Kinetics axial diffusion

The moments of the solutions thus obtained are then related to the individual mass transport diffusion mechanisms, dispersion mechanisms and the capacity of the adsorbent. The equation that results from this process is the model widely referred to as the three resistance model. It is written specifically for a gas phase driving force. Haynes and Sarma included axial diffusion, hence they were solving the equivalent of Eq. (9.10) with an axial diffusion term. Their results cast in the consistent nomenclature of Ruthven first for the actual coefficient responsible for sorption kinetics as ... 

Suen, S.-Y., Caracotsios, M., and Etzel, M. R. (1993). Sorption kinetics and axial diffusion in binary-solute affinity-membrane bioseparations. Chem. Eng. Sci. 48, 1801-1812. 

The second limiting case we consider is that of linear kinetics with negligible axial diffusion (Per 1). For this case, the averaged model can be written in... 

The conventional channel using a macroelectrode and fast-flow channel both operate in the limit of negligible axial diffusion. The enhanced kinetic discrimination of the channel microband electrode arises from using a small microband at slow flow rates thus incurring a significant amount of axial diffusion. 

Aris (1991a), in addition to the case of M CSTRs in series, has also analyzed two other homotopies the plug flow reactor with recycle ratio R, and a PFR with axial diffusivity and Peclet number P, but only for first-order intrinsic kinetics. The values M = 1(< ), R = >(0), and P = 0( o) yield the CSTR (PFR). The M CSTRs in series were discussed earlier in Section IV,C,1. The solutions are expressed in terms of the Lerch function for the PFR with recycle, and in terms of the Niemand function for the PFR with dispersion. The latter case is the only one that has been attacked for the case of nonlinear intrinsic kinetics, as discussed below in Section IV,C,7,b. Guida et al. (1994a) have recently discussed a different homotopy, which is in some sense a basically different one no work has been done on multicomponent mixture systems in such a homotopy. 

Both studies show that at relatively low temperatures, i.e., during ignition of the catalyst, the rate-limiting step shifts from chemical kinetics to diffusion in the washcoat. This is clear from Fig. 7, computed using a one-dimensional model by Nakhjavan et al. [54]. Figure 7A shows the Thiele modulus and Fig. 7B an external diffusion limiting factor F versus dimensionless axial position in the reactor at various times on-stream for the catalytic combustion of propene in monolith reactors. The time is defined as the time after injection of the fuel in a preheated air flow. 

Several models use the mass balance in Eq. 2.2 (ideal and equUibrimn-disper-sive models. Sections 2.2.1 and 2.2.2) as derived here without combining it with kinetic equations. In the latter case, Di in Eq. 2.2, which accounts only for axial diffusion, bed tortuosity, and eddy diffusion, is replaced with Da, which accoimts also for the effect of the mass transfer resistances. This is legitimate imder certain conditions, as explained later in Section 2.2.6. Other simple models account for a more complex mass transfer kinetics by coupling Eq. 2.2 with a kinetic equation (lumped kinetic models. Section 2.2.3) in which case Di is used. More complex models write separate mass balance equations for the stream of mobile phase percolating through the bed and for the mobile phase stagnant inside the pores of the particles (the general rate model and the lumped pore diffusion or FOR model, see later Sections 2.1.7 and 2.2.4). 

Therefore, Giddings [67] has demonstrated that nonequilibrium effects resulting from the finite rate of mass transfer kinetics can be treated as a contribution to the axial dispersion, itself the result of axial molecular diffusion, the tortuosity of the packing, the anastomosis of the network of interparticle channels where the stream of mobile phase flows, and the nonhomogeneity of the coliunn packing. The axial diffusion and column tortuosity account for the B term of the classical Knox equation ... 

There is a minimum of the SLT for an intermediate value of the mobile phase velocity, as in linear chromatography [11]. At low velocities, axial dispersion is large due to the long migration time during which axial diffusion proceeds constantly to relax the concentration gradients, while at high velocities, the finite rate of the mass transfer kinetics causes the SLT to increase in proportion to the velocity. If we assume as above the Van Deemter equation for the axial dispersion term (Eq. 14.3Q2), we obtain for the optimum velocity for minimum SLT in displacement (imder isotachic conditions)... 

