Most reacting systems of interest involve multiple reactions, and product selectivity – that is, the distribution of the various products – is a primary factor in the design. In this section we will consider a very realistic case. The following sequence of reactions, all of which proceed nearly irreversibly with mass-action kinetics, accounts for the production of more than 3× 109kg/year (3 million metric tons) in the United States alone, and more than 9× 109kg/year in the world:
A+ B → R, R + B → S, S + B → T. (7.16)
Some typical reactants and products within this class are listed inTable 7.4.
We will focus here on the reaction between water and ethylene oxide (EtO, also known by the official IUPAC name oxirane) to form mono-, di-, and triethylene glycol, as shown inFigure 7.2.The glycols are colorless, odorless liquids at room tem-perature. Sixty percent of the ethylene oxide used in the United States is consumed as a reactant in this reaction scheme. The monoglycol, usually called ethylene glycol (and sometimes MEG), is the primary product, but the di- and triglycols also have industrial uses. About two-thirds of the ethylene glycol manufactured worldwide is used as a chemical intermediate in the manufacture of polyester resins for fibers,
H H
Figure 7.2. Reaction scheme for the reaction of ethylene oxide and water to form ethylene glycols.
films, and bottles, while about one-fourth is used as antifreeze in engine coolants.
The distribution of products by weight is given by various sources as being in the neighborhood of 88/10/2 mono/di/tri and 90/9/1; as we shall see, the product distri-bution is determined by the reactor design. In one year for which specific figures are available, the recorded market demand in the United States was 90/8/2.
We assume that the reactants A (water) and B (ethylene oxide) are mixed prior to the reactor and enter in a single feedstream, and we assume that no product is present in the feedstream. With the usual assumptions regarding the density we then obtain the steady-state equations for a CFSTR:
A (water): 0= q(cAf − cA)− rA−V, (7.17a) B (EtO): 0= q(cBf− cB)− rB−V, (7.17b) R (MEG): 0= −qcR+ rR+V− rR−V, (7.17c) S (di-glycol): 0= −qcS+ rS+V− rS−V, (7.17d) T (tri-glycol): 0= −qcT+ rT+V. (7.17e) With mass action kinetics, which have been validated for this reaction system, the rates are as follows:
It is convenient to express the product concentrations in terms of xA= cA/cAf, which is the fraction of unreacted A; that is, one minus the fractional conversion of water.
For example, we can solve Equation7.18c for cBand substitute directly into Equation 7.18a to obtain, after some algebraic manipulation,
cR
Similarly, solving for cBin Equations7.18d and7.18e, respectively, and substituting into Equation7.17a,
By similar manipulations we obtain the expression for the conversion of B:
cBf− cB Note that these relations depend only on the conversion of A and the relative rate constants but are independent of the reactor volume and throughput. They are not stoichiometric relations, because the rates are explicitly included, but they lead to the following powerful conclusion: Any reactor conditions in a CFSTR that produce a given conversion of water will have the same product distribution.
The rate constant k1 for this system has a value of 6.37 × 10−7 L/(g-mol min) at 25◦C, and, to within experimental uncertainty, the ratios of the rate constants are k2/k1= k3/k1= 2.0. (We need only the latter result to evaluate the product distribution in terms of xA.) The distributions of the products by mass fraction are given inTable 7.5in terms of xA, where the mass fraction of species i is calculated as follows: can-not match the 90/8/2 product distribution exactly, but we can come close, for example, by operating with xAbetween 0.965 and 0.970. Note that this corresponds to conver-sion of only 3.0–3.5 percent of the water. It is clear that we will be operating with a very large excess of water, which must then be separated from the product.
Table 7.5. Product distribution for glycols and conversion of water and EtO in a continuous-flow stirred-tank reactor forvarious conversions of ethylene oxide. (Mass fractions may not sum to unity because of rounding.)
Water in effluent Product mass fractions EtO in effluent xA= cA
cAf
Conversion of water
1− xA Monoglycol Diglycol Triglycol
cBf− cB
cAf
0.990 0.010 0.966 0.033 0.001 0.010
0.985 0.015 0.950 0.048 0.002 0.016
0.980 0.020 0.934 0.063 0.004 0.021
0.975 0.025 0.918 0.077 0.006 0.026
0.970 0.030 0.902 0.090 0.008 0.032
0.965 0.035 0.887 0.102 0.011 0.038
0.960 0.040 0.872 0.115 0.014 0.043
0.955 0.045 0.857 0.126 0.017 0.049
0.950 0.050 0.842 0.137 0.020 0.055
0.945 0.055 0.828 0.148 0.024 0.061
0.900 0.100 0.710 0.221 0.069 0.122
0.850 0.150 0.599 0.267 0.133 0.199
0.800 0.200 0.507 0.289 0.204 0.289
0.750 0.250 0.429 0.294 0.277 0.390
We are now part way through the design process, and we have a considerable amount of information in hand. We presume from this point forward that the desired product distribution has been specified (as before, probably based on a market analysis). The first thing we note is that the amount of water that must be removed per mole of mixed product produced, cA/(cAf − cA)= xA/(1 − xA), is a constant for a given product distribution. The total production of glycols is given (with a bit of algebra to convert from total molar throughput to mass throughput with a fixed product distribution) by q(cBf− cB). We see from Equation 7.19d that the product distribution, which is fixed by xA, determines only the ratio (cBf− cB)/cAf, and we are free to choose the actual conversion of B (EtO), cB/cBf. Consideration of the results inTable 7.5give us some immediate insight into the extent to which we should try to convert B and the impact on the resulting process design. From Equation7.18a we can write effective first-order rate constant. If we take k1cBfθ to be of order unity, then cB/cBf
will be of orderε. We can thus reach some tentative conclusions about the design:
We probably want to choose the residence time to be of order 1/k1cBf. This will lead to a reactor that is designed for nearly complete conversion of EtO (cB/cBf ∼ ε), in which case it will not be necessary to have a separate unit to remove unreacted ethylene oxide. (We would most likely set θ to about 3/k1cBf or slightly more in order to achieve a conversion of ethylene oxide of 99 percent or better.)
