Volume as a function of conversion

We need (X) to We now need only to find volume as a function of conversion to obtain the obtain Cj - hj(x) species concentration as a fiinction of conversion. 

Batch Reactors with Variable Volume Although variable volume batch reactors are seldom encountered because they are usually solid steel containers, we will develop the concentrations as a function of conversion because (1) they have been u.sed to collect reaction data for gas-phase reactions, and (2) the development of the equations that express volume as a function of conversion will facilitate analyzing flow systems with variable volumetric flow rates. 

The ratio of volumes as a function of conversion is shown in Figure 16.2 (curves 2 and 3). The influence of parameter M is significant, especially for high conversions. 

The design equation for the CSTR reactor volume as a function of conversion... 

A typical example for a reaction with substantial contraction of volume is the synthesis of methanol from syngas. Formally, 1 mol CO and 2 mol of H2 react to form 1 mol of methanol. This means that at high degrees of conversion, the contraction in volume can be a factor of three. This has dramatic implications for the pressure as the gas volume drops, the total pressure also drops. As a surplus, the analytical evaluation of the reaction is also complicated owing to the change in volume as a function of the degree of conversion. 

Disposing the Flory-Huggins modified equation, including the free entropy of mixing per total volume, AS , as a function of conversion and the enthalpy term expressed with the interaction parameter [66-68,72] ... 

Figure 17.2. Relative volumes of maximum-mixed and segregated flow reactors with the same RTDs identified by n = 1 /< , as a function of conversion for second- and half-order reactions. For first-order reactions the ratio is unity throughout.

As an extension of Exercise 15, consider the reversible, elementary, gas phase reaction of A and B to form C occurring at 300 K in a variable volume (constant pressure) reactor with an initial volume of 1.0 L. For a reactant charge to the reactor of 1.0 mol of A, 2.0 mol of B, and no C, find the equilibrium conversion of A. Plot the composition in the reactor and the reactor volume as a function of time. 

The previous examples show that if we know the molar flow rate to the reactor and the reaction rate as a function of conversion, then we can calculate the reactor volume necessary to achieve a specified conversion. The reaction rate does not depend on conversion alone, however. It is also affected by the initial concentrations of the reactants, the temperature, and the pressure. Consequently, the experimental data obtained in the laboratory and presented in Table 2-1 as -ta for given values of X are useful only in the design of full-scale reactors that are to be operated at the same conditions as the laboratory experiments (temperature, pressure, initial reactant concentrations). This conditional relationship is generally true i.e., to use laboratory data directly for sizing reactors, the laboratory and full-scale operating conditions must be identical, Usually, such circumstances are seldom encountered and we must revert to the methods described in Chapter 3 to olrtain — ta as a function of X. 

We have shown that in order to calculate the time necessary to achieve a given conversion X in a batch system, or to calculate the reactor volume needed to achieve a conversion X in a flow system, we need to know the reaction rate as a function of conversion. In tins chapter we show how this functional dependence is obtained. First there is a brief discussion of chemical kinetics, emphasizing definitions, which illustrates how the reaction rate depends on the concentrations of the reacting species. This discussion is followed by instructions on how to convert the reaction rate law from the concentration dependence to a dependence on conversion. Once this dependence is achieved, we can design a number of isothermal reaction systems. 

Concentration as a function of conversion v/hen no volume change occurs with reaction... 

Individual concentrations can be determined by expressing the volume V for a batch system (or volumetric flow rate v for a flow system) as a function of conversion using the following equation of state ... 

One of the major objectives of this chapter is to learn how to express any given rate law as a function of conversion. The schematic diagram in Figure 3-3 helps to summarize our discussion on this point. The concentration of the key reactant, A (the basis of our calculations), is expressed as a function of conversion in both flow and batch systems, for various conditions of temperature, pressure, and volume. 

