TERMINACIÓN UNILATERAL DEL CONTRATO POR PARTE DEL SUSCRIPTOR O

The feed is charged all at once to a batch reactor, and the products are removed together, with the mass in the system being held constant during the reaction step. Such reactors usually operate at nearly constant volume. The reason for this is that most batch reactors are liquid-phase reactors, and liquid densities tend to be insensitive to composition. The ideal batch reactor considered so far is perfectly mixed, isothermal, and operates at constant density. We now relax the assumption of constant density but retain the other simplifying assumptions of perfect mixing and isothermal operation.

The component balance for a variable-volume but otherwise ideal batch reac- tor can be written using moles rather than concentrations:

dðVaÞ dt ¼

dNA

where NAis the number of moles of component A in the reactor. The initial con- dition associated with Equation (2.30) is that NA¼ ðNAÞ0at t ¼ 0. The case of a first-order reaction is especially simple:

dNA

dt ¼
Vka ¼
kNA so that the solution is

NA¼ ðNAÞ0e

kt _ð2:31Þ

This is a more general version of Equation (1.24). For a first-order reaction, the number of molecules of the reactive component decreases exponentially with time. This is true whether or not the density is constant. If the density happens to be constant, the concentration of the reactive component also decreases expo- nentially as in Equation (1.24).

Example 2.8: Most polymers have densities appreciably higher than their monomers. Consider a polymer having a density of 1040 kg/m3 that is formed from a monomer having a density of 900 kg/m3. Suppose isothermal batch experiments require 2 h to reduce the monomer content to 20% by weight. What is the pseudo-first-order rate constant and what monomer content is predicted after 4 h?

Right Solution: Use a reactor charge of 900 kg as a basis and apply Equation (2.31) to obtain YA¼ NA ðNAÞ0 ¼0:2ð900Þ=MA ð900Þ=MA ¼ 0:2 ¼ expð
2kÞ

This gives k ¼ 0.8047 h
1. The molecular weight of the monomer, MA, is not actually used in the calculation. Extrapolation of the first-order kinetics to a 4-h batch predicts that there will be 900 exp(–3.22) ¼ 36 kg or 4% by weight of monomer left unreacted. Note that the fraction unreacted, YA, must be defined as a ratio of moles rather than concentrations because the density varies during the reaction.

Wrong Solution: Assume that the concentration declines exponentially according to Equation (1.24). To calculate the concentration, we need the density. Assume it varies linearly with the weight fraction of monomer. Then  ¼ 1012 kg/m3 at the end of the reaction. To calculate the monomer concentrations, use a basis of 1 m3of reacting mass. This gives

a a0

¼0:2ð1012Þ=MA 900=MA

¼ 0:225 ¼ expð
2kÞ or k ¼0:746

This concentration ratio does not follow the simple exponential decay of first-order kinetics and should not be used in fitting the rate constant. If it were used erroneously, the predicted concentration would be 45.6/MA

(kgEmol)/m3at the end of the 4 h reaction. The predicted monomer content after 4 h is 4.4% rather than 4.0% as more properly calculated. The difference is small but could be significant for the design of the monomer recovery and recycling system.

For reactions of order other than first, things are not so simple. For a second- order reaction, dðVaÞ dt ¼ dNA dt ¼
Vka 2_¼
kN2A V ¼
kN2 A 0V0 ð2:32Þ

Clearly, we must determine V or as a function of composition. The integration will be easier if NAis treated as the composition variable rather than a since this avoids expansion of the derivative as a product: dðVaÞ ¼ Vda þ adV. The numerical methods in subsequent chapters treat such products as composite variables to avoid expansion into individual derivatives. Here in Chapter 2, the composite variable, NA¼ Va, has a natural interpretation as the number of moles in the batch system. To integrate Equation (2.32), V or must be deter- mined as a function of NA.Both liquid- and gas-phase reactors are considered in the next few examples.

Example 2.9: Repeat Example 2.8 assuming that the polymerization is second order in monomer concentration. This assumption is appropriate for a binary polycondensation with good initial stoichiometry, while the pseudo-first-order assumption of Example 2.8 is typical of an addition polymerization.

