Multiple reactions mass balance

Note that for multiple-reaction systems we can simply substitute 52 v,jr/ for vjr in the mass-balance expressions for a single-reaction mass-balance equation. The difference with multiple reactions is that now we must solve R simultaneous mass-balance equations rather than the single equation we had with a single reaction. 

The UASB tractor was modeled by the dispensed plug flow model, considering decomposition reactions for VFA componaits, axial dispersion of liquid and hydrodynamics. The difierential mass balance equations based on the dispersed plug flow model are described for multiple VFA substrate components considaed... 

The results actually showed a deracemization of the racemic hydroxyester 10 as opposed to enantioselective hydrolysis with formation of optically pure (R)-hydroxyester 10 and only 20 % loss in mass balance. Small quantities of ethyl 3-oxobutanoate 9 (<5%) were also detected throughout the reaction, leading the authors to suggest a multiple oxidation-reduction system with one dehydrogenase enzyme (DH-2) catalysing the irreversible reduction to the (R)-hydroxy-ester (Scheme 5). 

We took the 4- sign on the square root term for second-order kinetics because the other root would give a negative concentration, which is physically unreasonable. This is true for any reaction with nth-order kinetics in an isothermal reactor There is only one real root of the isothermal CSTR mass-balance polynomial in the physically reasonable range of compositions. We will later find solutions of similar equations where multiple roots are found in physically possible compositions. These are true multiple steady states that have important consequences, especially for stirred reactors. However, for the nth-order reaction in an isothermal CSTR there is only one physically significant root (0 < Ca < Cao) to the CSTR equation for a given T. ... 

In this chapter we consider the performance of isothermal batch and continuous reactors with multiple reactions. Recall that for a single reaction the single differential equation describing the mass balance for batch or PETR was always separable and the algebraic equation for the CSTR was a simple polynomial. In contrast to single-reaction systems, the mathematics of solving for performance rapidly becomes so complex that analytical solutions are not possible. We will first consider simple multiple-reaction systems where analytical solutions are possible. Then we will discuss more complex systems where we can only obtain numerical solutions. 

In this chapter we consider how we should design chemical reactors when we want to produce a specific product while converting most of the reactant and rninitnizing the production of undesired byproducts. It is clear that in order to design any chemical process, we need to be able to formulate and solve the species mass-balance equations in multiple-reaction systems to determine how we can convert reactants into valuable products efficiently and economically. 

For a batch reactor with multiple reactions the mass-balance equation on species j is... 

Most multiple-reaction systems are more comphcated series-parallel sequences with multiple reactants, some species being both reactant and product in different reactions. These simple rules obviously will not work in those situations, and one must usually solve the mass-balance equations to determine the best reactor configuration. 

In polymerization we play the standard game we use for any multiple-reaction system by writing the mass balance for each species using our standard mass-balance equation for a multiple-reaction system. 

For the sake of generality, we now develop most general mass-balance equations for a two phase system in which each phase has multiple inputs and multiple outputs and in which each phase is undergoing reactions within its boundaries. 

Notice that for a distributed system, the multiple input problem changes into an artificial idle stage mass balance with no reaction or mass transfer at the input for each phase as shown in Figure 6.13. 

For the same type of catalyst we have observed in a recirculation laboratory reactor multiplicity, periodic and chaotic behavior. Unfortunately, so far we are not able to suggest such a reaction rate expression which would be capable of predicting all three regimes [8]. However, there is a number of complex kinetic expressions which can describe periodic activity. One can expect that such kinetic expressions combined with heat and mass balances of a tubular nonadiabatic reactor may give rise to oscillatory behavior. Detailed calculations of oscillatory behavior of singularly perturbed parabolic systems describing heat and mass transfer and exothermic reaction are apparently beyond, the capability of both standard current computers and mathematical software. 

Before we leave this topic, it would be wise to note the results of some recent research on heterogeneously catalysed gas reactions. Here finite rates of adsorption and desorption had to be introduced into the reaction scheme in order to explain the occurrence of multiple steady states and oscillatory phenomena. This observed exotic behaviour could be reproduced by solving a set of coupled equations for the rates of adsorption/desorption, the rate of the surface reaction, and the mass balance relations [22, 23], Adsorption steps (ii) and (iv) may therefore need to be invoked for any heterogeneously catalysed solution reactions that are found to exhibit similar dynamic behaviour. 

In a review article on oscillatory reactions (294), Sheintuch discusses the effect of introducing a heat balance for the catalyst rather than a mass balance for the reactor into the differential equation system for a surface reaction with oxidation/reduction cycles. Although the coverage equations alone can yield oscillatory behavior, as was the case for the models discussed in the previous section, Sheintuch s model is discussed in this section because introduction of the heat balance adds qualitatively new features. In this extended system complex, multiple peak behavior and quasiperiodicity was observed as shown in Fig. 8. Sheintuch also investigated the interaction of two oscillators. This work, however, will be treated in detail in Section V, were synchronization and chaos are discussed. 

