Velocity profile tubular reactor

A tubular plug flow (Figure 5-28) reactor assumes that mixing of fluid does not take place, the velocity profile is flat, and both temperature and composition are uniform at any cross-section in the reactor. 

Example 8.9 Find the temperature distribution in a laminar flow, tubular heat exchanger having a uniform inlet temperature and constant wall temperature Twall- Ignore the temperature dependence of viscosity so that the velocity profile is parabolic everywhere in the reactor. Use art/P = 0.4 and report your results in terms of the dimensionless temperature... 

This section derives a simple version of the convective diffusion equation, applicable to tubular reactors with a one-dimensional velocity profile V (r). The starting point is Equation (1.4) applied to the differential volume element shown in Figure 8.9. The volume element is located at point (r, z) and is in the shape of a ring. Note that 0-dependence is ignored so that the results will not be applicable to systems with significant natural convection. Also, convection due to is neglected. Component A is transported by radial and axial diffusion and by axial convection. The diffusive flux is governed by Pick s law. 

The final idealized flow situation that we will consider is laminar flow in a tubular reactor in the absence of either radial or longitudinal diffusion. The velocity profile in such a reactor is given by... 

The PFR model assumes a flat velocity profile across the whole of the reactor cross-section in reality, this is impossible to achieve although in practice certain combinations of physical conditions are closely described by this assumption. If the Reynolds number, dupln, in a tubular reactor is less than about 2100, then the flow therein will be laminar and where the flow is fully developed, the velocity profile across the reactor will be parabolic in form. If one assumes that diffusion is negligible between adjacent radial layers of fluid, then it is relatively straightforward to derive the forms of E(t), E(0) and F(0) associated with this type of reactor [42]. These are given in the equations... 

Experimental data on multiple steady-state profiles in tubular packed bed reactors have been reported in the literature by Wicke et al. 51 -53) and Hlavacek and Votruba (54, 55) (Table VI). The measurements have been performed in adiabatic tubular reactors. In the following text the effects of initial temperature, inlet concentration, velocity, length of the bed, and reaction rate expression on the multiple steady state profiles will be studied. 

There are some fundamental investigations devoted to analysis of the flow in tubular polymerization reactors where the viscosity of the final product has a limit (viscosity < >) i.e., the reactive mass is fluid up to the end of the process. As a zero approximation, flow can be considered to be one-dimensional, for which it is assumed that the velocity is constant across the tube cross-section. This is a model of an ideal plug reactor, and it is very far from reality. A model with a Poiseuille velocity profile (parabolic for a Newtonian liquid) at each cross-section is a first approximation, but again this is a very rough model, which does not reflect the inherent interactions between the kinetics of the chemical reaction, the changes in viscosity of the reactive liquid, and the changes in temperature and velocity profiles along the reactor. 

The investigation by Lynn and Huff201 was the first one in which the true velocity profiles during polymerization in a tubular reactor were determined. The idea of the dependence of the hydrodynamic field on the varying rheokinetics during a chemical reaction is quite fruitful and has... 

Figure 4.25. Profiles of axial velocities in the flow of a rheokinetic liquid through a tubular reactor. Row rate increases from (a) to (c). The dashed lines denote the boundary between the high-viscosity product (high degree of conversion) and the low-viscosity reactant (breaking through stream).

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. 

FIGURE 8.3 First-order reaction with kt = 1 in a tubular reactor with a parabolic velocity profile. 

The laminar velocity profile in Figure 8-2la is approximated by a series of annuli, within each of which the velocity is constant as illustrated in Figure 8-2lb. Each annulus is considered to be a plug flow tubular reactor having its own space velocity. The velocities of the fluid elements at different radii are given by the parabolic velocity profile for fully developed laminar flow. The velocity is expressed as... 

Note that not all velocity profiles in a tubular reactor are parabolic. Power law fluids have the profile... 

Fick s diffusion law is used to describe dispersion. In a tubular reactor, either empty or packed, the depletion of the reactant and non-uniform flow velocity profiles result in concentration gradients, and thus dispersion in both axial and radial directions. Fick s law for molecular diffusion in the x-direction is defined by... 

Mixing in static mixers considered as chemical reactors was essentially studied by Nauman (165, 166). This author proposed a model which consists of a tubular reactor comprising N zones in laminar flow (parabolic velocity profile). Mixing between each zone is achieved accross a plane by a permutation of the radial position of fluid particles (r — , in this way the flowrate... 

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. 

A theoretical and experimental study of multiplicity and transient axial profiles in adiabatic and non-adiabatic fixed bed tubular reactors has been performed. A classification of possible adiabatic operation is presented and is extended to the nonadiabatic case. The catalytic oxidation of CO occurring on a Pt/alumina catalyst has been used as a model reaction. Unlike the adiabatic operation the speed of the propagating temperature wave in a nonadiabatic bed depends on its axial position. For certain inlet CO concentration multiplicity of temperature fronts have been observed. For a downstream moving wave large fluctuation of the wave velocity, hot spot temperature and exit conversion have been measured. For certain operating conditions erratic behavior of temperature profiles in the reactor has been observed. 

Thirty years later, Gerhard Damkohler (1937) in his historic paper, summarized various reactor models and formulated the two-dimensional CDR model for tubular reactors in complete generality, allowing for finite mixing both in the radial and axial directions. In this paper, Damkohler used the flux-type boundary condition at the inlet and also replaced the assumption of plug flow with parabolic velocity profile, which is typical of laminar flow in tubes. 

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