Solve the second order equation (1) using the boundary conditions a) and b)

1.73. Mass Transfer in Bubble Column Bubble columns are used for liquid aeration and gas-liquid reactions. Thus, finely suspended bubbles produce large interfacial areas for effective mass transfer, where the contact area per unit vol- ume of emulsion is calculated from the expression a= 6/db, where & is the volume fraction of injected gas. While simple to design and construct, bubble columns sustain rather large eddy dispersion coefficients, and this must be accounted for in the modeling process. For concurrent operation, liquid of superficial velocity UoL is injected in parallel with gas superficial velocity Uog. The liquid veloc- ity profile can be taken as plug shaped, and the gas voidage can be treated as uniform throughout the column. We wish to model a column used to aerate water, such that liquid enters with a composition Co. Axial dispersion can be modeled using a Fickian-like relationship dC J = -De dx moles liquid area time while the solubility of dissolved oxygen is denoted as C*. We shall denote distance from the bottom of the column as x. Derive the steady-state oxygen mole balance for an incremental volume of AAX (A being the col- umn cross-sectional area) and show that the liquid phase balance is dc - dC UOL + kca(C* C) = 0 dx (1 )De dx2 - Lump parameters by defining a new dimension- less length as koax Z UOL and define the excess concentration as y=(C C*), and so obtain the elementary, second-order, ordinary differential equation d?y dy (1) dz2 dz - y = 0) where : (1 )Deka ul (dimensionless) OL Note the usual conditions in practice are such that a