The ''bubble less'' shell-and-tube membrane aeration system shown below is used to transfer oxygen gas to liquid water. Water completely free of dissolved oxygen is added to the tube side at the entrance. Pure oxygen gas, maintained at a constant pressure of 1.0 atm, flows through the annular space. The inner diameter of the tube is 1.0 cm. The tube wall is made of silicone, a polymer that is highly permeable to O₂ gas but not to water vapor. The O₂transfer properties associated with the thin tube wall are neglected, so that the concentration of dissolved O₂ in the water at the inner surface of the wall is established by the O₂ partial pressure on the inside of the tube. The tube length is 500 cm, and the bulk velocity of liquid through the tube is 50 cm/s. The process is maintained at 25°C, wherethe Henry's law constant for O₂ gas dissolvedin water is 0.78 atm (mol/m³) and the diffusion coefficient for dissolved O₂ in water is 2:1 × 10^-5 cm²/s, and the kinematic viscosity of water is 9:12 × 10^-3 cm²/s.

a. What are the Sherwood and Schmidt numbers inside the tube?

b. What is the predicted exit concentration of dissolved oxygen? It will be wise to develop a material balance model on O₂ for the shell-and-tube mass exchanger.

c. What would be the required length of tubing to achieve 60% of dissolved O₂ saturation?