Plug flow reactors time-dependent

Space time ST is equal to the residence time in a plug flow reactor only if the volumetric flowrate remains constant throughout the reactor. The residence time depends on the change in the flowrate through the reactor, as well as V/u. The change in u depends on the variation in temperature, pressure, and the number of moles. The concept of SV with conversions in the design of a plug flow reactor is discussed later in this chapter. 

When the basic system was operated as a continuous packed bed reactor, the analytical model developed here allows us to describe the performance of all types of reactors, from a continuous stirred tank reactor (CSTR) to a plug flow reactor (PFR). It was shown that the information-processing function depends on the reactor type, the flow rate through the reactor, the concentration of the cofactor in the feed stream, the values of Vm,i, the presence of internal inhibitors, and the cycle time of the input signal. 

As mentioned in the previous discussions of deactivation, catalyst aging is very composition dependent. Thus, catalyst state at a given time on stream will vary with axial distance in a plug flow reactor. This is shown in Fig. 22. Benzene and methylcyclopentane compositions as a function of time on stream are shown at 20% through the catalyst bed and at the end of the bed. KINPTR predicts the catalyst state gradient in the reactor. 

For a tubular (plug flow) reactor, the conditions at any point in the reactor are independent of time, and the linear velocity v of the reacting mixture is the same at every point in a cross-section S perpendicular to the flow direction and equal to (G/pS). The composition of the reaction mixture depends on the distance L from the inlet point. 

The plug flow reactor is increasingly being used under transient conditions to obtain kinetic data by analysing the combined reactor and catalyst response upon a stimulus. Mostly used are a small reactant pulse (e.g. in temporal analysis of products (TAP) [16] and positron emission profiling (PEP) [17, 18]) or a concentration step change (in step-response measurements (SRE) [19]). Isotopically labeled compounds are used which allow operation under overall steady state conditions, but under transient conditions with respect to the labeled compound [18, 20-23]. In this type of experiments both time- and position-dependent concentration profiles will develop which are described by sets of coupled partial differential equations (PDEs). These include the concentrations of proposed intermediates at the catalyst. The mathematical treatment is more complex and more parameters are to be estimated [17]. Basically, kinetic studies consist of ... 

Examples of chemical process units in this category include plug flow reactors, laminar flow reactors, turbulent flow reactors, plasma reactors, and separation units that are described in terms of the mass transfer concept. To develop a numerical algorithm, the time and spatial derivatives are replaced by finite difference approximations. In general, the time derivative is represented by a forward difference, whereas the second order spatial derivatives are approximated by central differences as follows for the dependent variable Y in Cartesian coordinates ... 

A differential equation for a function that depends on only one variable is called an ordinary differential equation. The independent variable is frequently time, t, but for reactors it can also be length down a plug flow reactor. An example of an ordinary differential equation is... 

Reactors do not always run at steady state. In fact, many pharmaceuticals are made in a batch mode. Such problems are easily solved using the same techniques presented above because the plug flow reactor equations are identical to the batch reactor equations. Even CSTRs can be run in a transient mode, and it may be necessary to model a time-dependent CSTR to study the stability of steady solutions. When there is more than one solution, one or more of them will be unstable. Thus, this section considers a time-dependent CSTR as described by Eq. (8.51) ... 

Note that all the conditions are known at one time, t = 0. Thus it is possible to calculate the function on the right-hand side at f = 0 to obtain the derivative there. This makes the set of equations initial value problems. The equations are ordinary differential equations because there is only one independent variable. Any higher-order ordinary differential equation can be turned into a set of first-order ordinary differential equations they are initial value problems if all the conditions are known at the same value of the independent variable [Finlayson, 1980, 1997 (p. 3-54), 1990 (Vol. BI, p. 1-55)]. The methods for initial value problems are explained here for a single equation extension to multiple equations is straightforward. These methods are used when solving plug-flow reactors (Chapter 8) as well as time-dependent transport problems (Chapters 9-11). 

Numerical simulations and analyses were performed for both the continuous stirred-tank reactor (CSTR) and the plug-flow reactor (PER). A comparison between the microkinetic model predictions for an isothermal PFR and the experimental results [13], is presented in Fig. 2 for the following conditions commercial low temperature shift Cu catalyst loading of 0.14 g/cm total feed flow rate of 236 cm (STP) min residence time r = 1.8 s feed composition of H20(10%), CO(10%), C02(0%), H2(0%) and N2(balance). As can be seen, the model can satisfactorily reproduce the main features of the WGSR on Cu LTS catalyst without any further fine-tuning, e.g., coverage dependence of the activation energy, etc, which is remarkable and provides proof of the adequacy of the... 

