As society searches for technical solutions to global warming, vast ponds of photosynthetic algae are seen as one possible solution for the removal of CO₂ from combustion of fossil fuels. Consider the process shown in the figure (next page), where a flue gas containing a binary mixture 10 mole% CO₂ and 90 mole% N₂ is bubbled into a pond containing photosynthetic, single-celled algae that are uniformly suspended in the liquid water. The agitation provided by the rising bubbles keeps the liquid suspension well mixed. The algae consume dissolved CO₂ according to a first-order reaction of the form

where RA is the CO₂ consumption per unit volume of liquid suspension (gmole CO₂/m³ s, k₁ is the rate constant for CO₂ uptake by the algal cells (m³/g cells/h), X is the algal cell density (g cells/m³), c_AL is the concentration of dissolved CO₂ (gmole CO₂/m³), and k₁ is the apparent rate constant for CO₂ uptake (k₁ k₁ X h^–1). In the present process, there is no inflow or outflow of water into the pond. However, there is a constant delivery of CO₂ to the liquid suspension by the bubbled-in flue gas, and a constant consumption of dissolved CO₂ by the photosynthetic algae, with k_{1 =} 0 06435 m³/g cells h. Furthermore, since CO² is not very soluble in water, it was found that the composition of CO₂ gas inside the gas bubble decreased only slightly as it traveled up the water column. Therefore, the partial pressure of CO² gas inside the gas bubble (p_A) is considered constant. However, there is still a significant rate of transfer of CO₂ into the liquid phase. The open pond is maintained at 20 C. At 20 C, the Henry’s law constant for the dissolution of CO₂ gas in the liquid is 0.025 atm m³/gmole, the mass density of liquid is 1000 kg/m³, and the viscosity of the liquid is 993 x 10^–6 kg/m s. The aeration system is designed so that the average bubble size is 7.0 mm and the desired interphase area/liquid volume is a 5.0 m²/m³. The mass density of the gas mixture is 1.2 kg/m³. The total liquid volume of the pond is kept constant.
a. What is the molecular diffusion coefficient of CO₂ in water?
b. What is the mass-transfer coefficient for CO₂ on the liquid film side of the gas bubble, k_L?
c. Develop a material balance model for predicting the dissolved CO₂ concentration in the pond water. The model must be in algebraic form and contain the following variables: c_AL, p_A, H, k_L, a, k₁.
d. If the cell density in the pond is maintained at 50 g cells/m³,what is the predicted concentration of dissolved CO₂ in the pond, c_AL?
e. If the total pond volume is 1000 m³, what is the total CO₂ removal rate in kg CO₂/h?

Transcribed Image Text:

RA = -K₁X.CAL = -k₁. CAL