Modeling Tools for Environmental Engineers and Scientists

to the flow, the molar transport rate, J x,i (MT –1 ) by advection is thereforefound from the following:J x,i = A x N x,i = A x v x C i = QC i (4.22)where Q is the volumetric flow rate (L 3 T –1 ) of the bulk medium.4.5 INTERPHASE MASS TRANSPORTThe above sections dealt with intraphase transport processes. The transferof mass from one phase to another is an important transport process in environmentalsystems. Examples of such processes include aeration, reaeration,air stripping, soil emissions, etc. The following theories have been proposedto model such transfers: the Two-Film Theory proposed in the 1920s, thePenetration Theory proposed in the 1930s, and the Surface Renewal Theoryproposed in the 1950s. Among these, the first is the simplest, most understood,and most commonly used. As such, only the Two-Film Theory isreviewed here.4.5.1 TWO-FILM THEORYThe Two-Film Theory can best be illustrated using the classical exampleof transfer of oxygen in an air-water binary system as shown in Figure 4.3.According to this theory, the following are postulated:• There are two films at the interface, one on each side.• Concentration gradients exist only within the two films, and the bulkare well mixed.• Concentrations, C a,i and C w,i , at the interface are at equilibrium.Because the interfacial concentrations are at equilibrium, using linear partitioningCa,i = K a–w (4.23)CThe molar flow rate through the gas-side film, J x,i,a , is given byw,iJ x,i,a = D i,a A x d Cidx= D i,aA x ( C a , b – C a,i ) = k g A x (C a,b – C a,i ) (4.24)∆xand, the molar flow rate through the liquid-side film, J x,i,w , is given byaJ x,i,w = D i,w A x d Cidx= D i,w A (C w,i – C w,b )x = k l A x (C w,i – C w,b ) (4.25)∆x w© 2002 by CRC Press LLC