Reactor performance equations

Batch Reactor. Figure 18.15 sketches the main features of an experimental reactor which uses a batch of catalyst and a batch of fluid. In this system we follow the changing composition with time and interpret the results with the batch reactor performance equation. 

Substituting this model in the reactor-performance equations for Michaelis-Menten kinetics, the following equations can be obtained for the three main types of reactions Batch reactor ... 

The experimental study of solid catalyzed gaseous reactions can be performed in batch, continuous flow stirred tank, or tubular flow reactors. This involves a stirred tank reactor with a recycle system flowing through a catalyzed bed (Figure 5-31). For integral analysis, a rate equation is selected for testing and the batch reactor performance equation is integrated. An example is the rate on a catalyst mass basis in Equation 5-322. 

Ph = hydrogen partial pressure, psia Phs = hydrogen sulfide partial pressure, psia The reactor performance equations are as follows ... 

The plug flow reactor is probably the most commonly used reactor in catalyst evaluation because it is simply a tube filled with catalyst that reactants are fed into. However, for catalyst evaluation, it is difficult to measure the reaction rate because concentration changes along the axis, and there are frequently temperature gradients, too. Furthermore, because the fluid velocity next to the catalyst is low, the chance for mass transfer limitations through the film around the catalyst is high. Eq. (3) is the reactor performance equation for a plug flow reactor. 

Reactor performance equations for a plug flow reactor... 

Thus, the equation below is the reactor performance equation for a single stirred tank. 

Reactor Performance Equations for the Reactor-Regenerator System... 

Under steady-state conditions in a plug-flow tubular reactor, the onedimensional mass transfer equation for reactant A can be integrated rather easily to predict reactor performance. Equation (22-1) was derived for a control volume that is differentially thick in all coordinate directions. Consequently, mass transfer rate processes due to convection and diffusion occur, at most, in three coordinate directions and the mass balance is described by a partial differential equation. Current research in computational fluid dynamics applied to fixed-bed reactors seeks a better understanding of the flow phenomena by modeling the catalytic pellets where they are, instead of averaging or homogenizing... 

As a first step, based on laboratory-scale data at different times, a rate equation can be developed and then the batch reactor performance equation... 

Using flow reactors imder steady-state conditions, we can easily collect data for process optimization, record activity, and selectivity and study catalyst life and deactivation processes. If we know the contacting pattern in the reactor, then we can explore the kinetics from the reactor performance equation. All of the flow reactors described previously present data for the average reactor concentration versus time. Generally the activity, selectivity, and stability are presented as a function of different process variables such as temperature, pressure, and space velocity. From conversion we can calculate the rate from the performance equation of the reactor, for example, for a CSTR ... 

Unfortunately, the RTD of a CSTR does not provide the lower limit on reactor performance. Equation (30) is worthless unless one can guarantee that there is no bypassing and stagnancy in the reactor. To prove that, one needs an experimental RTD and the design is not truly predictive any more. We consider here the RTD based on the generalized tanks in series model, i.e. based on the gamma probability density function, which allows the following representation of the normalized exit age density functions ... 

PFR, boiling transfer mediurn, 714 PFR, single-phase transfer mediurn, 714 pressure drop in packed bed, equation, 715 reactor performance equation, 715 runaway reactions, 712 safety issues, 712... 

Computer simulation of the reactor kinetic hydrodynamic and transport characteristics reduces dependence on phenomenological representations and idealized models and provides visual representations of reactor performance. Modem quantitative representations of laminar and turbulent flows are combined with finite difference algorithms and other advanced mathematical methods to solve coupled nonlinear differential equations. The speed and reduced cost of computation, and the increased cost of laboratory experimentation, make the former increasingly usehil. 

Dimensional Analysis. Dimensional analysis can be helpful in analyzing reactor performance and developing scale-up criteria. Seven dimensionless groups used in generalized rate equations for continuous flow reaction systems are Hsted in Table 4. Other dimensionless groups apply in specific situations (58—61). Compromising assumptions are often necessary, and their vaHdation must be estabHshed experimentally or by analogy to previously studied systems. 

From diese various estimates, die total batch cycle time t(, is used in batch reactor design to determine die productivity of die reactor. Batch reactors are used in operations dial are small and when multiproducts are required. Pilot plant trials for sales samples in a new market development are carried out in batch reactors. Use of batch reactors can be seen in pharmaceutical, fine chemicals, biochemical, and dye industries. This is because multi-product, changeable demand often requues a single unit to be used in various production campaigns. However, batch reactors are seldom employed on an industrial scale for gas phase reactions. This is due to die limited quantity produced, aldiough batch reactors can be readily employed for kinetic studies of gas phase reactions. Figure 5-4 illustrates die performance equations for batch reactors. 

Reactor Performance Measures. There are four common measures of reactor performance fraction unreacted, conversion, yield, and selectivity. The fraction unreacted is the simplest and is usually found directly when solving the component balance equations. It is a t)/oo for a batch reaction and aout/ciin for a flow reactor. The conversion is just 1 minus the fraction unreacted. The terms conversion and fraction unreacted refer to a specific reactant. It is usually the stoichiometrically limiting reactant. See Equation (1.26) for the first-order case. 