The only new twist in calculating reactor volumes or conversions for a recycle reactor is a mole balance at the stream intersections (points P and Q) to express properly the species concentrations as a function of conversion. 

Make a plot of the mole fraction of each component as a function of conversion of pentamethylbenzene. Make a plot of the mole fraction of each component as a function of plug-flow reactor volume. Discuss any optimization that could be done. 

It must agree with the value calculated from thermodynamic value and it does 2. For a constant volume (V=Vq) batch system, C, = N/ /Vg and Cb = N /Vq. Substituting Equations (E3-8.2) and (E3-8.3) into the rate law, we obtain the rate of disappearance of A as a function of conversion ... 

Express P, and Cj for all species as functions of conversion for a constant-pressure batch reactor operated isothermally. Express volume as a function of X. 

We now need to determine the volume as a function of either conversion or time. An overall mass balance on all species gives... 

After completing this chapter you will be able to size CSTRs and PFRs given the rate of reaction as a function of conversion and to calculate the Overall conversion and reactor volumes for reactors arranged in series. 

Here, k is the specific reaction rate and is a function only of temperature, and Cao is the entering concentration. We note in Equations (2-13) and (2-16) the reactor volume in a function of the reciprocal of For this first-order dependence. a plot of the reciprocal rate of reaction IZ-r ) as a function of conversion yields a curve similar to the one shown in Figure 2-1, where... 

To summarize these last examples, we have seen that in the design of reactors that are to be operated at conditions (e.g.. temperature and initial concentration) identical to those at which the reaction rate data were obtained, we can size determine the reactor volume) both CSTRs and PFRs alone or in various combinations. In principle, it may be possible to scale up a laboratory-bench or pilot-plant reaction system solely from knowledge of as a function of X or Q. However, for most reactor systems in industry, a, scale-up proce.s.s cannot be achieved in this manner because knowledge of solely as a function of X is seldom, if ever, available under identical conditions. In Chapter 3. we shall see how we can obtain = yfX) from information obtained either in the laboratory or from the literature. This relationship will be developed in a two-step process. In Step 1, we will find the rate law that gives the rate as a function of concentration and in Step 2, we will find the concentrations as a function of conversion. Combining Steps 1 and 2 in Chapter 3. we obtain -/-.v =JiX). We can then use the method.s developed in this chapter along with integral and numerical methods to size reactors. 

Overview. In Chapter 2, we showed that if we had the rate of reaction as a function of conversion, = /(X), we could calculate reactor volumes necessary to achieve a specified conversion for flow systems and the time to achieve a given conversion in a batch system. Unfortunately, one is seldom, if ever, given = yiX) directly from raw data. Not to fear, in this chapter we will show how to obtain the rate of reaction as a function of conversion. This relationship between reaction rate and conversion will be obtained in two steps. In Step 1, Part 1 of this chapter, we define the rate law, which relates the rate of reaction to the concentrations of the reacting species and to temperature. In Step 2, Part 2 of this chapter, we define concentrations for fiow and batch systems and develop a stoichiometric table so that one can write concentrations as a function of conversion. Combining Steps 1 and 2, we see that one can then write the rate as a function conversion and use the techniques in Chapter 2 to design reaction systems. 

If the rate of reaction is not given explicitly as a function of conversion, we must proceed to level where the rate law must be determined by either finding it in books or journals or by determining it experimentally in the laboratory. Techniques for obtaining and analyzing rate data to determine the reaction order and rate constant are presented in Chapter 5. After the rate law has been established, one has only to use stoichiometry (level ) together with the conditions of the system (e.g.. constant volume, temperature) to express concentration as a function of conversion. 

For liquid-phase reactions and for gas-phase reactions with no pressure drop (F = Pq), one can combine the information in levels and , to express the rate of reaction as a function of conversion and arrive at level . It is now possible to detemiine either the time or reactor volume necessary to achieve the desired conversion by substituting the relationship linking conversion and rate of reaction into the appropriate design equation (level ). 

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