In Example 5-3 the temperature and conversion leaving the reactor were obtained by simultaneous solution of the mass and energy balances. The results for each temperature in Table 5-7 represented such a solution and corresponded to a diiferent reactor, i.e., a different reactor volume. However, the numerical trial-and-error solution required for this multiple-reaction system hid important features of reactor behavior. Let us therefore reconsider the performance of a stirred-tank reactor for a simple single-reaction system. 

After entering all reaction data, a SMART assessment can be performed. The program then performs a series of mass-balance calculations and provides the waste quantification, hazard classification, and a qualitative level of concern. The algorithms cover single-step reactions that produce a single chemical product the software is not applicable to reactions with multiple products or for polymer reactions. In this case, the individual reactions of a synthetic sequence have to be calculated sequentially. 

Salnikov specifically reported multiple singular points and a limit cycle establishing the existence of oscillations in chemical reactions. Bilous and Amundson (1955) referred to Salnikov s (1948) paper as the first work where periodic phenomenon in reaction systems was discussed. They also indicated that a reaction A -> B in CSTR is irreversible, exothermic, and kinetically first order. Considering mass balance and heat balance equations it is known that at the steady states, the heat consumption... 

Another interesting phenomenon can emerge under non-isothermal conditions for strongly exothermic reactions there will be multiple solutions to the coupled system of energy and mass balances even for the simplest first-order reaction. Such steady-state multiplicity results in the existance of several possible solutions for the steady state overall effectiveness factor, usually up to three with the middle point usually unstable. One should, however, note that the phenomenon is, in practice, rather rarely encountered, as can be understood from a comparison of real parameter values (Table 9.2). 

Several of these simple mass balances with basic rate expressions were solved analytically. In the case of multiple reactions with nonlinear rate expressions (i.e., not first-order reaction rates), the balances must be solved numerically. A high-quality ordinary differential equation (ODE) solver is indispensable for solving these problems. For a complex equation of state and nonconstant-volume case, a differential-algebraic equation (DAE) solver may be convenient. 

For multiple reactions, the set of the stoichiometric independent reactions must be integrated simultaneously. Molar extent of reaction is more convenient as reaction variable, at best by reference to the total mass flow =, / /w. The mass balance equations become ... 

The solution to this problem requires an analysis of multiple gas-phase reactions in a differential plug-flow tubular reactor. Two different solution strategies are described here. In both cases, it is important to write mass balances in terms of molar flow rates and reactor volume. Molar densities and residence time are not appropriate for the convective mass-transfer-rate process because one cannot assume that the total volumetric flow rate is constant in the gas phase, particularly when the total number of moles is not conserved. In each reaction, 2 mol of reactants generates 1 mol of product. Furthermore, an overall mass balance suggests that the volumetric flow rate is constant only when the overall mass density does not change. This is a reasonable assumption for liquid-phase reactors but not for gas-phase problems when the total volume is not restricted. The exception is a constant-volume batch reactor. 

When multiple reactions occur in the gas phase, the mass balance for component i is written for an ideal tubular reactor at high mass transfer Peclet numbers in the following form, and each term has units of moles per volume per time ... 

The solution strategy described above is based on writing a differential plug-flow reactor mass balance for each component in the mixture, and five coupled ODEs are solved directly for the five molar flow rates. The solution strategy described below is based on the extent of reaction for independent chemical reactions, and three coupled ODEs are solved for the three extents of reaction. Molar flow rates are calculated from the extents of reaction. The starting point is the same as before. The mass balance is written for component i based on molar flow rate and differential reactor volume in the presence of multiple chemical reactions ... 

This analysis begins with the unsteady-state mass balance for component i in the well-mixed reactor. At high-mass-transfer Peclet numbers, which are primarily a function of volumetric flow rate q, the rate processes of interest are accumulation, convective mass transfer, and multiple chemical reactions. Generic subscripts are... 

The mass balance for a differential plug-flow reactor with multiple chemical reactions that operates at high-mass-transfer Peclet numbers allows one to replace... 

Answer The unsteady-state macroscopic mass balance for each component in a constant-volume batch reactor with multiple chemical reactions. 

The open-system mass balance for component i with nnits of mass per time is stated qnalitatively as 1 = 2 H- 3 where 1 is the accnmnlation rate process or the unsteady-state contribution, 2 is the net rate of inpnt dne to mass flux acting across the snrface that surrounds the control volume, and 3 is the rate of production of component i due to multiple chemical reactions. In mathematical terms ... 

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