Cylindrical Control Volume
The cylindrical coordinate system and cylindrical control volume are illustrated in Figure 2.6. There are some differences in the development of a mass balance equation on a cylindrical control volume. Primarily, the rdOdx side of the control volume increases in area as r increases. For the control volume of Figure 2.6, the area normal to the r-coordinate would be which is a function of r, one of the independent variables. Then, analogous to equation 2.12 , the convective flux in the...
H Kuh
3. Hayduk-Laudie Relationship for Diffusion Coefficient in Water Hayduk and Laudie 1974 developed a relationship specifically for water. They eliminated some of the solvent-specific parameters from the Wilkie-Chang relationship, eliminated absolute temperature, and fit three coefficients in the relationship. The results were as follows 12 cm s ,1.14 V 0.589 3.20 where j,2 is in centipoise and Vb is the LeBas molar volume cm3 mole of the solute at the boiling point. EXAMPLE 3.5 Estimating...
Penetration Theory
The penetration theory is attributed to Higbie 1935 . In this theory, the fluid in the diffusive boundary layer is periodically removed by eddies. The penetration theory also assumes that the viscous sublayer, for transport of momentum, is thick, relative to the concentration boundary layer, and that each renewal event is complete or extends right down to the interface. The diffusion process is then continually unsteady because of this periodic renewal. This process can be described by a...
F Gas Film Coefficient
The gas film coefficient is dependent on turbulence in the boundary layer over the water body. Table 4.1 provides Schmidt and Prandtl numbers for air and water. In water, Schmidt and Prandtl numbers on the order of 1,000 and 10, respectively, results in the entire concentration boundary layer being inside of the laminar sublayer of the momentum boundary layer. In air, both the Schmidt and Prandtl numbers are on the order of 1. This means that the analogy between momentum, heat, and mass...
A Complete Mix Reactors
A complete mix reactor is one with a high level of turbulence, such that the fluid is immediately and completely mixed into the reactor. The outflow concentration and the reactor concentration are equal, and the diffusion term is zero due to the gradient being zero. Figure 6.1 shows an illustration of the concept. If we make the entire reactor into our control volume, then a mass balance on the reactor gives Rate of Flux rate Flux rate Source sink VddC QCi - QC VS 6.2 Figure 6.1. Illustration...
E Plug Flow with Dispersion
Dispersion is the enhanced mixing of material through spatial variations in velocity. When it is of interest when we are not keeping track of the three-dimensional mixing , dispersion is typically one or two orders of magnitude greater than turbulent diffusion. The process of dispersion is associated with a spatial mean velocity. The means used in association with diffusion, turbulent diffusion, and dispersion are identified in Table 6.2. The means by which diffusion and possibly turbulent...
ttr
Figure 6.7. Response of a plug flow reactor to a pulse and to a front in concentration at t 0. Top Front. Bottom Pulse. change in concentration occurs. Thus, a plug flow reactor serves similar to a bypass, except with a given residence time. Of course, zero mixing does not exist, and the plug flow reactor is used when the mixing is not important to the process of interest. Mixing can be simulated, however, with a set of complete mix reactors-in-series with a plug flow reactor. If, for example,...
Stagnant Film Theory
The stagnant film theory was developed by Nernst 1904 . In this theory, a stagnant film exists on both sides of the interface, as illustrated in Figure 8.8. The thickness of the film is controlled by turbulence and is constant. With the steady-state situation seen in Figure 8.8, dC dt 0, dC dx and d C d y 0, and w 0. Then, the diffusion equation in each media, as long as turbulence does not penetrate the concentration boundary layer, becomes that results in a constant gradient of concentration...
C The Product Rule
We will introduce the product rule through demonstrating its use in an example problem. The product rule can be used to expand a solution without source and sink terms to the unsteady, one-dimensional diffusion equation to two and three dimensions. It does not work as well in developing solutions to all problems and therefore is more of a technique rather than a rule. Once again, the final test of any solution is 1 it must solve the governing equation s and 2 it must satisfy the boundary...
Info Ola
Figure E5.3.2. Adjustment of the boundary conditions from a static pulse to a continuous pulse with mass flux rate, M. M A Example 2.2 M Uh Mass flux rate The fully mixed concentration can be taken care of by placing an image source in the solution at x,y 0,2b . This will meet the conditions of boundary conditions 1 and 2. The ambient concentration will be handled by assigning C C - C0. Then, equation E5.3.2 becomes
Reactor Mixing Assumptions
Solving the diffusion equation in environmental transport can be challenging because only specific boundary conditions result in an analytical solution. We may want to consider our system of interest as a reactor, with clearly defined mixing, which is more amenable to time dependent boundary conditions. The ability to do this depends on how well the conditions of the system match the assumptions of reactor mixing. In addition, the system is typically assumed as one dimensional. The common...