Extent of reaction balance

There are three different types of material balances that may be written when a chemical reaction is involved the molecular balance, the atomic balance, and the extent of reaction balance. It is a matter of convenience which of the three types is used. 

The extent of reaction balance gets its name from the fact that the amounts of the chemicals involved in the reaction are described in terms of how much of a particular reactant has been consumed or how much of a particular product has been generated. As an example, take the formation of ammonia from hydrogen and nitrogen ... 

Using the extent of reaction balances approach gives... 

One may apply extent of reaction balances to determine expressions for eaeh of the components leaving the reactor in terms of the amoimt of C2H4 converted to C2H4O and the amount converted to CO2. 

Choose a basis of 100 Ibmol/h C2H6 input. Perform extent of reaction balances on the two reactions. Only the C2H6, CH3CHO, and CO2 need be considered since information on the other components is not required in the solution ... 

If one starts with 10 moles of hydrogen and 10 moles of nitrogen, and let z represent the amount of nitrogen consumed by the end of the reaction, the output amounts of all three components are (10-z) for the nitrogen, (10-3z) for the hydrogen, and 2z for the ammonia. The following is a convenient way of representing the extent of reaction balance ... 

For a fixed extent of reaction, the presence of multifunctional monomers in an equimolar mixture of reactive groups increases the degree of polymerization. Conversely, for the same mixture a lesser extent of reaction is needed to reach a specified with multifunctional reactants than without them. Remember that this entire approach is developed for the case of stoichiometric balance. If the numbers of functional groups are unequal, this effect works in opposition to the multifunctional groups. 

The design equations for a CSTR do not require that the reacting mixture has constant physical properties or that operating conditions such as temperature and pressure be the same for the inlet and outlet environments. It is required, however, that these variables be known. Pressure in a CSTR is usually determined or controlled independently of the extent of reaction. Temperatures can also be set arbitrarily in small, laboratory equipment because of excellent heat transfer at the small scale. It is sometimes possible to predetermine the temperature in industrial-scale reactors for example, if the heat of reaction is small or if the contents are boiling. This chapter considers the case where both Pout and Tout are known. Density and Q ut wiU not be known if they depend on composition. A steady-state material balance gives... 

Nonisothermal stirred tanks are governed by an enthalpy balance that contains the heat of reaction as a significant term. If the heat of reaction is unimportant, so that a desired Tout can be imposed on the system regardless of the extent of reaction, then the reactor d5mamics can be analyzed by the methods of the previous section. 

The initial formulation of Epon 828 and NMA must be such that oxirane equivalents equal the concentration of anhydride groups. A balanced stoichiometric ratio enables the polymerization to develop macromolecules at high extents of reaction (10). Therefore... 

If one considers a batch reactor in which the chemistry is characterized by a single extent of reaction, the material balance analysis presented in Section 8.1 indicates that the holding time necessary to change the fraction conversion from fA1 to fA2 is given by... 

In general, when designing a batch reactor, it will be necessary to solve simultaneously one form of the material balance equation and one form of the energy balance equation (equations 10.2.1 and 10.2.5 or equations derived therefrom). Since the reaction rate depends both on temperature and extent of reaction, closed form solutions can be obtained only when the system is isothermal. One must normally employ numerical methods of solution when dealing with nonisothermal systems. 

As in the case of a batch reactor, the balance equation 2.3-3 or 2.3-4 may appear in various forms with other measures of flow and amounts. For a flow system, the fractional conversion of A (/A), extent of reaction (f), and molarity of A (cA) are defined in terms of Fa rather than nA ... 

The procedure developed by Joris and Kalitventzeff (1987) aims to classify the variables and measurements involved in any type of plant model. The system of equations that represents plant operation involves state variables (temperature, pressure, partial molar flowrates of components, extents of reactions), measurements, and link variables (those that relate certain measurements to state variables). This system is made up of material and energy balances, liquid-vapor equilibrium relationships, pressure equality equations, link equations, etc. 

