Kinetic mass balance equations

The SimuSolv program (Program B) which was written to simulate the reaction finishing process with extra initiator addition is similar to Program A and uses the monomer and initiators mass balance equations with optimized values of the kinetic parameters. The semibatch step had been experimentally optimized for obtaining... 

Based on the kinetic mechanism and using the parameter values, one can analyze the continuous stirred tank reactor (CSTR) as well as the dispersed plug flow reactor (PFR) in which the reaction between ethylene and cyclopentadiene takes place. The steady state mass balance equations maybe expressed by using the usual notation as follows ... 

At the end of 24 hours of continuous process the system was shut down. The knowledge of flowed buffer volumes and of the optical densities inside and downstream each ultrafiltration stage allowed to estimate product distribution (see appendix for mass-balance equations and the calculation procedure). The content of each cell was recovered and ffeeze-dried in order to be stored and used for subsequent kinetic experiments. A schematic flow-sheet of the whole procedure is illustrated in figure 1. 

This situation is one involving both a total and a component mass balance, combined with a kinetic equation for the rate of decomposition of the waste component. Neglecting density effects, the total mass balance equation is... 

The component mass balance equation, combined with the reactor energy balance equation and the kinetic rate equation, provide the basic model for the ideal plug-flow tubular reactor. 

Although the Lewis cell was introduced over 50 years ago, and has several drawbacks, it is still used widely to study liquid-liquid interfacial kinetics, due to its simplicity and the adaptable nature of the experimental setup. For example, it was used recently to study the hydrolysis kinetics of -butyl acetate in the presence of a phase transfer catalyst [21]. Modeling of the system involved solving mass balance equations for coupled mass transfer and reactions for all of the species involved. Further recent applications of modified Lewis cells have focused on stripping-extraction kinetics [22-24], uncatalyzed hydrolysis [25,26], and partitioning kinetics [27]. 

Evolution of the component concentrations in the two phases of the bioreactor are modeled by using an iteractive program (Runge-Kutta). All the kinetic equations and the experimentally determined constants are introduced into the program. The mass balance equation is included. 

Let us now consider the situation where [/] [E], We have here a situation that is analogous to our discussion of pseudo-first-order kinetics in Appendix 1. When [/] E in equilibrium binding studies, the diminution of [/]f due to formation of El is so insignificant that we can ignore it and therefore make the simplifying assumption that [/]f = [/]T. Combining this with the mass balance Equations (A2.1) and (A2.2), and a little algebra, we obtain... 

Our treatment of chemical kinetics in Chapters 2-10 is such that no previous knowledge on the part of the student is assumed. Following the introduction of simple reactor models, mass-balance equations and interpretation of rate of reaction in Chapter 2, and measurement of rate in Chapter 3, we consider the development of rate laws for single-phase simple systems in Chapter 4, and for complex systems in Chapter 5. This is... 

When species i disappears by either radioactive decay or chemical reaction with first-order kinetics, the mass balance equation must be changed according to... 

Pollutants emitted by various sources entered an air parcel moving with the wind in the model proposed by Eschenroeder and Martinez. Finite-difference solutions to the species-mass-balance equations described the pollutant chemical kinetics and the upward spread through a series of vertical cells. The initial chemical mechanism consisted of 7 species participating in 13 reactions based on sm< -chamber observations. Atmospheric dispersion data from the literature were introduced to provide vertical-diffusion coefficients. Initial validity tests were conducted for a static air mass over central Los Angeles on October 23, 1968, and during an episode late in 1%8 while a special mobile laboratory was set up by Scott Research Laboratories. Curves were plotted to illustrate sensitivity to rate and emission values, and the feasibility of this prediction technique was demonstrated. Some problems of the future were ultimately identified by this work, and the method developed has been applied to several environmental impact studies (see, for example, Wayne et al. ). 

Thus, for known kinetics and a specified residence time distribution, we can predict the fractional conversion of reactant which the system of Fig. 9 would achieve. Recall, however, that this performance is also expected from any other system with the same E(t) no matter what detailed mixing process gave rise to that RTD. Equation (34) therefore applies to all reactor systems when first-order reactions take place therein. In the following example, we apply this equation to the design of the ideal CSTR and PFR reactors discussed in Chap. 2. The predicted conversion is, of course, identical to that which would be derived from conventional mass balance equations. 

The chemical engineer almost never has kinetics for the process she or he is working on. The problem of solving the batch or continuous reactor mass-balance equations with known kinetics is much simpler than the problems encountered in practice. We seldom know reaction rates in useful situations, and even if these data were available, they frequently would not be particularly useful. 

It is obvious from this example why we quickly lose interest in solving mass-balance equations when the kinetics become high order and reversible. However, for any single reaction the mass-balance equation is always separable and soluble as... 

Note that this problem is even easier than for a batch reactor because for the CSTR we just have to solve an algebraic equation rather than a differential equation For second-order kinetics, r = kC, the CSTR mass-balance equation becomes... 

We have now developed mass balance equations for the three simple reactors in which we can easily calculate conversion versus time tbatch> residence time T, or position L for specified kinetics. For a first-order irreversible reaction with constant density we have solved the mass balance equations to yield... 

