Turbulent plug flow profile

The time-to-distance transfonnation requires fast mixing and a known flow profile, ideally a turbulent flow with a well-defined homogeneous composition perpendicular to the direction of flow ( plug-flow ), as indicated by tire shaded area in figure B2.5.1. More complicated profiles may require numerical transfomiations. 

The distribution of tracer molecule residence times in the reactor is the result of molecular diffusion and turbulent mixing if tlie Reynolds number exceeds a critical value. Additionally, a non-uniform velocity profile causes different portions of the tracer to move at different rates, and this results in a spreading of the measured response at the reactor outlet. The dispersion coefficient D (m /sec) represents this result in the tracer cloud. Therefore, a large D indicates a rapid spreading of the tracer curve, a small D indicates slow spreading, and D = 0 means no spreading (hence, plug flow). 

Dispersion models, as just stated, are useful mainly to represent flow in empty tubes and packed beds, which is much closer to the ideal case of plug flow than to the opposite extreme of backmix flow. In empty tubes, the mixing is caused by molecular diffusion and turbulent diffusion, superposed on the velocity-profile effect. In packed beds, mixing is caused both by splitting of the fluid streams as they flow around the particles and by the variations in velocity across the bed. 

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. 

Checks on the relationships between the axial coefficients were provided in empty tubes with laminar flow by Taylor (T2), Blackwell (B15), Bournia et al. (B19), and van Deemter, Breeder and Lauwerier (V3), and for turbulent flow by Taylor (T4) and Tichacek et al. (T8). The agreement of experiment and theory in all of these cases was satisfactory, except for the data of Boumia et al. as discussed previously, their data indicated that the simple axial-dispersed plug-flow treatment may not be valid for laminar flow of gases. Tichacek et al. found that the theoretical calculations were extremely sensitive to the velocity profile. Converse (C20), and Bischoff and Levenspiel (B14) showed that rough agreement was also obtained in packed beds. Here, of course, the theoretical calculation was very approximate because of the scatter in packed-bed velocity-profile data. 

We will now find the RDT for several models of tubular reactors. We noted previously that the perfect PFTR cannot in fact exist because, if flow in a tube is sufficiently fast for turbulence (Rco > 2100), then turbulent eddies cause considerable axial dispersion, while if flow is slow enough for laminar flow, then the parabolic flow profile causes considerable deviation from plug flow. We stated previously that we would ignore this contradiction, but now we will see how these effects alter the conversion from the plug-flow approximation. 

In practice, there is always some degree of departure from the ideal plug flow condition of uniform velocity, temperature, and composition profiles. If the reactor is not packed and the flow is turbulent, the velocity profile is reasonably flat in the region of the turbulent core (Volume 1, Chapter 3), but in laminar flow, the velocity profile is parabolic. More serious however than departures from a uniform velocity profile are departures from a uniform temperature profile. If there are variations in temperature across the reactor, there will be local variations in reaction rate and therefore in the composition of the reaction mixture. These transverse variations in temperature may be particularly serious in the case of strongly exothermic catalytic reactions which are cooled at the wall (Chapter 3, Section 3.6.1). An excellent discussion on how deviations from plug flow arise is given by DENBIGH and TURNER 5 . 

NMR imaging techniques were applied to the measurements of velocity field in opaque systems such as tomato juice and paper pulp suspensions [58-60]. In both cases, the particle concentrations are sufficiently high that widely applied techniques such as hot film and laser Doppler anemometry could not be used. The velocity profile for a 6 % tomato juice slurry clearly showed a power-law behavior [58, 59]. Row NMR images for a 0.5 % wood pulp suspension provided direct visual of three basic types of shear flow plug flow, mixed flow and turbulent flow as mean flow rate was increased. Detailed analysis of flow NMR image is able to reveal the complex interaction between the microstructure of suspensions and the flow [60]. 