Such a system was delicately balanced, and small changes in the arbitrary constants could greatly alter the shapes of the reaction curves. However, before claiming that this system represents the reactions which are occurring, a plausible explanation must be found for the termination reaction, which had a rate proportional to the extent of reaction. Two possible assumptions have been considered, but neither fully stands up to experimental test, and it is evident that the system is one of subtle balance and complexity. 

The general equations for chemical reaction in a turbulent medium are easy to write if not to solve (2). In addition to those for velocities (U = U + uJ and concentrations (Cj = Cj + Cj), balance equations for q = A u, the segregation ( , and the dissipations e and eg can be written (3). Whatever the shape of the reactor under consideration (usually a tube or a stirred tank), the solution of these equations poses difficult problems of closure, as u S, 5 cj, cj, and also c cj, c Cj in the reaction terms have to be evaluated. The situation is even more complicated when the temperature and the density of the reacting mixture are also fluctuating. Partial solutions to this problem have been proposed. In the case of instantaneous reactions (t << Tg) the "e-quilibrium assumption" applies the mixed reactants are immediately converted and the apparent rate of reaction is simply that of the decrease of segregation, with Corrsin s time constant xs. For instance, with a stoichiometric proportion of reactants, the extent of reaction X is given by 1 - /T ( 2), a simple result which can also be found by application of the IEM model (see (33)). 

With Eq. (19), the final form of the component mass balance in terms of extents of reactions is obtained ... 

In many systems, slow reactions occur to an appreciable extent in the presence of fast, reversible reactions. Given extents of reaction measured experimentally, or computed by integration of a kinetic model, it is desired to compute compositions of a system. This has been accomplished using EQUILK (19) given specifications for extents or temperature approaches for less than R reactions, or the moles of some species in the product. The RAND Method has been extended to permit these specifications, as well (38). In principle, these specifications are easily added to the mass balances for any nonlinear programming algorithm. 

When one of the above three resistances is predominant and the other two are negligible, the basic mass balance differential equation can be integrated and gives expressions for the relation between the extent of reaction and time, both being experimentally observable. Typical results... 

Mass balances for the process are recalculated using the extent-of-reaction values from above and assuming that all unreacted ethylbenzene (EB) is recycled and converted to products. On the basis of 1 lb-mol (104 lb) of styrene product, the calculations are ... 

Note that this equation is the material balance for a CSTR (see Table 3.5.1). Thus, when using any recycle reactor for the measurement of reaction rate data, the effect of stirring speed (that fixes recirculation rates) on extent of reaction must be investigated. If the outlet conditions do not vary with recirculation rates, then the recycle reactor can be evaluated as if it were a CSTR. 

In general, systems that involve chemical reactions may be analyzed using (a) molecular species balances (the approach always used for nonreactive systems), (b) atomic species balances, and (c) extents of reaction. Each approach leads to the same results, but any one of them may be more convenient for a given calculation so it is a good idea to become comfortable with all three. 

Finally, when you are using either molecular species balances or extents of reaction to analyze a reactive system, the degree-of-freedom analysis must account for the number of independent chemical reactions among the species entering and leaving the system. Chemical reactions are independent if the stoichiometric equation of any one of them cannot be obtained by adding and subtracting multiples of the stoichiometric equations of the others. 

Given that all three methods of carrying out material balances on reactive systems— molecular species balances, atomic species balances, and extents of reaction—necessarily yield the same results, the question is which one to use for a given process. There are no hard and fast rules but we suggest the following guidelines ... 

The feed to the reactor contains 7.80 mole% CHi, 19.4% O2, and 72.8% N2. The percentage conversion of methane is 90.0%, and the gas leaving the reactor contains 8 mol C02/mol CO. Carry out a degree-of-freedom analysis on the process. Then calculate the molar composition of the product stream using molecular species balances, atomic species balances, and extents of reaction. 

In the analysis that follows, we will count molecular species balances for all systems. (We could equally well use atomic species balances or the extent of reaction.) Note that the reaction occurs... 

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