##\alpha## is not the amount of surface area. It is the surface per unit volume. Even so, these results don't surprise me. All it is saying is that for that small value of the drop radius, the two phases equilibrate very rapidly.Derivation of Eqn. 3
The objective of Eqn. 3 is to estimate the mass flux at the interface between the drop and the gas in terms of (a) the difference between the average species concentration in the drop and the the species concentration in the liquid at the interface ##C_L-C_{LI}## and (b) the difference between the average species concentration in the gas and the species concentration in the gas at the interface ##C_G-C_{GI}##.
Within the scope of what we are trying to do, it is too computationally intensive to obtain an exact solution for these quantities (which involves solving partial differential equations for the compositions with respect to time and spatial position). Instead, we adopt an approximate, yet very accurate, method to express the interface flux algebraically in terms of the above concentration differences. This approach gives us a lower bound for the mass transfer coefficient and species flux at the interface, and thus, a lower bound to the concentration changes for a given value of the residence time at the exit of the scrubber. However, even though they give a lower bound, they still provide a close approximation to the true changes for cases of practical interest where the amount of mass transferred is on the order of (say) 1/3 of the equilibration amount or more. The approach we use is what an experienced heat and mass transfer expert (unabashedly, like myself) would do to attack this problem.
If we suddenly took a spherical drop of fresh liquid and placed it in an infinite ocean of fresh gas (and then allowed to diffusion to occur), the concentrations and mass fluxes initially would vary very rapidly with radial position close to the interface in both the liquid and gas, while away from the interface, at short times, the concentrations would hardly be disturbed. The mass flux at the interface would be very high.
As time progressed, the region where the concentrations and mass fluxes are affected would grow larger, and the mass flux at the interface would decrease (as a result of lower concentration gradients near the interface). Eventually, the concentration would be smoothly varying everywhere within the drop, and also in an effective region on the order of one diameter away from the interface in the gas. Beyond that time, the system would approach so called "asymptotic mass transfer behavior," where the basic shape of the spatial concentration variations don't change with time, and the mass transfer coefficients at the interface approach lower bound constant values. The amount of time it takes to reach this point would be on the order of ##R^2/D## (both on the liquid side of the interface and on the gas side of the interface), where R is the drop radius and D is the diffusion coefficient. This amount of time would typically be very short compared to the residence time in the scrubber. Therefore, the "asymptotic mass transfer behavior" would be characteristic of the vast majority of the scrubber.
Our game plan then is to determine (algebraic) asymptotic mass transfer relationships between the inward mass flux ##\phi## through the interface at r = R, and the concentration driving forces on the two sides of the interface.
I think I'll stop here for now, and get back to this tomorrow.
That was just a starting point. After working with the numbers all morning I'm currently looking at 0.12% (0.0012). As examples of the capture rates I'm getting:I'm very concerned about your suggesting that the volume fraction of liquid is only 0.0004. With that small a value, the liquid does not have enough capacity to remove much of the undesirable species from the gas, even if the two phases equilibrated in the scrubber. What does your Eqn. 20-eq tell you about the ratio of the final- to the initial concentration of CO2 in the gas, assuming that f = 0.0004 and CL0 = 0? I get virtually no change.