Solution: Equation (2.32) applies, and  must be found as a function of NA. A simple relationship is

 ¼ 1040
140NA=ðNAÞ0

The reader may confirm that this is identical to the linear relationship based on weight fractions used in Example 2.8. Now set Y ¼ NA=ðNAÞ0: Equation (2.32) becomes dY dt ¼
k 0_Y2 1040
140Y 900  

where k0¼ kðNAÞ0=V0¼ ka0: The initial condition is Y ¼ 1 at t ¼ 0. An analy- tical solution to this ODE is possible but messy. A numerical solution integrates the ODE for various values of k0 until one is found that gives Y ¼0.2 at t ¼ 2. The result is k0¼ 1.83.

Example 2.10: Suppose 2A
!k=2 B in the liquid phase and that the density changes from 0 to 1¼ 0þ
 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.

Solution: For a constant mass system,

V ¼ 0V0¼ constant

Assume, for lack of anything better, that the mass density varies linearly with the number of moles of A. Specifically, assume

 ¼ 1

 NA ðNAÞ0

 

Substitution in Equation (2.32) gives

dNA dt ¼
kN 2 A 1

NA=ðNAÞ0 0V0  

This messy result apparently requires knowledge of five parameters: k, V0, (NA)0, 1, and 0. However, conversion to dimensionless variables usually reduces the number of parameters. In this case, set Y ¼ NA=ðNAÞ0 (the fraction unreacted) and ¼ t=tbatch(fractional batch time). Then algebra gives dY d ¼
K_Y2_ 1

Y 0

This contains the dimensionless rate constant, K¼ a0ktbatch, plus the initial and final densities. The comparable equation for reaction at constant density is

dY0 d ¼
K

_Y02

where Y0 would be the fraction unreacted if no density change occurred. Combining these results gives

dY0 Y02¼
K _{d ¼} 0dY 1

YY2 or dY0 dY ¼ 0Y02 1

YY2

and even K drops out. There is a unique relationship between Y and Y0 that depends only on 1 and 0. The boundary condition associated with this ODE is Y ¼ 1 at Y0 ¼ 1. An analytical solution is possible, but numerical integration of the ODE is easier. Euler’s method works, but note

that the indepen-dent variable Y0starts at 1.0 and is decreased in small steps until the desired final value is reached. A few results for the case of1¼ 1000 and0¼ 900 are Y Y0 1.000 1.000 0.500 0.526 0.200 0.217 0.100 0.110 0.050 0.055 0.020 0.022 0.010 0.011

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% (Y0¼ 0.526).

Example 2.11: Suppose initially pure A dimerizes, 2A
!k=2 B, isothermally in the gas phase at a constant pressure of 1 atm. Find a solution to the batch design equation and compare the results with a hypothetical batch reactor in which the reaction is 2A ! B þ C so that there is no volume change upon reaction.

Solution: Equation (2.32) is the starting point, as in the previous example, but the ideal gas law is now used to determine V as a function of NA:

V ¼ ½NAþ NBRgT=P ¼ NAþ ðNAÞÞ0
NA 2   RgT=P ¼ Y þ1 2   ðNAÞ0RgT=P ¼ Y þ1 2   V0

where Y is the fraction unreacted. Substitution into Equation (2.32) gives

dNA dt ¼ ðNAÞ0 dY dt ¼
2kN2 A V0½Y þ 1 ¼
2a0kY2ðNAÞ0 ½1 þ Y Defining, K, and Y0as in Example 2.10 gives

dY0 Y02¼
K

_{d ¼}½Y þ 1dY 2Y2

An analytical solution is again possible but messy. A few results are Y Y0 1.000 1.000 0.500 0.542 0.200 0.263 0.100 0.150 0.050 0.083 0.020 0.036 0.010 0.019

The effect of the density change is larger than in the previous example, but is still not major. Note that most gaseous systems will have substantial amounts of inerts (e.g. nitrogen) that will mitigate volume changes at constant pressure.

The general conclusion is that density changes are of minor importance in liquid systems and of only moderate importance in gaseous systems at constant pressure. When they are important, the necessary calculations for a batch reactor are easier if compositions are expressed in terms of total moles rather than molar concentrations.

We have considered volume changes resulting from density changes in liquid and gaseous systems. These volume changes were thermodynamically determined using an equation of state for the fluid that specifies volume or density as a function of composition, pressure, temperature, and any other state variable that may be important. This is the usual case in chemical engineering problems. In Example 2.10, the density depended only on the composition. In Example 2.11, the density depended on composition and pressure, but the pressure was specified.

