Kinetic rate constants mass balance equations

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. 

Physiological Models for chemical bioaccumulation in fish are based on the same mass balance equations as the kinetic models for bioaccumulation, but the rate constants and chemical fluxes that quantify the rates of uptake and elimination of the substance are derived from Kow and a set of physiological parameters. The most well known model in this category is the FGETS (Food and Gill Exchange of Toxic Substances) model Barber et al. (1988, 1991) developed. This is a FORTRAN simulation model that predicts dynamics of a fish s whole body concentration of non-ionic, nonmetabolized, organic chemicals absorbed from the water only, or from water and food jointly. 

Suppose sys(f) is the total energy (internal + kinetic + potential) of a system, and ihm and /hout are the mass flow rates of the system input and output streams. (If the system is closed, these quantities each equal zero.) Proceeding as in the development of the transient mass balance equation, we apply the general energy balance equation (11.3-1) to the system in a small time interval from t to t + 1st, during which time the properties of the input and output streams remain approximately constant. The terms of the equation are as follows (see Section 7,4) ... 

In these models, the mass balance equation (Eq. 2.2) is combined vHth a kinetic equation (Eq. 2.5), relating the rate of variation of the concentration of each component in the stationary phase to its concentrations in both phases and to the equilibrium concentration in the stationary phase [80-93]. Although in principle kinetic models are more exact than the equilibrium-dispersive model, the difference between the individual band profiles calculated using the equilibrium-dispersive model or the linear driving force model, for example, is negligible when the rate constants are not very small i.e., when the column efficiency exceeds a few him-dred theoretical plates), as shown in Chapter 14 (Section 14.2). 

The same approaches that were successful in linear chromatography—the use of either one of several possible liunped kinetic models or of the general rate model — have been applied to the study of nonlinear chromatography. The basic difference results from the replacement of a linear isotherm by a nonlinear one and from the coupling this isotiienn provides between the mass balance equations of the different components of the mixture. This complicates considerably the mathematical problem. Analytical solutions are possible only in a few simple cases. These cases are limited to the band profile of a pure component in frontal analysis and elution, in the case of the reaction-kinetic model (Section 14.2), and to the frontal analysis of a pure component or a binary mixture, if one can assume constant pattern. In all other cases, the use of numerical solutions is necessary. Furthermore, in most studies found in the literature, the diffusion coefficient and the rate constant or coefficient of mass transfer are assumed to be constant and independent of the concentration. Actually, these parameters are often concentration dependent and coupled, which makes the solution of the problem as well as the discussion of experimental results still more complicated. 

Separate mass balance equations are written in the form of Section 10.6.2 for each of the two compartments. Variables and A2 represent the amount of drug in compartment 1 and compartment 2, respectively, and the total amount of drug in the body is given by the sum of Ai and A2. The rate of drug absorption is a function of a first-order absorption rate constant kj, the bioavailability (F), and the administered dose (D). Distribution between the compartments follows first-order kinetics as described previously. Elimination occurs only from compartment 1 in the standard model form, with the elimination rate equal to the amount of drug remaining in compartment... 

The temperature-dependent physical constants in the mass balance (i.e., the kinetic rate constant and the equilibrium constant) are expressed in terms of nonequilibrium conversion x using the linear relation (3-42). The concept of local equilibrium allows one to rationalize the definition of temperature and calculate an equilibrium constant when the system is influenced strongly by kinetic changes. In this manner, the mass balance is written with nonequilibrium conversion of CO as the only dependent variable, and the problem can be solved by integrating only one ordinary differential equation for x as a function of reactor volume. 

The dimensionless scaling factor in the mass transfer equation for reactant A with diffusion and chemical reaction is written with subscript j for the jth chemical reaction in a multiple reaction sequence. Hence, A corresponds to the Damkohler number for reaction j. The only distinguishing factor between all of these Damkohler numbers for multiple reactions is that the nth-order kinetic rate constant in the 7th reaction (i.e., kj) changes from one reaction to another. The characteristic length, the molar density of key-limiting reactant A on the external surface of the catalyst, and the effective diffusion coefficient of reactant A are the same in all the Damkohler numbers that appear in the dimensionless mass balance for reactant A. In other words. 