The flow patterns, composition profiles, and temperature profiles in a real tubular reactor can often be quite complex. Temperature and composition gradients can exist in both the axial and radial dimensions. Flow can be laminar or turbulent. Axial diffusion and conduction can occur. All of these potential complexities are eliminated when the plug flow assumption is made. A plug flow tubular reactor (PFR) assumes that the process fluid moves with a uniform velocity profile over the entire cross-sectional area of the reactor and no radial gradients exist. This assumption is fairly reasonable for adiabatic reactors. But for nonadiabatic reactors, radial temperature gradients are inherent features. If tube diameters are kept small, the plug flow assumption in more correct. Nevertheless the PFR can be used for many systems, and this idealized tubular reactor will be assumed in the examples considered in this book. We also assume that there is no axial conduction or diffusion. 

Dead time can result from measurement lag, analysis, and computation time, communication lag or the transport time required for a fluid to flow through a pipe. Figure 2.27 illustrates the response of a control loop to a step change, showing that the response started after a dead time (td) has passed and reaches a new steady state as a function of its time constant (t), defined in Figure 2.23. When material or energy is physically moved in a process plant, there is a dead time associated with that movement. This dead time equals the residence time of the fluid in the pipe. Note that the dead time is inversely proportional to the flow rate. For liquid flow in a pipe, the plug flow assumption is most accurate when the axial velocity profile is flat, a condition that occurs when Newtonian fluids are transported in turbulent flow. 

Plug Flow Reactor. A PFR is a continuous flow reactor. It is an ideal tubular type reactor. The assumption we make is that the reaction mixture stream has the same velocity across the reactor cross-sectional area. In other words, the velocity profile across the reactor is a flat one. In a PFR there is no axial mixing along the reactor. The condition of plug flow is met in highly turbulent flows, as is usually the case in chemical reactors. 

The plug-flow model indicates that the fluid velocity profile is plug shaped, that is, is uniform at all radial positions, fact which normally involves turbulent flow conditions, such that the fluid constituents are well-mixed [99], Additionally, it is considered that the fixed-bed adsorption reactor is packed randomly with adsorbent particles that are fresh or have just been regenerated [103], Moreover, in this adsorption separation process, a rate process and a thermodynamic equilibrium take place, where individual parts of the system react so fast that for practical purposes local equilibrium can be assumed [99], Clearly, the adsorption process is supposed to be very fast relative to the convection and diffusion effects consequently, local equilibrium will exist close to the adsorbent beads [2,103], Further assumptions are that no chemical reactions takes place in the column and that only mass transfer by convection is important. 

Axial dispersion is negligible. Whether this assumption is valid, can be seen from the Peclet number uJJD) for typical conditions = 1 m/sec, L = 0.5 m) and laminar flow, the Peclet number is larger than 1000, and even for turbulent flow it will be much larger than 10. In laminar flow, the radial flow profile within a subchannel will also result in deviation from plug flow. The effect of this deviation can be estimated by comparing the predictions from different mathematical models, one of which takes the flow profile into account and the other of which assumes plug flow. 

The gas flow through tubular reactors is of particular importance because the composition at any point is influenced by the linear velocity of the gas, the size of the reactor and the size of the catalyst particles. When gas flows through a pipe at low linear velocity (low Reynolds number), the radial velocity is not uniform. As the linear velocity increases, turbulence increases and the velocity profile approaches what is called plug flow. However, in a packed bed, plug flow can never be completely attained because of the high voidage near the reactor wall. 

Under plug flow conditions the convective transport is completely dominant over the diffusive mass transport term. The fluid moves like a plug and the diffusive term can be neglected. The conditions for plug flow are closely satisfied for narrow and long tubular reactors when the viscosity is low. However, this approximation is clearly best for fully developed turbulent flow, for which the velocity profiles are relatively fiat. For dynamic conditions, the species mass balance is a PDF with z and t as the independent variables. The Eulerian species mass balance (1.301) reduces further to ... 

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