2.9.When gas is injected into a column of water, a liquid circulation pattern develops. Thus, upflow at a rate Q, (m/s) rises in the central core and downflow occurs at a rate Q, in the annulus. If liquid of composition Co is also injected at the column base at a rate Q, with outflow at the same rate, then Q-Q+Qo (if density is constant). = (a) The injected gas contains a soluble component (with solubility C* moles/m) so that mass transfer occurs by way of a constant volu- metric mass transfer coefficient denoted as k,a. There is also an exchange of solute between upflowing and downflowing liquid at a rate per unit height equal to K(CC). If the flow areas for upflow and downflow areas are equal (4), perform a material bal- ance and show that dCu Qu dz dC Qs dz =kaA(C* - C) - KE(C - Ca) = kaA(CC) + KE(C-Ca) where z is distance from column base. (b) Define new variables to simplify matters as 8-Cu-C* = C-C 5=2(k,aAKE)/Q0 Qu = Qu/20 94 - Qa/Qo (dimensionless distance) (dimensionless upflow) (dimensionless downflow). and show that the coupled relations are de, = 4 d where a = KE/(KE + kaA) +8 = a (c) Use the operator method to find solutions for 8(5) and @(). (d) Show that the resulting four arbitrary constants are not independent and then write the solutions in terms of only two unknown integra- tion constants. (e) Apply the saturation condition 000 as