Notice that the molar density of key-limiting reactant A on the external surface of the catalytic pellet is always used as the characteristic quantity to make the molar density of component i dimensionless in all the component mass balances. This chapter focuses on explicit numerical calculations for the effective diffusion coefficient of species i within the internal pores of a catalytic pellet. This information is required before one can evaluate the intrapellet Damkohler number and calculate a numerical value for the effectiveness factor. Hence, 50, effective is called the effective intrapellet diffusion coefficient for species i. When 50, effective appears in the denominator of Ajj, the dimensionless scaling factor is called the intrapellet Damkohler number for species i in reaction j. When the reactor design focuses on the entire packed catalytic tubular reactor in Chapter 22, it will be necessary to calcnlate interpellet axial dispersion coefficients and interpellet Damkohler nnmbers. When there is only one chemical reaction that is characterized by nth-order irreversible kinetics and subscript j is not required, the rate constant in the nnmerator of equation (21-2) is written as instead of kj, which signifies that k has nnits of (volume/mole)"" per time for pseudo-volumetric kinetics. Recall from equation (19-6) on page 493 that second-order kinetic rate constants for a volnmetric rate law based on molar densities in the gas phase adjacent to the internal catalytic surface can be written as... 

The heterogeneous rate law in (22-57) is dimensionalized with pseudo-volumetric nth-order kinetic rate constant k that has units of (volume/mol)" per time. k is typically obtained from equation (22-9) via surface science studies on porous catalysts that are not necessarily packed in a reactor with void space given by interpellet. Obviously, when axial dispersion (i.e., diffusion) is included in the mass balance, one must solve a second-order ODE instead of a first-order differential equation. Second-order chemical kinetics are responsible for the fact that the mass balance is nonlinear. To complicate matters further from the viewpoint of obtaining a numerical solution, one must solve a second-order ODE with split boundary conditions. By definition at the inlet to the plug-flow reactor, I a = 1 at = 0 via equation (22-58). The second boundary condition is d I A/df 0 as 1. This is known classically as the Danckwerts boundary condition in the exit stream (Danckwerts, 1953). For a closed-closed tubular reactor with no axial dispersion or radial variations in molar density upstream and downstream from the packed section of catalytic pellets, Bischoff (1961) has proved rigorously that the Danckwerts boundary condition at the reactor inlet is... 

Additional parameters needed for the mass-balance equations are the physico-chemical properties of all transformation products considered, the degradation rate constants, and the fractions of formation of all transformation reactions. The fractions of formation account for the generation of several transformation products in parallel and for yields of less than 100%. For example, if two products are formed in roughly equal amounts and about 80% of the precursor is known to be converted into these two products, their fractions of formation are 0.4. Fractions of formation can be derived from kinetic information about a transformation pathway (see Sect. 4.1). However, because this information is often missing, most fractions of formation have to be estimated. 

Kinetic approaches represent realistic and comprehensive description of the mechanism of network formation. Under this approach, reaction rates are proportional to the concentration of unreacted functional groups involved in a specific reaction times an associated proportionality constant (the kinetic rate constant). This method can be applied to the examination of different reactor types. It is based on population balances derived from a reaction scheme. An infinite set of mass balance equations will result, one for each polymer chain length present in the reaction system. This leads to ordinary differential or algebraic equations, depending on the reactor type under consideration. This set of equations must be solved to obtain the desired information on polymer distribution, and thus instantaneous and accumulated chain polymer properties can be calculated. In the introductory paragraphs of Section... 

Figure 5. Schematic representation and equations defining the 2-box (top) and 3-box (bottom) kinetic models. X = dissolved metal Y, Yi and Y2 = reversibly sorbed metal on Freundlich sorption sites f = fraction of Freundlich sorption sites reaching equilibrium instantaneously K and n are the Freundlich-isotherm constants r and k are the reversible and irreversible rate constant, respectively Z = irreversibly sorbed metal. The subscript 0 in the mass balance equations denotes concentrations at time zero, and Cp = particle concentration. (Adapted from ref. 15)...

