Tube Flows with Diffusion

3 X 10 according to Equation 8.3, then radial diffusion is unimportant and reactor performance is similar to that for laminar flow without diffusion. If is large, 

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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... 

Consider a well-mixed ensemble of particles flowing through a tube of radius R with no other factors but diffusion tending to remove the particles from the flow. With diffusion velocity considered as a net movement of particles to the tube surface, in an interval of 1 s there will be J(2raR) particles deposited per unit length of tube. In a time dt = dxl u0, a 1-cm length of aerosol traverses a distance dx. Thus in this time J 2mR) dx/u0 particles are removed, and the change in concentration is the number of particles removed divided by the volume from which they are removed, or... 

When a sample is injected into the carrier stream it has the rectangular flow profile (of width w) shown in Figure 13.17a. As the sample is carried through the mixing and reaction zone, the width of the flow profile increases as the sample disperses into the carrier stream. Dispersion results from two processes convection due to the flow of the carrier stream and diffusion due to a concentration gradient between the sample and the carrier stream. Convection of the sample occurs by laminar flow, in which the linear velocity of the sample at the tube s walls is zero, while the sample at the center of the tube moves with a linear velocity twice that of the carrier stream. The result is the parabolic flow profile shown in Figure 13.7b. Convection is the primary means of dispersion in the first 100 ms following the sample s injection. 

Flow in a Tube. Laminar flow with a flat velocity profile and slip at the walls can occur when a viscous fluid is strongly heated at the walls or is highly non-Newtonian. It is sometimes called toothpaste flow. If you have ever used Stripe toothpaste, you will recognize that toothpaste flow is quite different than piston flow. Although Vflr) = u and z(7) = 1, there is little or no mixing in the radial direction, and what mixing there is occurs by diffusion. In this situation, the centerline is the critical location with respect to stability, and the stability criterion is... 

The simulation of reacting flows in packed tubes by CFD is still in its earliest stages. So far, only isothermal surface reactions for simplified geometries and elementary reactions have been attempted. Heterogeneous catalysis with diffusion, reaction, and heat transfer in solid particles coupled to the flow, species, and temperature fields external to the particles remains a challenge for the future. 

The first expression here is very similar to the Damkohler result for A and B equal to 1. Since the turbulent exchange coefficient (eddy diffusivity) correlates well with IqU for tube flow and, indeed, /0 is essentially constant for the tube flow characteristically used for turbulent premixed flame studies, it follows that... 

The combustion efficiency of the benzene was beyond 99.999% even at an overall equivalence ratio of 1.0 (including the entrainment air which was 30% of the total air flow). With the controller off, the flame was extremely sooty and, in fact, sooting as the quartz tube would be blackened and the mass spectrometer sampling probe clogged in a matter of seconds, and soot was sucked into the scrubber system. This comparison between controller off and controller on conditions with benzene fuel is extremely dramatic and shows the efficacy of active combustion control in vortices to eliminating soot from diffusion flames. 

When a tube or pipe is long enough and the fluid is not very viscous, then the dispersion or tanks-in-series model can be used to represent the flow in these vessels. For a viscous fluid, one has laminar flow with its characteristic parabolic velocity profile. Also, because of the high viscosity there is but slight radial diffusion between faster and slower fluid elements. In the extreme we have the pure convection model. This assumes that each element of fluid slides past its neighbor with no interaction by molecular diffusion. Thus the spread in residence times is caused only by velocity variations. This flow is shown in Fig. 15.1. This chapter deals with this model. 

Taylor (T2) and Westhaver (W5, W6, W7) have discussed the relationship between dispersion models. For laminar flow in round empty tubes, they showed that dispersion due to molecular diffusion and radial velocity variations may be represented by flow with a flat velocity profile equal to the actual mean velocity, u, and with an effective axial dispersion coefficient Djf = However, in the analysis, Taylor... 

