Species Balances and Design Equations

The general conservation statement for species j over a stationary system is 

Molar flow( Rate moles rate of I I of species j species J [ ] formed 

Appendix B provides the derivation of the design equation from the species continuity equation. In Section 4.1, we carry out macroscopic species balances to derive the species-based design equation of any chemical reactor. In Section 4.2, 

Principles of Chemical Reactor Analysis and Design, Second Edition. By Uzi Mann Copyright 2009 John Wiley Sons, Inc. 

1 MACROSCOPIC SPECIES BALANCES—GENERAL SPECIES-BASED DESIGN EQUATIONS 

---

The genesis of the reactor design equations is the conservation of mass. Since reactor operations involve changes in species compositions, the mass balance is written for individual species, and it is expressed in terms of moles rather than mass. Species balances and the reactor design equations are discussed in detail in Chapter 4. To obtain a complete description of the reactor operation, it is necessary to know the local reaction rates at all points inside the reactor. This is a formidable task that rarely can be carried out. Instead, the reactor operation is described by idealized models that approximate the actual operation. Chapters 5-9 cover the applications of reactor design equations to several ideal reactor conflgurations that are commonly used. 

Case II Quantitative Treatment of a Single Stirred-Tank Reactor. When reactions (9.3.3) and (9.3.4) take place in a single continuous-flow stirred-tank reactor, the route to a quantitative relation describing the product distribution involves writing the material balances or design equations for species V and A ... 

The starting point for the development of the basic design equation for a well-stirred batch reactor is a material balance involving one of the species participating in the chemical reaction. For convenience we will denote this species as A and we will let (— rA) represent the rate of disappearance of this species by reaction. For a well-stirred reactor the reaction mixture will be uniform throughout the effective reactor volume, and the material balance may thus be written over the entire contents of the reactor. For a batch reactor equation 8.0.1 becomes... 

In principle, one can carry out a four-dimensional optimization in which the four parameters are varied subject to constraints (< 1 and P4 < 1 ), to minimize the deposition time with the non-uniformity bounded e.g., MN < 3. However, objective function evaluations involve solutions of the Navier-Stokes and species balance equations and are computationally expensive. Instead, Brass and Lee carry out successive unidirectional optimizations, which show the key trends and lead to excellent designs. A summary of the observed trends is shown in Table 10.4-1. Both the deposition rate and the non-uniformity are monotonic functions of the geometric parameters within the bounds considered, with the exception that the non-uniformity goes through a minimum at optimal values of P3 and P4. 

Model Equations to Describe Component Balances. The design of PVD reacting systems requires a set of model equations describing the component balances for the reacting species and an overall mass balance within the control volume of the surface reaction zone. Constitutive equations that describe the rate processes can then be used to obtain solutions to the model equations. Material-specific parameters may be estimated or obtained from the literature, collateral experiments, or numerical fits to experimental data. In any event, design-oriented solutions to the model equations can be obtained without recourse to equipment-specific fitting parameters. Thus translation of scale from laboratory apparatus to production-scale equipment is possible. 

For each of the ideal reactor types, viz. ideal batch reactor, plug-flow reactor (PFR), and continuous-flow stirred-tank reactor (CSTR), continuity equations or design equations can be derived using mass (or rather molar) balance equations for each species involved. 

Basically, the processes taking place in a chemical reactor are chemical reaction, and mass, heat and momentum transfer phenomena. The modeling and design of reactors are therefore sought from emplo3dng the governing equations describing these phenomena [1] the reaction rate equation, and the species mass, continuity, heat (or temperature) and momentum balance equations. 

Species Balance Equation and Reactor Design Equation... 

To obtain useful expressions from the general species-based design equation, we should know the formation rate of species j, (rj), at any point in the reactor. To express (vj), the local concentrations of aU species as well as the local temperatures should be provided. To obtain these quantities, we should solve the overall continuity equation, the individual species continuity equations, and the energy balance equation. This is a formidable task, and, in most situations, we cannot reduce those... 

The design equations and the species concentration relations contain another dependent variable, 6, the dimensionless temperature, whose variation during the reactor operation is expressed by the energy balance equation. For ideal batch reactors with negligible mechanical shaft work, the energy balance equation, derived in Section 5.2, is... 

We start the analysis of plug-flow reactors by considering isothermal operations with single reactions. For isothermal operations, rf0/rfT = O, and we have to solve only the design equations. The energy balance equation provides the heating (or cooling) load necessary to maintain isothermal conditions. Furthermore, for isothermal operations, the reaction rate depends only on the species concentrations, and Eq. 7.1.5 reduces to... 

The design equations, the energy balance equation, and the auxihary relations for species concentrations... 

In this chapter, the analysis of chemical reactors is expanded to additional reactor configurations that are commonly used to improve the yield and selectivity of the desirable products. In Section 9.1, we analyze semibatch reactors. Section 9.2 covers the operation of plug-flow reactors with continuous injection along their length. In Section 9.3, we examine the operation of one-stage distillation reactors, and Section 9.4 covers the operation of recycle reactors. In each section, we first derive the design equations, convert them to dimensionless forms, and then derive the auxiliary relations to express the species concentrations and the energy balance equation. 

To derive the design equation of a recycle reactor, we consider a differential reactor element, dV, and write a species balance equation over it for species j ... 

With the concentration relations and an expression for 0i, we can now complete the design formulation of recycle reactors. Substituting the species concentrations and 6 in the individual reactions rates, r s and r s, we obtain a set of first-order, nonlinear differential equations that should be solved simultaneously with the energy balance equation for the initial condition that at t = 0, Z s = 0 and 0 =... 

The mass and energy balance equations developed in Secs. 14.1 and 14.2 are the basic equations used in reactor design and analysis. In many cases, however, our needs are much more modest than in engineering design. In particular, we may not be interested in such details as the type of reactor used and the concentration and temperature profiles or time history in the reactor, but merely in the species mass and total energy balances for the reactor. In such situations one can use the general black-box equations of Table 8.4-1 ... 

---