Case A. Constant-Volume Batch Reactor

A constant volume batch reactor is used to convert reactant. A, to product, B, via an endothermic reaction, with simple stoichiometry, A — B. The reaction kinetics are second-order with respect to A, thus 

From the reaction stoichiometry, product B is formed at exactly the same rate as that at which reactant A, is decomposed. 

For the second-order reaction, the term representing the rate of heat production by reaction, simplifies to 

This gives the resultant heat balance equations as 

The component mass balance, when coupled with the heat balance equation and temperature dependence of the kinetic rate coefficient, via the Arrhenius relation, provide the dynamic model for the system. Batch reactor simulation examples are provided by BATCHD, COMPREAC, BATCOM, CASTOR, HYDROL and RELUY. 

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A variable-volume batch reactor is a constant-pressure (piston-like) closed tank. On the other hand, a variable-pressure tank is a constant-volume batch reactor (Fogler, 1999). Thus, in batch reactors, the expansion factor is used only in the case of a constant-pressure tank whereas and not in a constant-volume tank, even if the reaction is realized with a change in the total moles. However, in continuous-flow reactors, the expansion factor should be always considered. In the following section and for the continuous-flow reactors, the volume V can be replaced by the volumetric flow rate Q, and the moles N by the molar flow rate F in all equations. 

P3-14b Reconsider the decomposition of nitrogen tetroxide discussed in Example 3-8. The reaction is to be Carried out in PER and aiso in a constant-volume batch reactor at 2 atm and 340 K. Only Nj04 and an inert I are to be fed to the reactors, Plot the equilibrium conversion as a function of inert mole fraction in the feed for both a constant-volume batch reactor and a plug Sow reactor, Why is the equilibrium conve ton lower for the batch system than the flow system in Example 3-87 Will this lower equilibrium conversion result always be the case for batch systems ... 

Rewrite the design equation in terms of the measured variabte. When there is a net increase or decrease in the totai number of moles in a gas phase reaction, the reaction order may be determined from experiments performed with a constant-volume batch reactor by monitoring the total pressure as a function of time. The total pressure data should not be converted to conversion and then analyzed as conversion-time data just because the design equations are written in terms of the variable conversions. Rather, transform the design equation to the measured variable, which in this case is pressure. Consequently, we need to express the concentration in terms of total pressure and then substitute for the concemtation of A in Equation (E5-I.1),... 

For the case of a constant-volume batch reactor, we recall Equations (3-26) and (3-38) ... 

The simplest case to consider by far is that of first-order or linear kinetics in a constant volume batch reactor. If the rate of reaction is directly proportional to the rate of the reaction, then we call this the first order in the concentration of reactant, and the right-hand side becomes ... 

The solution to this problem requires an analysis of multiple gas-phase reactions in a differential plug-flow tubular reactor. Two different solution strategies are described here. In both cases, it is important to write mass balances in terms of molar flow rates and reactor volume. Molar densities and residence time are not appropriate for the convective mass-transfer-rate process because one cannot assume that the total volumetric flow rate is constant in the gas phase, particularly when the total number of moles is not conserved. In each reaction, 2 mol of reactants generates 1 mol of product. Furthermore, an overall mass balance suggests that the volumetric flow rate is constant only when the overall mass density does not change. This is a reasonable assumption for liquid-phase reactors but not for gas-phase problems when the total volume is not restricted. The exception is a constant-volume batch reactor. 

In Chapter 3, the analytical method of solving kinetic schemes in a batch system was considered. Generally, industrial realistic schemes are complex and obtaining analytical solutions can be very difficult. Because this is often the case for such systems as isothermal, constant volume batch reactors and semibatch systems, the designer must review an alternative to the analytical technique, namely a numerical method, to obtain a solution. For systems such as the batch, semibatch, and plug flow reactors, sets of simultaneous, first order ordinary differential equations are often necessary to obtain the required solutions. Transient situations often arise in the case of continuous flow stirred tank reactors, and the use of numerical techniques is the most convenient and appropriate method. 

First, consider constant-volume batch reactors (reactors whose volumes do not change during the operation), Vr(t) = V/ (0). In practice, this condition is satisfied either for gas-phase reactions when the walls of the reactor are stationary or when the reaction takes place in a liquid phase. In the latter case, the assumption is that the density of the liquid does not vary during the operation. For most liquid-phase reactions, the density variations are indeed quite small. 