Taylor (T4, T6), in two other articles, used the dispersed plug-flow model for turbulent flow, and Aris s treatment also included this case. Taylor and Aris both conclude that an effective axial-dispersion coefficient Dzf can again be used and that this coefficient is now a function of the well known Fanning friction factor. Tichacek et al. (T8) also considered turbulent flow, and found that Dl was quite sensitive to variations in the velocity profile. Aris further used the method for dispersion in a two-phase system with transfer between phases (All), for dispersion in flow through a tube with stagnant pockets (AlO), and for flow with a pulsating velocity (A12). Hawthorn (H7) considered the temperature effect of viscosity on dispersion coefficients he found that they can be altered by a factor of two in laminar flow, but that there is little effect for fully developed turbulent flow. Elder (E4) has considered open-channel flow and diffusion of discrete particles. Bischoff and Levenspiel (B14) extended Aris s theory to include a linear rate process, and used the results to construct comprehensive correlations of dispersion coefficients. 

In fact, the plug-flow approximation is even better than this calculation indicates because of radial mixing, which wiU occur in a laminar-flow reactor. A fluid molecule near the wall will flow with nearly zero velocity and have an infinite residence time, while a molecule near the center will flow with velocity 2u. However, the molecule near the wall will diffuse toward the center of the tube, and the molecule near the center will diffuse toward the wall, as shown in Figure 8-7. Thus the tail on the RTD will be smaller, and the spike at t/2 will be broadened. We will consider diffusion effects in the axial direction in the next section. 

Any flow with a nonuniform velocity profile will, when spatial mean velocity and concentration are taken, result in dispersion of the chemical. For laminar flow, the well-described velocity profile means that we can describe dispersion analytically for some flows. Beginning with the diffusion equation in cylindrical coordinates (laminar flow typically occurs in small tubes) ... 

Next we consider a fluid flowing through a circular tube with material at the wall diffusing into the moving fluid. This situation is met with in the analysis of the mass transfer to the upward-moving gas stream in wetted-wall-tower experiments. Just as in the discussion of absorption in falling films, we consider mass transfer to a fluid moving with a constant velocity profile and also flow with a parabolic (Poiseuille) profile (see Fig. 5). 

Design Parameters. In a diffusion denuder an airstream, in laminar flow, contacts a surface (e.g., the inner wall of a tube) coated with a sorptive or reactive trapping medium for the gaseous pollutant of interest. Fine particles, which have diffusion coefficients more than 4 orders of magnitude smaller than those for gases, penetrate the denuder with high efficiency. Forrest et al. (40) reported losses of 0.2-2.2% for particles between 0.3 and 0.6 xm and about 4-5% for 1- to 2- xm particles. 

Inertial forces of the fluid increase with density and the square of velocity (pv2) while viscous forces decrease with increasing diameter of tube (nv/d) and increase with viscosity and velocity. High Reynolds numbers (Re>4000) result in turbulent flow with low Reynolds number (Re<2000) the flow is laminar. Laminar flow results from formation of layers of fluid with different velocities after a certain flow distance, as illustrated in Figure 2.10A. Flow at the walls is zero and increases approaching the center of the tubes. The laminar flow pattern results from layers of mobile phase with different velocities travelling parallel to each other. The maximum flow at the center is twice the average flow velocity of the fluid. Molecules in the fluid can exchange between fluid layers by molecular diffusion. Most open tubular columns operate under laminar flow conditions. 

Take the case of flow and diffusion in a long tube with a flat velocity profile but no reaction. The concentration of solute is governed by... 

Wall-coated flow tube reactors have been used to study the uptake coefficients onto liquid and solid surfaces. This method is sensitive over a wide range of y (10" to 10 1). For liquids this method has the advantage that the liquid surface is constantly renewed, however if the uptake rate is fast, the liquid phase becomes saturated with the species and the process is limited by diffusion within the liquid, so that corrections must be applied [70,72,74]. Many experiments were designed to investigate the interaction of atmospheric species on solid surfaces. In this case the walls of the flow tube were cooled and thin films of substrate material were frozen on the wall. Most of the reaction probabilities were obtained from studies on flow tubes coated with water-ice, NAT or frozen sulfate. Droplet train flow tube reactors have used where liquid droplets are generated by means of a vibrating orifice [75]. The uptake of gaseous species in contact with these droplets has been measured by tunable diode laser spectroscopy [41]. 

Non-isothermal and non-adiabatic conditions. A useful approach to the preliminary design of a non-isothermal fixed bed reactor is to assume that all the resistance to heat transfer is in a thin layer near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the approximate design of reactors. Neglecting diffusion and conduction in the direction of flow, the mass and energy balances for a single component of the reacting mixture are ... 

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