Pseudo-differential reactor

The conversion per reaction cycle is a differential amount, but it is at the high level of conversion typical of an integral reactor. The loop reactor is therefore referred to as a pseudo-differential reactor or gradient-free reactor. The identity indicated in Equ. 4.16 can be more easily recognized when the two parts that result from the mass equivalence of a recycling reactor (left side) and set equal to that of an ideal stirred vessel (right side), and the resulting equation is then solved for ... 

Fig. 4.14. A special type of integral reactor (pseudo-integral reactor) is one constructed with taps at various distances along the length so that samples may be removed and the actual concentration profile measured. A disadvantage of integral reactors is that the balance equations are a system of coupled differential equations. The measured conversion often is due to a complex interaction of transport and reaction processes. For quick, empirical, and pragmatic process development, the integral reactor may be well suited, especially now that fast digital computers and effective integration algorithms facilitate parameterization.

For fast reactions Da becomes large. Based on that assumption and standard correlations for mass transfer inside the micro channels, both the model for the micro-channel reactor and the model for the fixed bed can be reformulated in terms of pseudo-homogeneous reaction kinetics. Finally, the concentration profile along the axial direction can be obtained as the solution of an ordinary differential equation. 

The packed bed reactors section of this volume presents topics of catalyst deactivation and radial flow reactors, along with numerical techniques for solving the differential mass and energy balances in packed bed reactors. The advantages and limitations of various models (e.g., pseudo-homogeneous vs. heterogeneous) used to describe packed bed reactors are also presented in this section. 

To determine reaction rate parameters from the experimental data, the following differential equation was used to describe the reaction system in a constant-volume batch reactor assuming a pseudo-first-order equation for propylene epoxidation ... 

The non-isothermal, pseudo-homogeneous model A simple non-isothermal model that takes into consideration the temperature variation along the length of the reactor tube is used in this section. An energy balance on the reformer tube element, assuming constant skin temperature, gives the following differential equation (see section 6.3.4). 

It is clear from these values and Figure 7.14 that p 1 / is an excellent approximation for this reactor. Substituting this equation for p into the mass balance and solving the differential equation produces the results shown in Figure 7.2 7. The concentration of O2 is nearly constant, which Justifies the pseudo-first-order rate expression. Reactor volume... 

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... 

The differential rate law described by this equatioa is mathematically equivalent to the differential rate law described by equatioa 16. Thus the reaction will appear to follow first-order kinetics but is in reali a second-order reaction. Such a reaction is referred to as a pseudo ferst-order reaction and the rate constant BCobs determmed fiom this reaction is a pseudo first-order rate constant The second-order rate constant may be evaluated by plotting BCobs versus Cb for different concentrations of B obtained fiom a batch reactor system. The slope of the linear trace obtained fiom such a plot will have a value numerically equal to K. 

CPBR behavior under pseudo-steady-state and plug-flow regime is described by the resolution of the system of differential Eqs. 5.76 and 5.79. This model was experimentally validated in a laboratory packed-bed reactor with chitin-immobilized... 

Heterogeneous reactions. Components of water or air pollution are usually in the fluid phase. Hence we may write equations such as equations 6.2, 6.6, 6.8, and 6.12 for the fluid. The fluid may have non-permeable boundaries (the reactor walls) and permeable boundaries (entrances and exits of the system as well as catalytic surfaces where mass fluxes must be equal to the superficial reaction rates). Usually, these reaction rates are modeled as pseudo-homogeneous and, moreover, concentration measurements are almost always made in the fluid phase. Heterogeneous reactions are the result of a process that occurs at phase interfaces. This means that for the differential equation written for the fluid phase, heterogeneous reactions (surface reactions, for example) are just boundary conditions. The problem is very simple to formulate at steady state and at the boundary of an active surface, the normal mass or molar fluxes must be made equal to the heterogeneous, superficial reaction rate. Then,... 

For fixed-bed catalytic reactors, a PFR model with a pseudo-homogeneous kinetic equation is usually adequate and is referred to as a 1-D (one-dimensional) model. However, if the reactor is nonadiabatic with heat transfer to or from the wall, the PFR model is not usually adequate and a 2-D model, involving the solution of partial differential equations for variations in temperature and composition in both the axial and radial directions, is necessary. Simulators do not include 2-D models, but they can be generated by the user and inserted into the simulator. 

Pilavakis (20, 29) investigated the esterification of methanol by acetic acid in a packed column. He assumed the reaction to be pseudo-first-order with respect to either methanol or acid over certain specified concentration ranges and incorporated the effect of heat of reaction not only in the enthalpy balances but also in the flux equations. The column was calculated by numerical solution of a set of differential equations. The top product was an azeotropic mixture of methanol and ester which could, however, be broken by introduction of acetic acid high up in the column rather than further down as a mixed feed with methanol. Consequently, in practice such a column will consist of a rectifying section, an extractive distillation section with acetic acid as the extractive solvent and a distillation reactor section. Good agreement was obtained between theory and experiment which, however, suffered from the fact that the hold-up of liquid in the column was small in comparison to the reboiler hold-up so that most of the reaction occurred in the latter location. 

Through a preliminary study they demonstrated that, starting from the L-H kinetic model, the rate of the photocatalytic degradation followed a pseudo-zero order kinetics (Equation [21.1a]). Moreover, performing dialysis experiments (without irradiation), they assumed that mass transport of solutes was due to diffusion only and no exchange between the two compartments occurred. The authors described also the variation of concentration in the feed tank and in the reactor by differential equations (Equation [21.1b]). 

At the high recycling ratios the loop reactor operates as an ideal stirred-tank reactor. Therefore, the reaction rate can immediately be determined from the difference in concentration between the feed and the outlet, the throughput and the quantity of catalyst.The rate equation, describing the consumption of xylene and the formation of the reaction products, are considered to be pseudo first order. The parameter of the rate equations, which are the frequency factors and the activation energies, are determined by least square methods. In the above function (Fig. 6b) r is the measured rate, r is calculated with estimated parameters, w represent appropriate weight factors and N is the number of measured values. Because the rate equations could be differentiated v/ith respect to the unknown kinetic parameters, the objective function was minimized by a step-wise regression. 

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