Solution Example 4.5 was a reverse problem, where measured reactor performance was used to determine constants in the rate equation. We now treat the forward problem, where the kinetics are known and the reactor performance is desired. Obviously, the results of Run 1 should be closely duplicated. The solution uses the method of false transients for a variable-density system. The ideal gas law is used as the equation of state. The ODEs are... 

The method of lines formulation for solving Equation (8.52) does not require that T aii be constant, but allows T aiiiz) to be an arbitrary function of axial position. A new value of T aii may be used at each step in the calculations, just as a new may be assigned at each step (subject to the stability criterion). The design engineer is thus free to pick a T au z) that optimizes reactor performance. 

A substantial investment in algebra is needed to evaluate the six constants, but the result is remarkable. The exit concentration from an open system is identical to that from a closed system. Equation (9.20), and is thus independent of Dt and Dou The physical basis for this result depends on the concentration profile, a(z), for z<0. When Z) = 0, the concentration is constant at a value if until z = 0+, when it suddenly plunges to u(0+). When D >0, the concentration begins at when z = —oo and gradually declines until it reaches exactly the same concentration, u(0+), at exactly the same location, z = 0+. For z>0, the open and closed systems have the same concentration profile and the same reactor performance. 

When the residence time distribution is known, the uncertainty about reactor performance is greatly reduced. A real system must lie somewhere along a vertical line in Figure 15.14. The upper point on this line corresponds to maximum mixedness and usually provides one bound limit on reactor performance. Whether it is an upper or lower bound depends on the reaction mechanism. The lower point on the line corresponds to complete segregation and provides the opposite bound on reactor performance. The complete segregation limit can be calculated from Equation (15.48). The maximum mixedness limit is found by solving Zwietering s differential equation. ... 

Equation (15.48) governs the performance of the completely segregated reactor, and Equation (15.49) governs the maximum mixedness reactor. These reactors represent extremes in the kind of mixing that can occur between molecules that have different ages. Do they also represent extremes of performance as measured by conversion or selectivity The bounding theorem provides a partial answer ... 

Boundary layer similarity solution treatments have been used extensively to develop analytical models for CVD processes (2fl.). These have been useful In correlating experimental observations (e.g. fi.). However, because of the oversimplified fiow description they cannot be used to extrapolate to new process conditions or for reactor design. Moreover, they cannot predict transverse variations In film thickness which may occur even In the absence of secondary fiows because of the presence of side walls. Two-dimensional fully parabolized transport equations have been used to predict velocity, concentration and temperature profiles along the length of horizontal reactors for SI CVD (17,30- 32). Although these models are detailed, they can neither capture the effect of buoyancy driven secondary fiows or transverse thickness variations caused by the side walls. Thus, large scale simulation of 3D models are needed to obtain a realistic picture of horizontal reactor performance. 

For the various reactor types this equation simplifies in one way or another, and the resultant expression when integrated provides the basic performance equation for that type of unit. Since in batch reactor or operation, no stream is entering or leaving the reactor,... 

For semibatch or semiflow reactors all four of the terms in the basic material and energy balance relations (equations 8.0.1 and 8.0.3) can be significant. The feed and effluent streams may enter and leave at different rates so as to cause changes in both the composition and volume of the reaction mixture through their interaction with the chemical changes brought about by the reaction. Even in the case where the reactor operates isothermally, numerical methods must often be employed to solve the differential performance equations. 

The units of rv are moles converted/(volume-time), and rv is identical with the rates employed in homogeneous reactor design. Consequently, the design equations developed earlier for homogeneous reactors can be employed in these terms to obtain estimates of fixed bed reactor performance. Two-dimensional, pseudo homogeneous models can also be developed to allow for radial dispersion of mass and energy. 

The solution of problems in chemical reactor design and kinetics often requires the use of computer software. In chemical kinetics, a typical objective is to determine kinetics rate parameters from a set of experimental data. In such a case, software capable of parameter estimation by regression analysis is extremely usefiil. In chemical reactor design, or in the analysis of reactor performance, solution of sets of algebraic or differential equations may be required. In some cases, these equations can be solved an-... 

In this chapter, we first consider uses of batch reactors, and their advantages and disadvantages compared with continuous-flow reactors. After considering what the essential features of process design are, we then develop design or performance equations for both isothermal and nonisothermal operation. The latter requires the energy balance, in addition to the material balance. We continue with an example of optimal performance of a batch reactor, and conclude with a discussion of semibatch and semi-continuous operation. We restrict attention to simple systems, deferring treatment of complex systems to Chapter 18. 

The continuity equations for the three main regions lead eventually to the performance equation for the reactor model. 

Equation 23.4-6 is one form of the performance equation for the bubbling-bed reactor model. It can be transformed to determine the amount of solid (e.g., catalyst) holdup to achieve a specified /A or cA ... 

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