Volume changes also can be mechanically determined, as in the combustion cycle of a piston engine. If V ¼ V(t) is an explicit function of time, Equations like (2.32) are then variable-separable and are relatively easy to integrate, either alone or simultaneously with other component balances. Note, however, that reaction rates can become dependent on pressure under extreme conditions. See Problem 5.4. Also, the results will not really apply to car engines since mixing of air and fuel is relatively slow, flame propagation is important, and the spatial distribution of the reaction must be considered. The cylinder head is not perfectly mixed.

It is possible that the volume is determined by a combination of thermo- dynamics and mechanics. An example is reaction in an elastic balloon. See Problem 2.20.

The examples in this section have treated a single, second-order reaction, although the approach can be generalized to multiple reactions with arbitrary

kinetics. Equation (2.30) can be written for each component: dðVaÞ dt ¼ dNA dt ¼ VRAða, b, . . .Þ ¼ VRAðNA, NB,. . . , VÞ dðVbÞ dt ¼ dNB dt ¼ VRBða, b, . . .Þ ¼ VRBðNA, NB,. . . , VÞ ð2:33Þ

and so on for components C, D,. . . . An auxiliary equation is used to determine V. The auxiliary equation is normally an algebraic equation rather than an ODE. In chemical engineering problems, it will usually be an equation of state, such as the ideal gas law. In any case, the set of ODEs can be integrated numerically starting with known initial conditions, and V can be calculated and updated as necessary. Using Euler’s method, V is determined at each time step using the ‘‘old’’ values for NA, NB,. . . . This method of integrating sets of ODEs with various auxiliary equations is discussed more fully in Chapter 3.

2.6.2 Fed-Batch Reactors

Many industrial reactors operate in the fed-batch mode. It is also called the semi- batch mode. In this mode of operation, reactants are charged to the system at various times, and products are removed at various times. Occasionally, a heel of material from a previous batch is retained to start the new batch.

There are a variety of reasons for operating in a semibatch mode. Some typi- cal ones are as follows:

1. A starting material is subjected to several different reactions, one after the other. Each reaction is essentially independent, but it is convenient to use the same vessel.

2. Reaction starts as soon as the reactants come into contact during the charging process. The initial reaction environment differs depending on whether the reactants are charged sequentially or simultaneously.

3. One reactant is charged to the reactor in small increments to control the composition distribution of the product. Vinyl copolymerizations discussed in Chapter 13 are typical examples. Incremental addition may also be used to control the reaction exotherm.

4. A by-product comes out of solution or is intentionally removed to avoid an equilibrium limitation.

5. One reactant is sparingly soluble in the reaction phase and would be depleted were it not added continuously. Oxygen used in an aerobic fermentation is a typical example.

All but the first of these has chemical reaction occurring simultaneously with mixing or mass transfer. A general treatment requires the combination of transport equations with the chemical kinetics, and it becomes necessary to solve sets of partial differential equations rather than ordinary differential equa- tions. Although this approach is becoming common in continuous flow systems, it remains difficult in batch systems. The central difficulty is in developing good equations for the mixing and mass transfer steps.

The difficulty disappears when the mixing and mass transfer steps are fast compared with the reaction steps. The contents of the reactor remain perfectly mixed even while new ingredients are being added. Compositions and reaction rates will be spatially uniform, and a flow term is simply added to the mass balance. Instead of Equation (2.30), we write

dNA

dt ¼ ðQaÞinþ VRAðNA, NB,. . . , VÞ ð2:34Þ where the term ðQaÞinrepresents the molar flow rate of A into the reactor. A fed- batch reactor is an example of the unsteady, variable-volume CSTRs treated in Chapter 14, and solutions to Equation (2.34) are considered there. However, fed-batch reactors are amenable to the methods of this chapter if the charging and discharging steps are fast compared with reaction times. In this special case, the fed-batch reactor becomes a sequence of ideal batch reactors that are reinitialized after each charging or discharging step.

Many semibatch reactions involve more than one phase and are thus classi- fied as heterogeneous. Examples are aerobic fermentations, where oxygen is sup- plied continuously to a liquid substrate, and chemical vapor deposition reactors, where gaseous reactants are supplied continuously to a solid substrate. Typically, the overall reaction rate will be limited by the rate of interphase mass transfer. Such systems are treated using the methods of Chapters 10 and 11. Occasionally, the reaction will be kinetically limited so that the trans- ferred component saturates the reaction phase. The system can then be treated as a batch reaction, with the concentration of the transferred component being dictated by its solubility. The early stages of a batch fermentation will behave in this fashion, but will shift to a mass transfer limitation as the cell mass and thus the oxygen demand increase.