Reactor variable density

We will work out the solution to variable-density reactors for a simple example where the number of moles varies with conversion. 

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. 

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). 

Flow reactors usually operate at nearly constant pressure, and thus at variable density when there is a change of moles of gas or of temperature. An appai ent l e.sidence time is the ratio of reactor volume and the inlet volumetric flow rate. 

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. 

Solution It is easy to begin the solution. In piston flow, molecules that enter together leave together and have the same residence time in the reactor, t. When the kinetics are first order, the probabiUty that a molecule reacts depends only on its residence time. The probability that a particular molecule will leave the system without reacting is exp(— F). For the entire collection of molecules, the probability converts into a deterministic fraction. The fraction unreacted for a variable density flow system is... 

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. 

Solution In a real problem, the individual values for k, V, and Qi would be known. Their values are combined into the dimensionless group, kVIQi . This group determines the performance of a constant-density reactor and is one of the two parameters needed for the variable-density case. The other parameter is the density ratio, r = Pmommer/Ppoiymer- Setting kVIQtn = 1 gives T = 0.5 as the fraction unreacted for the constant-density case. The individual values for k, V, Qi , Pmommer, and Ppoiymer can be assigned as convenient, provided the composite values are retained. The following... 

Solution Example 4.5 was a reverse problem, where measured reactor performance was used to determine constants in the rate equation. We now treat the forward problem, where the kinetics are known and the reactor performance is desired. Obviously, the results of Run 1 should be closely duplicated. The solution uses the method of false transients for a variable-density system. The ideal gas law is used as the equation of state. The ODEs are... 

The material balance around the mixing point of a loop reactor is given by Equation (4.21) for the case of constant fluid density. How would you work a recycle problem with variable density Specifically, write the variable-density counterpart of Equation (4.21) and explain how you would use it. 

ILLUSTRATION 8.4 DETERMINATION OF REQUIRED PLUG FLOW REACTOR VOLUME UNDER ISOTHERMAL OPERATING CONDITIONS—VARIABLE DENSITY CASE... 

The classical CRE model for a perfectly macromixed reactor is the continuous stirred tank reactor (CSTR). Thus, to fix our ideas, let us consider a stirred tank with two inlet streams and one outlet stream. The CFD model for this system would compute the flow field inside of the stirred tank given the inlet flow velocities and concentrations, the geometry of the reactor (including baffles and impellers), and the angular velocity of the stirrer. For liquid-phase flow with uniform density, the CFD model for the flow field can be developed independently from the mixing model. For simplicity, we will consider this case. Nevertheless, the SGS models are easily extendable to flows with variable density. 

The consequences of using equation 4.3-1 depend on the context constant or variable density and type of reactor. 

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). 

We note the use of t as a scaling factor for reactor size or capacity. In Example 15-2, neither V nor qa is specified. For a given r, if either V or q0 is specified, then the other is known. If either V or q0 is changed, the other changes accordingly, for the specified t and performance (cA or /A). This applies also to a CSTR, and to either constant- or variable-density situations. The residence time t may similarly be used for constant-density, but not variable-density cases. 

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. 

While this equation is correct for the steady-state CSTR with variable density, it does not give a correct description of the transient CSTR for variable density because Na (meaning the number of moles of species A in the reactor) is not given by the preceding expression unless the density is constant. 

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 conclusion to this section, research in the RTD area is always active and the initial concepts of Danckwerts are gradually being completed and extended. The population balance approach provides a theoretical framework for this generalization. However, in spite of the efforts of several authors, simple procedures, easy to use by practitioners, would still be welcome in the field of unsteady state systems (variable volumes and flow rates), multiple inlet/outlet reactors, variable density mixtures, systems in which the mass-flowrate is not conserved, etc... On the other hand, the promising "generalized reaction time distribution" approach could be developed if suitable experimental methods were available for its determination. 

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