Trickle Bed Reactors. If one reactant is in large excess, the mass balance equation for that component may be removed from the system, because its concentration will be practically constant throughout the bed and not kinetically controlling the mass transfer and chemical rate phenomena. Good catalyst wetting enables the contacting effectiveness to be taken as unity or close to 1,... 

The definition of reaction rate can be nsed to interpret kinetic data with respect to liquid phase or with respect to catalyst volume or mass (Levespiek 1999). In the case of packed-bed reactors, the definition of reaction rate based on mass or volume of catalyst is useful, and superficial velocity can be used instead of intrinsic velocity. Thus, in the balance equations using superficial velocity the intrinsic kinetic rate constant is better related to the apparent rate constant by anploying the wetting efficiency factor (/J as follows ... 

The kinetic equilibrium constant is estimated from the thermodynamic equilibrium constant using Equation (7.36). The reaction rate is calculated and compositions are marched ahead by one time step. The energy balance is then used to march enthalpy ahead by one step. The energy balance in Chapter 5 used a mass basis for heat capacities and enthalpies. A molar basis is more suitable for the current problem. The molar counterpart of Equation (5.18) is... 

Vectors, such as x, are denoted by bold lower case font. Matrices, such as N, are denoted by bold upper case fonts. The vector x contains the concentration of all the variable species it represents the state vector of the network. Time is denoted by t. All the parameters are compounded in vector p it consists of kinetic parameters and the concentrations of constant molecular species which are considered buffered by processes in the environment. The matrix N is the stoichiometric matrix, which contains the stoichiometric coefficients of all the molecular species for the reactions that are produced and consumed. The rate vector v contains all the rate equations of the processes in the network. The kinetic model is considered to be in steady state if all mass balances equal zero. A process is in thermodynamic equilibrium if its rate equals zero. Therefore if all rates in the network equal zero then the entire network is in thermodynamic equilibrium. Then the state is no longer dependent on kinetic parameters but solely on equilibrium constants. Equilibrium constants are thermodynamic quantities determined by the standard Gibbs free energies of the reactants in the network and do not depend on the kinetic parameters of the catalysts, enzymes, in the network [49]. 

When deriving a material balance equation, the rate of transformation of each component in a reactor is normally governed by the mass action law. However, unlike for the reactions in which only low molecular weight substances are involved, the number of such components in a polymer system and, consequently, the number of the corresponding kinetic equations describing their evolution are enormous. The same can be said about the number of the rate constants of the reactions between individual components. The calculation of such a system becomes feasible because certain general principle can be invoked under the description of the kinetics of the majority of macromolecular reactions. Let us discuss this principle in detail. 

The experimental technique controls how the mass transport and rate law are combined (and filtered, e.g. by removing convective transport terms in a diffusion-only CV experiment) to form the overall material balance equation. Migration effects may be eliminated by addition of supporting electrolyte steady-state measurements eliminate the need to solve the equation in a time-dependent manner excess substrate can reduce the kinetics from second to pseudo-first order in a mechanism such as EC. The material balance equations (one for each species), with a given set of boundary conditions and parameters (electrode/cell dimensions, flow rate, rate constants, etc.), define an I-E-t surface, which is traversed by the voltammetric technique. 

To calculate the amount of catalyst for a particular case, mass and heat balance have to be considered they can be described by two differential equations one gives the differential CO conversion for a differential mass of catalyst, and the other the associated differential temperature increase. As analytical integration is not possible, numerical methods have to be used for which today a number of computer programs are available with which the calculations can be performed on a powerful PC in the case of shift conversion. Thus the elaborate stepwise and graphical evaluation by hand [592], [609] is history. For the reaction rate r in these equations one of the kinetic expressions discussed above (for example, Eq. 83) together with the function of the temperature dependence of the rate constant has to be used. 

Most of the known IE kinetic problems have been solved by the use of a single mass-balanced diffusion equation [1-3,7-11,14-24,34-43]. They are, on this basis, identified as one component systems and the diffusion rate for the invading B ion is controlled by the concentration gradient of this ion alone. In these cases the effective interdiffusion coefficient depends on the ion concentrations and the equilibrium constants of the chemical reaction between both ions in the ion exchanger [2-3,7-12,16-22,23,23,30,32,34,42,32-34]. 

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