.$ The gas environment outside is pure oxygen at a total pressure of $1\mathrm{.}9g$ am is at a condo flow rate entering the device is $V^{}=2\mathrm{.}5{10}^{-5}\frac{m^3}{s}.$ The whole de vice with an oxyg en mole lemperature of $298K,$ and operates at steady state. Blood enters the device wio $1{10}^{-5}$ fraction of $9{10}^{-8},$ and it is desired to increase the oxygen mole fraction to flected, calculate the

a$.$ If the resistance to oxygen transfer through the tube walls is neglected, state all your length of each tube required to achieve the desired level of oxygena $,$ the resistance to assumptions please.

b$.$ Explain how your analysis would change if you were to account for the rull analysis, but oxygen transfer through the tube walls. You do not need to perform the full and how your must correctly explain how you would account for this additional ph analysis would change.

Potentially useful data: For any physical property not given, assume that blood behave the tube wall is and look up the relevant properties. The diffusion coefficient for oxygen through the environment can $D_{AB}=1\mathrm{.}5{10}^{-7}c\frac{m^2}{s}.$ Oxygen equilibrium between the tube wall and the gas is the solubility of be described by a linear relationship $C_A=S.A,$ where $.$ atm$)$ is the gas environment. oxygen in the polymeric membrane, and pa is the oxygen partial pressure in the gas a Henry's law with Oxygen equilibrium between water and a gas environment is described by a flow of liquids through Henry's constant $H=0\mathrm{.}78$atm.$m^3$ water $\frac{?}{\text{m}\text{o}\text{l}}.$ For fully developed lam that $\frac{?}{NU}=3\mathrm{.}658.$