Single reaction-progress variable

For simple chemistry, we have seen in Section 5.5 that limiting cases of general interest exist that can be described by a single reaction-progress variable, in addition to the mixture fraction.131 For these flows, the chemical source term can be closed by assuming a form for the joint PDF of the reaction-progress variable Y and the mixture fraction f. In general, it is easiest to decompose the joint PDF into the product of the conditional PDF of Y and the mixture-fraction PDF 132 

Since acceptable forms are available for the mixture-fraction PDF, the chemical source term (Sy(Y, i-)) can be closed by modeling fy y If x, t). 

By definition, the reaction-progress variable is bounded below by zero. Likewise, we have seen in Section 5.5 that a single reaction-progress variable is bounded above by a 

130 A fluid particle that has spent a significant time in the flow will have reacted further than a particle that has only recently entered. Thus, for the same value of the mixture fraction, their concentrations can be very different  

131 For premixed flows, the mixture fraction is not applicable. Nonetheless, the methods in this section can still be employed to model the PDF of the reaction-progress variable. 

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J2.2.2 Methods of Following the Course of a Reaction. A general direct method of measuring the rate of a reaction does not exist. One can only determine the amount of one or more product or reactant species present at a certain time in the system under observation. If the composition of the system is known at any one time, then it is sufficient to know the amount of any one species involved in the reaction as a function of time in order to be able to establish the complete system composition at any other time. This statement is true of any system whose reaction can be characterized by a single reaction progress variable ( or fA). In practice it is always wise where possible to analyze occasionally for one or more other species in order to provide a check for unexpected errors, losses of material, or the presence of side reactions. 

The chemical composition of many systems can be expressed in terms of a single reaction progress variable. However, a chemical engineer must often consider systems that cannot be adequately described in terms of a single extent of reaction. This chapter is concerned with the development of the mathematical relationships that govern the behavior of such systems. It treats reversible reactions, parallel reactions, and series reactions, first in terms of the mathematical relations that govern the behavior of such systems and then in terms of the techniques that may be used to relate the kinetic parameters of the system to the phenomena observed in the laboratory. 

The partial pressures of the various species are numerically equal to their mole fractions since the total pressure is one atmosphere. These mole fractions can be expressed in terms of a single reaction progress variable-the degree of conversion-as indicated in the following mole table. 

For reactor design purposes, the distinction between a single reaction and multiple reactions is made in terms of the number of extents of reaction necessary to describe the kinetic behavior of the system, the former requiring only one reaction progress variable. Because the presence of multiple reactions makes it impossible to characterize the product distribution in terms of a unique fraction conversion, we will find it most convenient to work in terms of species concentrations. Division of one rate expression by another will permit us to eliminate the time variable, thus obtaining expressions that are convenient for examining the effect of changes in process variables on the product distribution. 

Assume for the moment that there is a single irreversible reaction. Let be a progress variable describing the extent of reaction CKE 1. As Ej increases in infinitesimal increments d , the solution s analytical concentrations are perturbed and the activities of the species in solution change. Thus for each solute species sj, aj is a function of . 

For a single reaction this was called the fractional conversion X (or Xa), a number between zero and unity, because in a single reaction there is always a single variable that describes the progress of the reaction (we used Ca or X). For multiple reactants and multiple reactions there is not always a single species common to aU reactions to designate as A. However, there is fiequently a most valuable reactant on which to base conversion. We emphasize that by conversion Xj we mean the fractional conversion of reactant species j in all reactions. 

When the density varies, we need to find another variable to express the progress of a reaction. Earlier we defined the fractional conversion X for a single reaction, and in this chapter we defined the conversion of a reactant species for reactant A and Xj for reaction j. For the conversion in a reaction we need a different variable, and we shall use Xj (bold type), with the index i describing the reaction. We will first work our series and parallel reactions with these variables and then consider a variable-density problem. 

We divide the chapter into two parts Part 1 Mote Balances in Terms of Conversion, and Part 2 Mole Balances in Terms of Concentration, C,. and Molar Flow Rates, F,." In Pan 1, we will concentrate on batch reactors, CSTRs, and PFRs where conversion is the preferred measure of a reaction s progress for single reactions. In Part 2. we will analyze membrane reactors, the startup of a CSTR. and semibatch reactors, which are most easily analyzed using concentration and molar How rates as the variables rather than conversion. We will again use mole balances in terms of these variables (Q. f,) for multiple reactors in Chapter 6. 

The fraction conversion/is an intensive measure of the progress of a reaction. It is a variable that is simply related to the extent of reaction. The fraction conversion of a reactant Aj in a closed system in which only a single reaction is occurring is given by... 

For a discussion of progress variables and chemical equilibria see, e.g., F. T. Wall, Chemical Thermodynamics, 3rd ed. (San Francisco Freeman, 1974), Chapter 10. The pressure dependence of k in single-step reactions is usually small this problem is treated by C. A. Eckart, Ann. Rev. Phys. Chem. 23,239 (1972). Gas-phase unimolecular reaction, for which A is a sensitive function of p, is not a single-step process (Section 5.4). 

The second term is the chemical contribution and the last one accounts for coupling with the surroundings is the progress variable for chemical reaction, Q is the enthalpy release per mole of reaction, and y accounts for the effectiveness of thermal coupling between system and surroundings. Solution of (7.1) requires knowing how the rate of reaction, d jdty depends upon r. Assume a simple situation and consider a single, irreversible... 

Describing the Progress of a Reaction There are three variables that are commonly used to describe the composition of a reacting system, when a single reaction takes place. These three variables are the concentration of a species (usually the limiting reactant), the fractional conversion of a species (usually the limiting reactant), and the extent of reaction. The concentration of salicylic acid and the extent of reaction were used in Example 4-1. The application of each of these variables to Reaction (4-A) is discussed below. 

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