Variable density reactions

The general case of variable density reactions is applicable mostly to gas-phase reactions and seldom to liquid-phase reactions. Because the gas law gives the precise relationship between P, V, T, and N, we start with that equation. Based... 

Table 4.5 Analytical solutions (design equations) for some simple variable density reactions in a PFR...

BR (Table 4.2). Solutions to a few variable-density reactions are presented in Table 4.5. Graphical integration is straightforward and gives the reactor volume directly, as shown in Figure 4.9. The reciprocal rate is plotted as a function of either (Equation 4.61) or [ 4] (Equation 4.62). Alternatively, any of several numerical methods can be used, and this is perhaps the most attractive. 

Conversion-concentration relationships Variable-density reactions Reactors Batch reactor... 

Adesina [14] considered the four main types of reactions for variable density conditions. It was shown that if the sums of the orders of the reactants and products are the same, then the OTP path is independent of the density parameter, implying that the ideal reactor size would be the same as no change in density. The optimal rate behavior with respect to T and the optimal temperature progression (T p ) have important roles in the design and operation of reactors performing reversible, exothermic reactions. Examples include the oxidation of SO2 to SO3 and the synthesis of NH3 and methanol CH3OH. 

Example 3.1 Find the fraction unreacted for a first-order reaction in a variable density, variable-cross-section PER. 

The fraction unreacted is /< > . Set z = L to obtain it at the reactor outlet. Suppose = 1 and that kai /Ui = 1 in some system of units. Then the variable-density case gives z = 0.3608 at = 0.5. The velocity at this point is 0.75m . The constant density case gives z = 0.5 at a = 0-5 and the velocity at the outlet is unchanged from The constant-density case fails to account for the reduction in u as the reaction proceeds and thus underestimates the residence time. 

Example 4.3 represents the simplest possible example of a variable-density CSTR. The reaction is isothermal, first-order, irreversible, and the density is a linear function of reactant concentration. This simplest system is about the most complicated one for which an analytical solution is possible. Realistic variable-density problems, whether in liquid or gas systems, require numerical solutions. These numerical solutions use the method of false transients and involve sets of first-order ODEs with various auxiliary functions. The solution methodology is similar to but simpler than that used for piston flow reactors in Chapter 3. Temperature is known and constant in the reactors described in this chapter. An ODE for temperature wiU be added in Chapter 5. Its addition does not change the basic methodology. 

We define the rate of reaction verbally for a species involved in a reacting system either as a reactant or as a product. The system may be single-phase or multiphase, may have fixed density or variable density as reaction proceeds, and may have uniform or varying properties (e.g., p, CA, T, P) with respect to position at any given time. The extensive rate of reaction with respect to a species A, RA, is the observed rate of formation of A ... 

The general characteristics of a batch reactor (BR) are introduced in Chapter 2, in connection with its use in measuring rate of reaction. The essential picture (Figure 2.1) in a BR is that of a well-stirred, closed system that may undergo heat transfer, and be of constant or variable density. The operation is inherently unsteady-state, but at any given instant, the system is uniform in all its properties. 

If the system is not of constant density, we must use the more general form of the equation for reaction time (12.3-2) to determine t for a specified conversion, together with a rate law, equation 12.3-3, and an equation of state, equation 2.2-9. Variable density implies that the volume of the reactor or reacting system is not constant. This may be visualized as a vessel equipped with a piston V changes with the position of the piston. Systems of variable density usually involve a gas phase. The density may vary if any one of T, P or n, (total number of moles) changes (so as to alter the position of the piston). 

For a CSTR, variable density means that the inlet and outlet streams differ in density. A significant difference in density occurs only for gas-phase reactions in which there is a change in at least one of total moles, T, or P, although the last is usually very small. 

P3.05.10. HALF TIME. REACTION PITH VARIABLE DENSITY... 

At the end of the chemical-reaction step, all particle properties (w n>, X(n), fl(n>) have been advanced in time to t + At. Particle-field estimates of desired outputs can now be constructed, and the MC simulation is ready to perform the next time step. For a constant-density flow, the particle-field estimates are not used in the FV code. Thus, for stationary flow, the particle properties can be advanced without returning to the FV code. For unsteady or variable-density flow, the FV code will be called first to advance the turbulence fields before calling the PDF code (see Fig. 7.3). 

In the reactors studied so far, we have shown the effects of variable holdups, variable densities, and higher-order kinetics on the total and component continuity equations. Energy equations were not needed because we assumed isothermal operations. Let us now consider a system in which temperature can change with time. An irreversible, exothermic reaction is carried out in a single perfectly mixed CSTR as shown in Fig. 3.3. 

In a variable-density reactor the residence time depends on the conversion (and on the selectivity in a multiple-reaction system). Also, in ary reactor involving gases, the density is also a function of reactor pressure and temperature, even if there is no change in number of moles in the reaction. Therefore, we frequently base reactor performance on the number of moles or mass of reactants processed per unit time, based on the molar or mass flow rates of the feed into the reactor. These feed variables can be kept constant as reactor parameters such as conversion, T, and P are varied. 

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. 

It is evident that for multiple reactions with variable density, we rapidly arrive at rather complex expressions that require considerable manipulation even to formulate the expressions, which can be used to calculate numerical values of the reactor volume required for a given conversion and selectivity to a desired product. 

In the case of A as limiting reactant and a variable-density system, the solution of the model is the same as for first-order irreversible homogeneous reactions of the form A - products (Levenspiel, 1972) ... 

The density change in this example increases the reaction rate since the volume goes down and the concentration of the remaining A is higher than it would be if there were no density change. The effect is not large and would be negligible for many applications. When the real, variable-density reactor has a conversion of 50%, the hypothetical, constant-density reactor would have a conversion of 47.4% (7 = 0.526). 

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