A schematic of a multiport plug flow reactor. At each successive port the space time is longer, while conversion and temperature will depend on the intervening reaction conditions... 

Coal Residence Time. Because the oxidation rate is fast, there is essentially no residence time required for chemical reaction. In a plug flow reactor, coal feed rate would depend only on the oxygen input rate and the unit oxygen consumption. Moreover, if gas distribution is uniform, the oxidation would be uniform. In a continuous fluidized-bed reactor, however, residence time must be long enough to minimize the effect of the short-circuiting of untreated feed into the product. The... 

Balancing of all space- and time-dependent variables over an infinitesimal element of a plug-flow reactor leads to the subsequent system of partial differential equations ... 

Because of the different paths taken by elements of a fluid to pass through a packed column, a residence-time distribution is generally obtained using a tracer signal at the bed input (salt, dyes), and by analysis of the output response [11]. A ical downstream signal is obtained, dependent on the kind of flow and mixing in the reactor (plug-flow reactor, mixed-flow reactor). The experimental distribution curve may be characterized in terms of mean and variance by ... 

A type of continuous reactor with performance similar to a batch reactor is the plug-flow reactor, a tubular or pipeline reactor with continuous feed at one end and product removal at the other end. The conversion is a function of the residence time, which depends on the flow rate and the reactor volume. The data for plug-flow reactors are analyzed in the same way as for batch reactors. The conversion is compared with that predicted from an integrated form of an assumed rate expression. A trial-and-error procedure may be needed to determine the appropriate rate equations. 

Figure 9.5 Dependence of effluent species concentrations on reactor space time for both a plug flow reactor and an individual continuous flow stirred-tank reactor.

The model calculations were performed with CHEMKIN II [117]. A single run calculated the concentrations of all species in dependence of the reaction time for a radial homogeneous plug-flow reactor and for overall isothermal conditions. In addition, sensitivity calculations and flow analysis were performed in order to gain information about important reaction steps. (For details see [128].)... 

For the first criterion, one compares the reactor volumes based on the average residence time for a given extent of reaction or final conversion. The average residence time depends on the reaction kinetics and therefore the reaction rate, which in turn depends on whether the reaction takes place at constant volume or variable volume. In a system at constant volume, one obtains directly a ratio between the volumes, because the average residence time is equal to space time which is defined as the ratio between reactor volume and inlet volumetric flow in the reactor. For the same conversion, the ratio between volumes is proportional. Since the average residence time in a PFR reactor is similar to the reaction time in a batch reactor, we may assume that they have similar behaviors and then we compare only the ideal tubular reactors (PFR — plug flow reactor) to the ideal tank reactors (CSTR—continuous stirred-tank reactor). 

Although the flow in the microchannels is laminar, a uniform radial concentration profile and consequently a narrow residence time distribution were obtained. Depending on the method used for manufacturing the microchannels, the Boden-stein number was found to be Bo = ud/Dj 70 and consequently the microreactor behaves almost like a plug-flow reactor. The catalytic coating had no influence on this distribution, indicating a uniform deposition of the catalyst within the microchannels (see Figure 14.3). 

FIGURE 51.4 Time dependences of the logarithmic isotope fraction in the reaction product for different types of reaction mechanisms in a CSTR (B) and plug-flow reactor (C) in comparison with a differential reactor (A). 

The above equations are for a steady-state flame at constant pressure. For time-dependent problems, such as pressure oscillations or ignition, the above equations must be modified to include the time dependent derivatives of the energy, species, and momentum (see, e.g.. Ref. 45 for a detailed description). Examples of the equations governing other chemically reacting systems, such as for a plug flow reactor or detonation, are given elsewhere. ... 

It will be clear from the above that the space-velocity (equation (2.85)) and the space-time yield (equations(2.105) and (2.Ml)) are dependent upon the degree of conversion and upon the initial (batch reactor) or inlet (plug flow reactor) oramtrattolu. In order to compare the perfomuiaoe of reactors involving different (in> values, it is therefore useful to define mmiglbed s and... 

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