Modeling Tools for Environmental Engineers and Scientists

the VOC inside the bubble (ML –3 ); and C * g is the gas phase concentrationof the VOC that would be in equilibrium with the liquid phase concentration ofthe VOC in the reactor, C. An expression for the bubble travel time and risevelocity can be derived as follows for solving the above MB equation:zt b = → d Cg = d Cgdz = d Cgv b (5.44)v dt b dzd t dzbwhere v b is the terminal rise velocity of the bubble (LT –1 ), and z is the depthof the sparge tank (L). Hence, combining Equations (5.43) and (5.44),b d Cd gz= K GaVV (C * g – C g ) = K GaV (C * g – C g ) (5.45)bVbzGwhere G is the volumetric gas flow rate (L 3 T –1 ). To eliminate C * g, the airwaterpartition coefficient, K a–w (Henry’s Constant), can be used:C * g = K a–w C (5.46)where C is the liquid-phase concentration of VOC in the reactor. The MBequation for VOCs in gas phase can now be solved as follows: d Cd gz= K GaVz G (K a–w C – C g ) C g,outC g =C g,in (Ka–wdC – C g ) K GaVzGBecause C g,in = 0, the final result is: zC g,out = K a–w C 1 – exp – K GaVG (5.47)Finally, an overall steady state MB equation for the VOCs across the reactorcan be written, assuming no other removal mechanism, and solved as follows:0 = QC in – QC – GC g,out0 = QC in – QC – G K a–wC 1 – exp K GaVG (5.48)to yield the concentration of VOCs that can be expected in the effluent ofthe CMFR:C inC = (5.49) 1 + GK a–wQ 1 – exp K GaV G0dz© 2002 by CRC Press LLC