Consider the case where this reaction is carried out isothermaUy at a constant pressure of 800 torr in a variable volume batch reactor. At the temperature of interest, the value of the rate constant is 1.30 M" /s. The initial mixture consists of equal moles of CjFg and CF3OF. Determine the time necessary for the concentration of C3Fg to decrease to... 

In this case (see equation 2.10), for a constant volume, isothermal batch reactor... 

The information required here is not concentration versus time, but rate of reaction versus concentration. As will be seen later, some types of chemical reactors give this information directly, but the constant-volume, batch systems discussed here do not [ What does it profit you, anyway —F. Villon], In this case it is necessary to determine rates from conversion-time data by graphical or numerical methods, as indicated for the case of initial rates in Figure 1.25. In Figure 1.27 a curve is shown representing the concentration of a reactant A as a function of time, and we identify the two points Cai and Ca2 for the concentration at times q and t2- The mean value for the rate of reaction we can approximate algebraically by... 

A gas-phase Ziegler-Natta polymerization is carried out in a constant-volume, isothermal batch reactor. Assume that surface monomer concentration is always proportional to bulk monomer concentration, [My] = / [Mj]. Obtain expressions for conversion as a function of time for two cases ... 

In a formal sense, Equation (2.38) applies to all batch reactor problems. So does Equation (2.42) combined with Equation (2.40). These equations are perfectly general when the reactor volume is well mixed and the various components are quickly charged. They do not require the assumption of constant reactor volume. If the volume does vary, ancillary, algebraic equations are needed as discussed in Section 2.6.1. The usual case is a thermodynamically imposed volume change. Then, an equation of state is needed to calculate the density. 

As an elaboration of point (3), if a batch reactor is used for a liquid-phase reaction, as indicated in Figure 2.1, we may usually assume that the volume per unit mass of material is constant (i.e., constant density), but if it is used for a gas-phase reaction, this may not be the case. 

Independently of enzyme stability an enzyme reaction at constant temperature and pH should be run with the smallest possible contribution by inhibition. For this reason, a CSTR is most suitable in the case of substrate inhibition because the substrate concentration is evened out across the reactor volume and thus minimized. In the case of product inhibition, however, a batch reactor or a PFR is preferred, as the reactor volume required for complete or nearly complete conversion is much smaller than in the case of a CSTR. 

The CSTR operator, Rc, has an identical term to describe accumulation under transient operation. The algebraic sum of the two other terms indicates the difference of in-flow and out-flow of that species. This operator also describes semibatch or semicontinuous operation in cases where the volume can be assumed to be essentially constant. In the more general case of variable volume, V must be included within the differential accumulation term. At steady state, it is a difference equation of the same form as the differential equation for a batch reactor. 

At constant pressure and granted ideal plug flow, the behavior of a tubular reactor at steady state is mathematically analogous to that of a batch reactor A volume element of the reaction mixture has no means of knowing whether it is suspended tea bag-style in a batch reactor or rides elevator-style through a tubular reactor being exposed to the same conditions it behaves in the same way in both cases. As in a batch reactor, what is measured directly are concentrations—here in the effluent—and a finite-difference approximation is needed to obtain the rate from experiments with different reactor space times and otherwise identical conditions. For a reaction without fluid-density variation ... 

The formal similarity allows us to carry over the equations for mass and energy balances in the tubular reactor, Eqs. (3.4.11)-(3.4.14). The momentum equation has no meaning. Care must be taken however to distinguish between a batch reactor working at constant volume and one that works at constant pressure. The latter has the Eqs. (3.4.12) or (3.4.14) which were derived from an enthalpy balance. In the former case the heat added would be equated to the internal energy change. Thus in this case c should replace Cp and the internal energy of reaction replace the heat of reaction. These... 

A special case of batch reactors is constant-volume or constant-density operation typical of liquid-phase reactions, with volume invariant with time ... 

On the other hand, an industrial process may be operated in a continuous mode, rather than in a batch mode. To achieve this, either a single or a series of interconnected vessels may be used. The required raw materials are continuously fed into this vessel or the first vessel and the reaction products continuously removed from the last so that the volume of material in the reactor(s) stays constant as the reaction proceeds. The concentrations of starting materials and products in the reactor eventually reach a steady state. One or more tanks in series may be used to conduct the continuous process. Another option for a continuous process is to use a pipe or tube reactor, in which the starting material(s) is fed into the tube at one end, and the product(s) is removed at the other. In this case, the reaction time is determined by the rate of flow of materials into the tube divided by the length of the tube. 

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