A pond containing a suspension of microorganisms is used to biologically degrade dissolved organic materials in wastewater, as shown in the figure (right column). The pond contains 1000 m3 of liquid. Recently, inflow and outflow pipes were installed. The inlet volumetric flow rate of water is 0.05 m³/s, and the outflow is the same to maintain a constant liquid volume. The concentration of dissolved oxygen in the inlet liquid water is 10.0 mmole/m³. The present process uses an air sparger to deliver 2.0 mm air bubbles into the wastewater. The rising bubbles uniformly mix the entire liquid phase of the pond. Only a very small portion of the O₂ gas within the air bubbles (containing 0.21 atm O₂) dissolves into the liquid, where it is consumed by the microorganisms. The Henry’s law constant to estimate the solubility of O₂ dissolved in the wastewater that is in equilibrium with the partial pressure of O₂ the aeration gas is H = 8.0 × 10^–4 atm-m³/mmole (1000 mmole = 1.0 gmole). The interphase mass-transfer area of the bubbles per unit volume of liquid is equal to 10 m²/m³ for this process. The process is 100% liquid film controlling. At the current conditions of operation, the current biological oxygen consumption demand, or “BOD,”vassociated with the microbial respiration in the pond is 0.200 mmole O₂/m³-s.

a. Develop a steady-state material balance model to predict the dissolved concentration of O₂ in the holding pond.

b. What is the liquid-phase mass-transfer coefficient k_L for O₂?

c. What is the steady-state concentration of dissolved oxygen in the pond?

Potentially useful data (all at process temperature and pressure): D_AB = 2.0 × 10^–9 m²/s (A = dissolved O₂, B = water solvent); r_B,liq = 1000 kg/m³; r_air = 1.2 kg/m³; v_air = 1.6 × 10^–5 m²/s; H ˆ 8.0 104 atm · m³/mmole; µ_B,liq = 825 × 10^–6 kg/m·s.