Tubular reactors definition

We now formalize the definition of piston flow. Denote position in the reactor using a cylindrical coordinate system (r, 6, z) so that the concentration at a point is denoted as a(r, 9, z) For the reactor to be a piston flow reactor (also called plug flow reactor, slug flow reactor, or ideal tubular reactor), three conditions must be satisfied ... 

The importance of dilfusion in a tubular reactor is determined by a dimensionless parameter, SiAt/S = QIaLKuB ), which is the molecular diffusivity of component A scaled by the tube size and flow rate. If SiAtlB is small, then the elfects of dilfusion will be small, although the definition of small will depend on the specific reaction mechanism. Merrill and Hamrin studied the elfects of dilfusion on first-order reactions and concluded that molecular diffusion can be ignored in reactor design calculations if... 

We will consider only the batch reactor in this chapter. This is a type of reactor that does not scale up well at all, and continuous reactors dominate the chemical industry. However, students are usually introduced to reactions and kinetics in physical chemistry courses through the batch reactor (one might conclude fi om chemistry courses that the batch reactor is the only one possible) so we wiU quickly summarize it here. As we vrill see in the next chapter, the equations and their solutions for the batch reactor are in fact identical to the plug flow tubular reactor, which is one of our favorite continuous reactors so we will not need to repeat all these definitions and derivations in the section on the plug flow tubular reactor. 

The second issue for cooled tubular reactors is how to introduce the coolant. One option is to provide a large flowrate of nearly constant temperature, as in a recirculation loop for a jacketed CSTR. Another option is to use a moderate coolant flowrate in countercurrent operation as in a regular heat exchanger. A third choice is to introduce the coolant cocurrently with the reacting fluids (Borio et al., 1989). This option has some definite benefits for control as shown by Bucala et al. (1992). One of the reasons cocurrent flow is advantageous is that it does not introduce thermal feedback through the coolant. It is always good to avoid positive feedback since it creates nonmonotonic exit temperature responses and the possibility for open-loop unstable steady states. 

The polymerization time in continuous processes depends on the time the reactants spend in the reactor. The contents of a batch reactor will all have the same residence time, since they are introduced and removed from the vessel at the same times. The continuous flow tubular reactor has the next narrowest residence time distribution, if flow in the reactor is truly plug-like (i.e., not laminar). These two reactors are best adapted for achieving high conversions, while a CSTR cannot provide high conversion, by definition of its operation. The residence time distribution of the CSTR contents is broader than those of the former types. A cascade of CSTR s will approach the behavior of a plug flow continuous reactor. 

It should not be forgotten that all the examples of this chapter are patient of two interpretations, for the same equations apply to both tubular and batch reactors. Indeed, as we shall see in the final chapter, when considering control, it is perhaps easier to attain the optimal conditions in the batch reactor. For definiteness however it will be convenient to speak in terms of the tubular reactor. 

The heterogeneous rate law in (22-57) is dimensionalized with pseudo-volumetric nth-order kinetic rate constant k that has units of (volume/mol)" per time. k is typically obtained from equation (22-9) via surface science studies on porous catalysts that are not necessarily packed in a reactor with void space given by interpellet. Obviously, when axial dispersion (i.e., diffusion) is included in the mass balance, one must solve a second-order ODE instead of a first-order differential equation. Second-order chemical kinetics are responsible for the fact that the mass balance is nonlinear. To complicate matters further from the viewpoint of obtaining a numerical solution, one must solve a second-order ODE with split boundary conditions. By definition at the inlet to the plug-flow reactor, I a = 1 at = 0 via equation (22-58). The second boundary condition is d I A/df 0 as 1. This is known classically as the Danckwerts boundary condition in the exit stream (Danckwerts, 1953). For a closed-closed tubular reactor with no axial dispersion or radial variations in molar density upstream and downstream from the packed section of catalytic pellets, Bischoff (1961) has proved rigorously that the Danckwerts boundary condition at the reactor inlet is... 

Convergence is obtained when the appropriate guess for d p./di at the reactor inlet predicts the correct Danckwerts condition in the exit stream, within acceptable tolerance. To determine the range of mass transfer Peclet numbers where residence-time distribution effects via interpellet axial dispersion are important, it is necessary to compare plug-flow tubular reactor simulations with and without axial dispersion. The solution to the non-ideal problem, described by equation (22-61) and the definition of Axial Grad, at the reactor outlet is I/a( = 1, RTD). The performance of the ideal plug-flow tubular reactor without interpellet axial dispersion is described by... 

At high-mass-transfer Peclet numbers, sketch the relation between average residence time divided by the chemical reaction time constant (i.e., r/co) for a packed catalytic tubular reactor versus the intrapeUet Damkohler number Aa, intrapeiiet for zeroth-, first-, and second-order irreversible chemical kinetics within spherical catalytic pellets. The characteristic length L in the definition of Aa, intrapeiiet is the sphere radius R. The overall objective is to achieve the same conversion in the exit stream for all three kinetic rate laws. Put all three curves on the same set of axes and identify quantitative values for the intrapeiiet Damkohler number on the horizontal axis. 

The reactor stability decreases with increasing values of a, since the fraction converted at the peak temperature is lower when ATa j is higher. One study showed that the allowable value of 9 for a first-order reaction ranged from 2.4 to 1.1 as a increased from about 7 to 70 [11,12]. There have been many other studies of the stability of tubular reactors and batch reactors, and some complex correlations for the stability limit allowing for changes in coolant temperature with length and the thermal capacity of the reactor wall [13]. However, it is generally not necessary to get the exact stability limit. The conservative criterion that 6> < 1 is often used unless calculations for different conditions show that even with 9 > the reactor is definitely stable to all likely disturbances. 

As an aid in the precise definition of integral and differential reactors, Fig. 4.13 shows a conversion versus time diagram, and the transfer of these data to a continuous tubular reactor (Moser and Lafferty, 1976). 

This last one, as a distribution, requires the definition of a DISTRIBUTION DOMAIN, and it is particularly useful for distributed variables such as the ones used in tubular reactors. The syntax is as follows ... 

For the modeling of a reactor we need solutions of the equations of the balances of mass, energy, and impulse. For isothermal operation the energy balance is not needed. The impulse balance mostly only serves to calculate the pressure drop of a reactor. The definition of a suitable control space for balancing is important. In the simplest case, the variables - such as temperature and concentrations - are constant within the control space (stirred tank reactor). However, in many cases the system variables depend on the location, for example, in the axial direction in a tubular reactor. Then infinitesimal balances (differential equations) have to be solved to obtain integral data. 

Suitable criteria, mostly originated in the context of thermal explosion theory [cf. Semenov (1928) Zeldovich et al. (1985)], have been adopted in chemical reactor theory to identify the nature of reactor behavior. These are based on a definition of runaway in mathematical terms which allows us to identify in the reactor parameter space the boundary separating runaway and nonrunaway operating conditions. Most of these criteria are based on some geometrical characteristic of the temperature profile in the system. In particular, in the context of thermal explosion (cf. the batch system considered in the classical Semenov problem) we refer to the temperature profile in time, while in the case of tubular reactors we refer to the temperature profile along the reactor axis. 

In addition to the above cross-flow reactors of extended definition, the block of thin-walled spiral-tubular-membrane catalyst reactor and the double-spiral coiled-plate-membrane reactor may be included [30-33]. 

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