A “bubbleless” shell-and-tube membrane aeration system is used to transfer oxygen gas (O₂) to liquid water. Water containing no dissolved oxygen is added to the tube side at the entrance at a bulk velocity of 5.0 cm/s. The inner diameter of the tube is 1.0 cm, and the tube wall thickness is 2.0 mm (0.20 cm). The tube wall is made of silicone, a polymer that is highly permeable to O₂ gas, but not to water vapor. Pure oxygen gas (100% O₂) is maintained at a constant pressure of 2.00 atm in the annular space surrounding the tube. The O₂ gas partitions into silicone polymer, and then diffuses through the tube wall to reach the flowing water. As the fluid flows down the length of the tube, the absorption of oxygen will increase the concentration of the dissolved oxygen. Determine the length of tubing necessary for the dissolved oxygen concentration to reach 30% of saturation with respect to the O₂ gas. The process is maintained at 25 C. At 25 C, the Henry’s law constant for O₂ gas dissolved in water is H 0.78 atm m³/gmole, the molecular diffusion coefficient for dissolved O₂ (A) in water (B) is DAB 2.1 10 5 cm²/s, and the kinematic viscosity of water is νB 9.12 10^–3 cm²/s. The solubility constant of O₂ gas in the silicone polymer is Sm 0.029 atm m³-silicone/gmole, and effective diffusion coefficient of O₂ in the silicone polymer is D_Ae 5.0 10^–6 cm²/s.