With cBf cB, it also follows fromTable 7.5that the molar feed ratio of water to ethylene oxide will be about 30 (cBf/cAf ∼ 0.035), which is a very large excess relative to the stoichiometric ratio, and a very large amount of water will have to be separated. In this case, the unreacted EtO will be taken off in trace amounts with the unreacted water; the specific numerical coefficient in the selection of the residence time will be based on the maximum concentration of unreacted B that is permitted in the unreacted water stream. The flowrate q will be determined from the required production rate. (The feed can be assumed to be a mixture of pure ethylene oxide and water, so the feed concentration is known.) The reactor volume is then determined from the residence time.
The rigorous solution of the design problem requires that we carry out an analysis similar to the optimal design in the preceding section, including the costs of all separations (products, water, and possibly ethylene oxide), and this can of course be done, but the only relevant question once the product distribution and production rate are set, other than whether the process can show a positive return, is whether we should operate with less than nearly complete conversion of B (ethylene oxide).
Complete analyses have been done industrially, and the conclusion is always that the cost of operating with partial conversion and separation of B far outweighs the cost of a larger reactor to achieve nearly complete conversion.
7.5 Concluding Remarks
This is an exceedingly important chapter, in some ways perhaps the single most important chapter in the text in terms of engineering practice. Using the principles developed in the preceding chapters we have arrived at the logical culmination of the analysis process: a practical engineering design. The reactor is clearly the key to the process, since what happens in the reactor affects every other downstream element of the process. The primary point to take away from this chapter is that the equations involving the design variables, such as reactor volume and flow rate, must be combined with the process economics and other constraints in order to obtain a meaningful solution. Furthermore, considerable insight can be obtained with a relatively straightforward analysis based on a few reasonable assumptions.
The design examples in this chapter are quite realistic in form, and the glycol example is, in fact, taken from an actual industrial design study. A “real” design problem will, of course, be more complex. We have assumed here that the reac-tor will be a single CFSTR; other reacreac-tor configurations are possible and must be considered. Reactor temperature and associated energy costs are important vari-ables. We have completely ignored the details of separation. There will often be a larger set of chemical reactions and products, with different selectivity issues. The rate expressions will frequently be more complex than the elementary mass action kinetics applicable to the glycol reactions. The linear cost functions are a gross sim-plification. Yet nothing changes in principle; frequently, only the algebra becomes more difficult, although in many cases considerable computational effort is required.
This introduction, if well understood, can provide an intellectual framework for all
future studies of the components of a process and the implementation of a process design.
Bibliographical Notes
The continuation and expansion of the subject matter introduced in this chapter may receive some coverage in the core chemical engineering course in kinetics and reaction engineering, and it is touched upon in some textbooks for that course, but in a traditional curriculum the topic is more likely to be covered in a capstone course in process design.
PROBLEMS
7.1. Consider the system studied in Example 7.1. Suppose that the feedstream is specified to be q= 6 × 10−3m3/min. Find the production rate qcMas a function of the reactor volume.
7.2. An irreversible first-order decomposition reaction A→ M is carried out in a CFSTR, with k= 0.005 min−1. The feed composition cAf is 0.2 g-mol/L, and the desired production rate of product is 50 g-mol/min.
a. What is the minimum possible flow rate?
b. Consider flow rates up to four times the minimum, and calculate the reactor volumes and the effluent concentrations.
7.3. Kinetic data for the reaction between sulfuric acid and diethyl sulfate are given in Examples 6.1 and 6.4. Suppose that V= 25.4 L, qAf = qBf, cAf= 11.0 g-mol/L, cBf= 5.5 g-mol/L, and the required effluent concentration of sulfuric acid is cA= 4.0 g-mol/L. Find the flow rates (a) assuming that that the reaction may be taken to be irreversible and (b) taking the reverse reaction into account.
7.4. Consider the optimal design problem in Section 7.3, but now suppose that the capital cost of the reactor increases with volume as Vα,α < 1. Derive the algebraic equation for the optimal conversion, but do not attempt to solve the equation. What is fundamentally different about the solution forα < 1?
7.5. The chemical reaction sequence A→ M → S takes place in a CFSTR. You may assume that the reactions are irreversible and first order, with rate constants k1for the reaction A→ M and k2for the reaction M→ S. M is the desired product. Find the residence time θ = V/q that maximizes the concentration of M in the reactor effluent, and find the maximum concentration.
7.6. The irreversible reaction A → products is believed to be nth order in the concentration of A. Devise a strategy for obtaining the order and rate constant by carrying out steady-state experiments in a CFSTR.
7.7. The decomposition of the carcinogen N-nitrosodiethylamine (NDEA) in water