Flow reaction, with axial diffusion

A plug flow or tubular flow reactor is tubular in shape with a high length/diameter (1/d) ratio. In an ideal case (as in the case of an ideal gas, this only approached reality) flow is orderly with no axial diffusion and no difference in velocity of any members in the tube. Thus, the time a particular material remains within the tube is the same as that for any other material. We can derive relationships for such an ideal situation for a first-order reaction. One that relates extent of conversion with mean residence time, t, for free radical polymerizations is ... 

If the flow rate is increased so that Peclet number Pe l, then there is a timescale at which transversal molecular diffusion smears the contact discontinuity into a plug. In Taylor (1993), Taylor found an effective long-time axial diffusivity proportional to the square of the transversal Peclet number and occurring in addition to the molecular diffusivity. After this pioneering work of Taylor, a vast literature on the subject developed, with over 2000 citations to date. The most notable references are the article (Aris, 1956) by Aris, where Taylor s intuitive approach was explained through moments expansion and the lecture notes (Caflisch and Rubinstein, 1984), where a probabilistic justification of Taylor s dispersion is given. In addition to these results, addressing the tube flow with a dominant Peclet number and in the absence of chemical reactions, there is... 

The transient reaction and diffusion in a packed bed with axial dispersion is governed by the following equations. Begin with an initial concentration in the bed of zero. At time zero start flowing with the inlet concentration = 1.0. Integrate to steady state. Use the parameters Pe = 100, 0 < x < 1, Da = 2, i/ = 2. 

The tube is packed with catalyst pellets. Flow may be either laminar or turbulent. The velocity profile is assumed to be flat. Transfer of heat and mass in the radial direction is modeled using empirical diffusion coefficients that combine the effects of convection and true diffusion in the radial direction. There is no axial diffusion. Details are given in Chapter 9. This model is important only for nonisothermai reactors. It reduces to piston flow if the reaction is isothermal. 

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 ... 

Model of ideal desaturation (model with plug flow regime) is the favorable approximation for calculation of reactor parameters [3,4,6] any cross-section normal for flow, weight hour space velocity w and flow s properties (pressure, temperature and reaction mixture structure) are uniform narrow distribution of reagents residence times in reaction zone Xpr diffusion in the axial line (coplanar mixing or turbulence) in comparison with weight hour space velocity is negligible low. 

The approach of reactor point effectiveness, however, does require some approximations. The adequacy of these approximations will be examined in detail. This will be followed by a section on the equivalence between the plug-flow model and the axial dispersion model, and that between the plug-flow and the radial dispersion model. With these equivalences established, it is possible to concentrate on the plug-flow model. Using this model, the design and analysis problem for reactions affected by diffusion, those affected by both diffusion and chemical deactivation, and those affected by catalyst sintering will be treated in detail. Detailed design and analysis procedures result from this treatment. 

It has been shown in Section 10.3 that the solution of the plug-flow model leads directly to that of the one-dimensional model with axial dispersion. These results can be applied to reactions affected by diffusion. Use of the results leads to the design equations for the fixed-beds with axial dispersion given in Table 10.3. Note that Co is the solution for the plug-flow model and that Eq. (E) in Table 10.3 is used for kb in Eq. (B), whereas Eq. (G) is in Eq. (F) (see Problem 10.9 for the case of = 0). 

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