Equations Batch reactor

Differential equations Batch reactor with first-order kinetics. Analytical or numerical solution with analytical or numerical parameter optimisation (least squares or likelihood). Batch reactor with complex kinetics. Numerical integration and parameter optimisation (least squares or likelihood). 

In TBP extraction, the yeUowcake is dissolved ia nitric acid and extracted with tributyl phosphate ia a kerosene or hexane diluent. The uranyl ion forms the mixed complex U02(N02)2(TBP)2 which is extracted iato the diluent. The purified uranium is then back-extracted iato nitric acid or water, and concentrated. The uranyl nitrate solution is evaporated to uranyl nitrate hexahydrate [13520-83-7], U02(N02)2 6H20. The uranyl nitrate hexahydrate is dehydrated and denitrated duting a pyrolysis step to form uranium trioxide [1344-58-7], UO, as shown ia equation 10. The pyrolysis is most often carried out ia either a batch reactor (Fig. 2) or a fluidized-bed denitrator (Fig. 3). The UO is reduced with hydrogen to uranium dioxide [1344-57-6], UO2 (eq. 11), and converted to uranium tetrafluoride [10049-14-6], UF, with HF at elevated temperatures (eq. 12). The UF can be either reduced to uranium metal or fluotinated to uranium hexafluoride [7783-81-5], UF, for isotope enrichment. The chemistry and operating conditions of the TBP refining process, and conversion to UO, UO2, and ultimately UF have been discussed ia detail (40). 

The hquid-phase chlorination of benzene is an ideal example of a set of sequential reactions with varying rates from the single-chlorinated molecule to the completely chlorinated molecule containing six chlorines. Classical papers have modeled the chlorination of benzene through the dichlorobenzenes (14,15). A reactor system may be simulated with the relative rate equations and flow equation. The batch reactor gives the minimum ratio of... 

Unsteady material and energy balances are formulated with the conservation law, Eq. (7-68). The sink term of a material balance is and the accumulation term is the time derivative of the content of reactant in the vessel, or 3(V C )/3t, where both and depend on the time. An unsteady condition in the sense used in this section always has an accumulation term. This sense of unsteadiness excludes the batch reactor where conditions do change with time but are taken account of in the sink term. Startup and shutdown periods of batch reactors, however, are classified as unsteady their equations are developed in the Batch Reactors subsection. For a semibatch operation in which some of the reactants are preloaded and the others are fed in gradually, equations are developed in Example 11, following. 

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. 

Assuming that the reactions are first order in a constant volume batch reactor, the rate equations for components A, B, C, and D, respectively, are ... 

For a well-mixed batch reactor, the design equation is... 

From the mass balance equation for a batch reactor... 

When heating a batch reactor with an external heat exchanger and non-isothermal heating, the following equation applies ... 

Consider the scale-up of a batch reactor from a pilot plant reactor to a full-scale reactor. Rewriting Equation 13-82 to the full-scale reactor yields ... 

Aspect ratio R2 (L/D)2 for zoning scale-up of a batch reactor using Equation 13-119... 

Continuous stirred-tank reactors can behave very differently from batch reactors with regard to the number of particles formed and polymerization rate. These differences are probably most extreme for styrene, a monomer which closely follows Smith-Ewart Case 2 kinetics. Rate and number of particles in a batch reactor follows the relationship expressed by Equation 13. 

Balance equations for batch reactors may all be viewed as special cases of the following general equation... 

In this work, the characteristic "living" polymer phenomenon was utilized by preparing a seed polymer in a batch reactor. The seed polymer and styrene were then fed to a constant flow stirred tank reactor. This procedure allowed use of the lumped parameter rate expression given by Equations (5) through (8) to describe the polymerization reaction, and eliminated complications involved in describing simultaneous initiation and propagation reactions. 

A batch reactor has no input or output of mass after the initial charging. The amounts of individual components may change due to reaction but not due to flow into or out of the system. The component balance for component A, Equation (1.6), reduces to... 

The most important characteristic of an ideal batch reactor is that the contents are perfectly mixed. Corresponding to this assumption, the component balances are ordinary differential equations. The reactor operates at constant mass between filling and discharge steps that are assumed to be fast compared with reaction half-lives and the batch reaction times. Chapter 1 made the further assumption of constant mass density, so that the working volume of the reactor was constant, but Chapter 2 relaxes this assumption. 

The component balance for a batch reactor. Equation (1.21), still holds when there are multiple reactions. However, the net rate of formation of the component may be due to several different reactions. Thus,... 

Suppose there are N components involved in a set of M reactions. Then Equation (1.21) can be written for each component using the rate expressions of Equations (2.7) or (2.8). The component balances for a batch reactor are... 

Example 2.2 Derive the batch reactor design equations for the reaction set in Example 2.1. Assume a liquid-phase system with constant density. 

Example 2.10 Suppose 2A —> B in the liquid phase and that the density changes from po to Poo = Po + Ap upon complete conversion. Find an analytical solution to the batch design equation and compare the results with a hypothetical batch reactor in which the density is constant. 

The circumflex over a and b allows for spatial variations. It can be ignored when the contents are perfectly mixed. Equation (2.36) is the form normally used for batch reactors where d = a t). It can be applied to piston flow reactors by setting ao = Ui and d = a z), and to CSTRs by setting ao = and d = Uout-... 

Using V, we can write the design equations for a batch reactor in very compact form ... 

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. 

Chapter 2 developed a methodology for treating multiple and complex reactions in batch reactors. The methodology is now applied to piston flow reactors. Chapter 3 also generalizes the design equations for piston flow beyond the simple case of constant density and constant velocity. The key assumption of piston flow remains intact there must be complete mixing in the direction perpendicular to flow and no mixing in the direction of flow. The fluid density and reactor cross section are allowed to vary. The pressure drop in the reactor is calculated. Transpiration is briefly considered. Scaleup and scaledown techniques for tubular reactors are developed in some detail. 

Chapter 1 treated the simplest type of piston flow reactor, one with constant density and constant reactor cross section. The reactor design equations for this type of piston flow reactor are directly analogous to the design equations for a constant-density batch reactor. What happens in time in the batch reactor happens in space in the piston flow reactor, and the transformation t = z/u converts one design equation to the other. For component A,... 

The extension to multiple reactions is done by writing Equation (3.1) (or the more complicated versions of Equation (3.1) that will soon be developed) for each of the N components. The component reaction rates are found from Equation (2.7) in exactly the same ways as in a batch reactor. The result is an initial value problem consisting of N simultaneous, first-order ODEs that can be solved using your favorite ODE solver. The same kind of problem was solved in Chapter 2, but the independent variable is now z rather than t. 

This is the piston flow analog of the variable-volume batch reactor. Equation (2.30). 

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