For the CSTR the mass-balance equations for these kinetics for Cbo ( Co e... 

We have used the first-order irreversible reaction as an example, but this is easy to generalize for any reaction, irreversible or reversible, with any kinetics. In a PFTR the mass-balance equation for an arbitrary reaction becomes... 

We summarize the rates we need in a catalytic reactor in Table 74. We always need r to insert in the relevant mass-balance equation. We must be given i- or P as functions of Cj and T from kinetic data. 

Thus we see that environmental modeling involves solving transient mass-balance equations with appropriate flow patterns and kinetics to predict the concentrations of various species versus time for specific emission patterns. The reaction chemistry and flow patterns of these systems are sufficiently complex that we must use approximate methods and use several models to try to bound the possible range of observed responses. For example, the chemical reactions consist of many homogeneous and catalytic reactions, photoassisted reactions, and adsorption and desorption on surfaces of hquids and sohds. Is global warming real [Minnesotans hope so.] How much of smog and ozone depletion are manmade [There is considerable debate on this issue.]... 

The ideal model and the equilibrium-dispersive model are the two important subclasses of the equilibrium model. The ideal model completely ignores the contribution of kinetics and mobile phase processes to the band broadening. It assumes that thermodynamics is the only factor that influences the evolution of the peak shape. We obtain the mass balance equation of the ideal model if we write > =0 in Equation 10.8, i.e., we assume that the number of theoretical plates is infinity. The ideal model has the advantage of supplying the thermodynamical limit of minimum band broadening under overloaded conditions. 

Several mass balance equations are written for the kinetics of each step as the analyte is passing through the porous stationary phase. For the bulk mobile phase in the interstitial volume, the following differential mass balance equation is written... 

The lumped kinetic model can be obtained with further simplifications from the lumped pore model. We now ignore the presence of the intraparticle pores in which the mobile phase is stagnant. Thus, p = 0 and the external porosity becomes identical to the total bed porosity e. The mobile phase velocity in this model is the linear mobile phase velocity rather than the interstitial velocity u = L/Iq. There is now a single mass balance equation that is written in the same form as Equation 10.8. 

The solution of the simplest kinetic model for nonlinear chromatography the Thomas model [9] can be calculated analytically. The Thomas model entirely ignores the axial dispersion, i.e., 0 =0 in the mass balance equation (Equation 10.8). For the finite rate of adsorption/desorption, the following second-order Langmuir kinetics is assumed... 

The above equation can be used as a formal kinetic approach without assuming a mechanism. This equation should be incorporated into the mass balance equations for oxygen in which the enhanced oxygen transfer rate due to the dispersed phase should also be considered. 

Constant pattern condition This condition reduces the mass balance equation (4.128) to the simple relation C/C0=q/qm lx (see the section A look into the constant pattern condition). Practically, the constant pattern assumption holds if the equilibrium is favorable, and at high residence times (Perry and Green, 1999 Wevers, 1959 Michaels, 1952 Hashimoto et al., 1977). However, the constant pattern assumption is weak if the system exhibits very slow kinetics (Wevers, 1959). 

The direction of a reaction can be assessed straightforwardly by comparing the equilibrium constant (Keq) and the ratio of the product solubility to the substrate solubility (Zsat) [39]. In the case of the zwitterionic product amoxicillin, the ratio of the equilibrium constant and the saturated mass action ratio for the formation of the antibiotic was evaluated [40]. It was found that, at every pH, Zsat (the ratio of solubilities, called Rs in that paper) was about one order of magnitude greater in value than the experimental equilibrium constant (Zsat > Keq), and hence product precipitation was not expected and also not observed experimentally in a reaction with suspended substrates. The pH profile of all the compounds involved in the reaction (the activated acyl substrate, the free acid by-product, the antibiotic nucleus, and the product) could be predicted with reasonable accuracy, based only on charge and mass balance equations in combination with enzyme kinetic parameters [40]. 

Our aim is to determine the concentration of A in the reactor as a function of time and in terms of the experimental conditions (inflow concentrations, pumping rates, etc.). We need to obtain the equation which governs the rate at which the concentration of A is changing within the reactor. This mass-balance equation will have contributions from the reaction kinetics (the rate equation) and from the inflow and outflow terms. In the simplest case the reactor is fed by a stream of liquid with a volume flow rate of q dm3 s 1 in which the concentration of A is a0. If the volume of the reactor is V dm3, then the average time spent by a molecule in the reactor is V/q s. This is called the mean residence time, tres. The inverse of fres has units of s-1 we will call this the flow rate kf, and see that it plays the role of a pseudo-first-order rate constant. We denote the concentration of A in the reactor itself by a. 

Ozone combined with ultraviolet radiation (A, = 254 nm) has been shown to oxidize atrazine in water. The process can be used to oxidize different organic compounds such as volatile organochlorine substances (e.g., pesticides). Mass transfer and kinetic data have been applied to the mass balance equations of atrazine to obtain corresponding concentrations under varying... 

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