An engineer proposes to use the mass-transfer equipment shown in the figure (next page) to prepare a water stream containing dissolved oxygen. Liquid water containing no dissolved oxygen is fed to the tank at a rate of 50 gmole/s. The inlet water is passed through a flow diffuser that makes the velocity of the liquid uniform in the tank. The tank contains 10 silicone-walled tubes of 100 cm length and 2.0 cm outer diameter. The silicone walls are permeable to O₂ gas but not to water. The long width of the tank (L) is also 100 cm, and the depth of the tank is 50 cm, so that the cross-sectional area for liquid flow is 5000 cm₂. You may assume that the concentration of dissolved O₂ in the tank is equal to the concentration of dissolved oxygen in the liquid outlet stream—i.e., the liquid volume is well mixed.
Potentially useful data: c*_AL = 5.0 gmole O₂ = m³ (4.0 atm 100% O₂ gas on tube side); ρ_L = 998.2 kg/m³; nL = 0.995 × 10^6 m²/s; D_AB = 2.0 × 10^9 m²/s (A = O₂, B = H₂O).
a. Develop a material balance model to predict the concentration of dissolved oxygen in the outlet liquid (c_AL). State all assumptions. Your final model must be in algebraic form. Your model development should contain the following variables: c_AL,o, inlet concentration of dissolved oxygen; c*_AL, concentration of dissolved oxygen in liquid at the tube surface; D, outer diameter of tube; v∞, bulk velocity of liquid; L, length of tube, long width of tank; W, width of tank, short dimension; kL, liquid film mass-transfer coefficient; Nt, number of tubes.
b. What is the mass-transfer coefficient kL? Does masstransfer process represents an external flow convection or internal flow convection?
c. Based on your results for parts (a) and (b) above, estimate the outlet concentration of dissolved oxygen. Based upon your analysis, do you think that this mass-transfer device works very well? Mention two doable options to increase c_AL.