Axial dispersion, D When a band migrates along a column packed with non-porous particles, it spreads axially because of the combination effects of axial diffusion and the inhomogeneity of the pattern of flow velocity in a packed bed. This combination of effects is accounted for by a single term, proportional to the axial dispersion coefficient. It is independent of the mass transfer resistance and of the other contributions of kinetic origin to band broadening. 

Diffusion with a convection and simultaneous first order reaction in a rectangular plate can be simulated using the program described above by using minor modifications. Consider the composition profile in a packed tube reactor undergoing isothermal linear kinetics with axial diffusion. The governing equation is... 

For larger catalyst particles, some gas by-pass close to the reactor walls is possible. When the reactants move along the reactor the compostion of the mixture changes and axial diffusion can become prominent. To avoid the possible impact of axial diffusion in kinetic experiments, the level of conversion should be below 10-15%. 

Now, the coupled mass and thermal energy balances can be combined and integrated analytically to obtain a linear relation between temperature and conversion under nonequilibrium (i.e., kinetic) conditions because it is not necessary to consider the temperature and conversion dependence of (Cp mixture)- At high-mass-transfer Peclet numbers, axial diffusion can be neglected relative to convective mass transfer, and the mass balance is expressed in terms of molar flow rate F, and differential volume dV for a gas-phase tubular reactor with one chemical reaction ... 

In processes where large macromolecules with slow associated kinetics are involved, the adsorption kinetics between the ligand and the dissolved molecule in the feed solution is the rate-limiting step. Not often, the pressure drop or the membrane mechanical strength is the limiting factor. When thin membranes are applied, the axial diffusion becomes also more dominant and requires a lower linear flow rate through the matrix. Beside this, also inhomogeneities in porosity and... 

The determination of the total rate constant k of the process RO2 + NO -> RO + N02/nitrate from the shape of iQ 02 ) is essentially based on the first-order law [NO2KO = [N02]oo [1 exp(-fc [NO] 01- Corrections were made for the (slight) [NO]-decrease and for axial diffusion effects. Moreover, side-reactions such as RO2 + NO2 pernitrate, RO + NO nitrite, RO + NO2 nitrate, decomposition of RO and subsequent reactions, as well as contributions of nitrates and pernitrates to the total N02 signal, were duly taken into account by kinetic... 

The equation tries to account for all the kinetic processes that occur when a separation is undertaken. A, B, and C are coefficients that relate to the three different processes A being the Eddy diffusion, B/u the longitudinal axial diffusion, Cu the mass transfer term, and u the mean linear velocity. 

Flow Pattern Ideality. A straightforward interpretation of the observed kinetics can only be made if the flow pattern in the reactor used corresponds to an ideal flow pattern. In particular for plug flow reactors, deviations from the ideal reactor behavior can be encountered. For perfectly mixed reactors such as a batch reactor and a continuous stirred tank reactor, the rotation speed of the stirrer is the key parameter that needs to be set sufficiently high to ensure complete mixing. Deviations from the ideal plug flow pattern can, for example, be caused by a less-dense packing of the catalyst pellets near the reactor wall, by a too high dilution of the catalyst bed with inert pellets or by the importance of effective axial diffusion compared to convection (15). 

This equation passes through a minimum which, for a small molecule and a well packed column, is found to occur at a value for h of about 2. For a conventional 4.6 mm i.d. column filled with a 5 micron reversed-phase packing in methanol/water mixtures the flow-rate of this minimum is about 0.5 ml/min, and v of the order 3—5. The A term describes the contribution to dispersion due to tortuous flow through a packed bed, and is a measure of how well the column is packed. The B term describes the effects of axial diffusion (in the direction of flow) on dispersion at normal flow-rates the influence of this term is relatively unimportant. The C term is concerned with the effects of slow equilibration between the mobile and stationary regions due to, for instance, the diffusion of solute into and out of the particle and the kinetics of adsorption and desorption. For a well packed column and a small solute A < I, B 2 and C S 0.1. 

---