Point reactor model

This simplified picture is known as the point reactor model. 

All these steps can influence the overall reaction rate. The reactor models of Chapter 9 are used to predict the bulk, gas-phase concentrations of reactants and products at point (r, z) in the reactor. They directly model only Steps 1 and 9, and the effects of Steps 2 through 8 are lumped into the pseudohomoge-neous rate expression, a, b,. ..), where a,b,. .. are the bulk, gas-phase concentrations. The overall reaction mechanism is complex, and the rate expression is necessarily empirical. Heterogeneous catalysis remains an experimental science. The techniques of this chapter are useful to interpret experimental results. Their predictive value is limited. 

In this chapter, we describe several ideal types of reactors based on two modes of operation (batch and continuous), and ideal flow patterns (backmix and tubular) for the continuous mode. From a kinetics point of view, these reactor types illustrate different ways in which rate of reaction can be measured experimentally and interpreted operationally. From a reactor point of view, the treatment also serves to introduce important concepts and terminology of CRE (developed further in Chapters 12 to 18). Such ideal reactor models serve as points of departure or first approximations for actual reactors. For illustration at this stage, we use only simple systems. 

In this chapter, we first cite examples of catalyzed two-phase reactions. We then consider types of reactors from the point of view of modes of operation and general design considerations. Following introduction of general aspects of reactor models, we focus on the simplest of these for pseudohomogeneous and heterogeneous reactor models, and conclude with a brief discussion of one-dimensional and two-dimensional models. 

We have chosen to concentrate on a specific system throughout the chapter, the methanation reaction system. Thus, although our development is intended to be generally applicable to packed bed reactor modeling, all numerical results will be obtained for the methanation system. As a result, some approximations that we will find to apply in the methanation system may not in other reaction systems, and, where possible, we will point this out. The methanation system was chosen in part due to its industrial importance, to the existence of multiple reactions, and to its high exothermicity. 

Although the finite difference technique is generally easily implemented and is quite robust, the procedure often becomes numerically prohibitive for packed bed reactor models since a large number of grid points may be required to accurately define the solution. Thus, since the early 1970s most packed bed studies have used one of the methods of weighted residuals rather than finite differences. 

These methods of weighted residuals are generally recommended for packed bed reactor modeling since solution computing time is usually low since the solution can usually be accurately defined with only a few grid points. 

At this point, there is an opportunity to analyze the effect of dissolution of reacting species in a second fluid in the overall material balance in a reactor. The overall material balance is convenient in the case where the conversion of one species is known and we want to estimate the conversion of the other species without solving the reactor models. 

Fig. 14.4 Comparison between experimented data (points) and modeling predictions (curves) for methane (CH4), ethane (C2H6), ethylene (C2H4), and acetylene (C2H2) oxidation in a flow reactor under very dilute, slightly fuel-rich conditions [148]. The excess air ratio X is about 0.9, and the residence time is of the order of 100 ms.

From these examples it is apparent that one needs to be cautious when using steady-state methods and continuation procedures near turning points. While the solutions may converge rapidly and even appear to be physically reasonable, there can be significant errors. Fortunately, a relatively simple time-stepping procedure can be used to identify the nonphysical solutions. Beginning from any of the solutions that are shown in Fig. 15.9 as shaded diamonds, a transient stirred-reactor model can be solved. If the initial solution (i.e., initial condition for the transient problem) is nonphysical, the transient procedure will march toward the physical solution. If the initial condition is the physical solution, the transient computational will remain stationary at the correct solution. 

Vertical CVD Reactors. Models of vertical reactors fall into two broad groups. In the first group, the flow field is assumed to be described by the one-dimensional similarity solution to one of the classical axisymmetric flows rotating-disk flow, impinging-jet flow, or stagnation point flow (222). A detailed chemical mechanism is included in the model. In the second category, the finite dimension of the susceptor and the presence of the reactor walls are included in a detailed treatment of axisymmetric flow phenomena, including inertia- and buoyancy-driven recirculations, whereas the chemical mechanism is simplified to a few surface and gas-phase reactions. 

The nuclear reactor kinetics was modelled using simple point kinetics. The point kinetics model utilised in the calculation was developed as an analogue to the point kinetics module of the RELAP5 code. The number of delayed neutron groups considered was six. A Doppler feedback coefficient of -0.0095 was used. Xenon feedback was also modelled, although due to the time scales considered in this document the xenon feedback is not relevant and has almost no impact on the results. 

A transient control volume model of the S-I and HyS cycle is presented. An important conclusion based on the results of this model is that the rate-limiting step of the entire S-I cycle is the HI decomposition section. In the HyS cycle, the rate-limiting step is the H2S04 decomposition. A generalised methodology for coupling these thermochemical cycle models to a nuclear reactor model is overviewed. The models were coupled to a THERMIX-DIREKT thermal model of a PBMR-268 and a point kinetics model. Key assumptions in the PBMR-268 model include flattening of the core and parallelisation of the flow channels. 

JAEA conducted an improvement of the RELAP5 MOD3 code (US NRC, 1995), the system analysis code originally developed for LWR systems, to extend its applicability to VHTR systems (Takamatsu, 2004). Also, a chemistry model for the IS process was incorporated into the code to evaluate the dynamic characteristics of process heat exchangers in the IS process (Sato, 2007). The code covers reactor power behaviour, thermal-hydraulics of helium gases, thermal-hydraulics of the two-phase steam-water mixture, chemical reactions in the process heat exchangers and control system characteristics. Field equations consist of mass continuity, momentum conservation and energy conservation with a two-fluid model and reactor power is calculated by point reactor kinetics equations. The code was validated by the experimental data obtained by the HTTR operations and mock-up test facility (Takamatsu, 2004 Ohashi, 2006). 

The mechanism for heat transfer includes the following steps (1) conduction in the catalyst particle (2) convection from the particle to the gas phase (3) conduction at contact points between particles (4) convection between the gas and vessel wall (5) radiation heat transfer between the particles, the gas, and the vessel wall (6) conduction in the wall and (7) convection to the coolant. There are a number of ways, through reactor models, that these steps are correlated to provide design and analysis estimates and criteria for preventing runaway in exothermic reactors. 

The completely mixed model succeeds in representing part of the experimental data and predicts that at industrial conditions the reactor is open-loop unstable. Initiator productivity decreases are accounted quite accurately only by the second reactor model which details the mixing conditions at the initiator feed point. Independent estimates of the model parameters result in an excellent match with experimental data for several initiator types. Imperfect mixing is shown to have a tendency to stabilize the reactor. 

Ihe chemical and catalytic reactor modeling topics included in this 21-chapter volume are illustrative of the current research emphasis from academic and industrial points of view. Most chapters present new research results and the others provide brief and timely user-oriented tutorials. 

Experimental Data. While the emphasis in this session was on reactor modeling, models can only ultimately prove successful if they are compared to experimental data. This point may seem obvious, but it is worth making since modeling efforts too often seem to be intellectual exercises rather than efforts to represent reality. While there is a need to verify some of the models presented at this symposium, it is gratifying that three of the papers (11,15,17) have already been exposed to the test of experimental data. 

Another common property of multireaction networks is stiffness, that is, the presence of kinetic steps with widely different rate coefficients. This property was pointed out by Curtiss and Hirschfelder (1952), and has had a major impact on the development of numerical solvers such as BASSE (Petzold 1983) and DDAPLUS of Appendix B. Since stiff equations take added computational effort, there is some incentive to reduce the stiffness of a model at the formulation stage this can be done by substituting Eq. (2.5-2b) or (2.5-3) for some of the reaction or production rate expressions. This strategy replaces some differential equations in the reactor model by algebraic ones to expedite numerical computations. 

Macroscopic mass, energy, and momentum balances provide the simplest starting point for reactor modeling. These equations give little spatial detail, but provide a first approximation to the performance of chemical reactors. This section builds on Chapter 22 of Bird, Stewart, and Llghtfoot (2002). A table of notation is given at the end of the current chapter. 

In this chapter, modeling of monolith reactors will be considered from a first-principles point of view, preceded by a discussion of the typical phenomena in monoliths that should be taken into account. General model equations will be presented and subsequently simplified, depending on the subprocesses that should be described by a model. A main lead will be the time scales at which these subprocesses occur. If they are all small, the process operates in the steady state, and all time-dependent behavior can be discarded. Unsteady-state behavior is to be considered if the model should include the time scale of reactor startup or if deactivation of the catalyst versus time-on-stream has to be addressed. A description of fully dynamic reactor operation, as met when cycling of the feed is applied, requires that all elementary steps of a kinetic model with their corresponding time scales are incorporated in the reactor model. 

Summary If flhe reaction is not first-order and a more precise estimate of reactor conversion is required than can be obtained from the boimds, a reactor model must be assumed. The choice of a proper model is almost pure art requiring creativity and engineering judgment. The flow pattern of the model must possess the most important characteristics of that in the real reactor. Standard models are available that have been used with some success, and these can be used as starting points. Models of tank reactors usually consist of combinations of PFRs, perfectly mixed CSTRs, and dead spaces in a configuration that matches as well as possible the flow pattern in the reactor. For tubular reactors, the simple dispersion model has proven most popular. 

The parameters in the model, which with rare exception should not exceed two in number, are obtained from the RTD. Once the parameters are evaluated, the conversion in the model, and thus in the real reactor, can be calculated. For typical tank-reactor models, this is the conversion in a series-parallel reactor system. For the dispersion model, the second-order differential equation must be solved, usually numerically. Analytical solutions exist for the first-order situation, but as pointed out previously, no model has to be assumed for the first-order system if the RTD is available. 

At this point, it is worth enquiring if these calculations have any relationship to reality. While it is very difficult to obtain information on crack extension in reactor coolant circuits for a variety of reasons, Tang etal. [63] published the data shown in Fig. 35. The data refer to the extension of a crack adjacent to the H-3 weld on the inner surface of the core shroud of a GE BWR in Taiwan. The authors had monitored the growth of the crack as a function of time after the eleventh outage for refueling. The reactor model was the same as that employed in our previous modeling and the coolant chemistry conditions could be estimated with sufficient accuracy to make a comparison between the observed and calculated crack extensions... 

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