Lumping kinetic parameters

The rate expression can be rearranged into the following form in terms of three lumped kinetic parameters, /[Pg.177]

These lumped kinetic parameters are defined by the following relationships ... 

These equations mean that a linear dependence of the reaction rate with light intensity (LVRPA) is observed when intensities are small, while square root dependence is observed when intensity is high. This latter dependence occurs for high intensities because the recombination of electron-hole pairs starts to limit the efficient use of the available photons (Alfano et al., 1997). The intensity at which the crossover between these two types of behavior occurs depends on the value of the lumped kinetic parameter 7, which in turn depends on the specific reaction under consideration. 

From a plausible reaction sequence in Table 1 and reliable approximations, a local expression for the reaction kinetics in terms of observable and independent variables has been obtained. a and are lumped kinetic parameters. 

In Equations (51)-(54), it must be noted that (i) e (x) = e [(x), Can] to recall that the LVRPA is a strong fimction of the catalyst loading and (ii) there are eight lumped kinetic parameters that depend on intrinsic properties of the system according to... 

Activation energies are derived from the dynamic mechanical experiments assuming that the kinetic mechanism does not change with temperature. The thus arrived at activation energy represents a lumped kinetic parameter for the reactions which have led to the point in time of the dynamics mechanical dispersion being measured. Activation energies for the first dispersion by DSA... 

The model was completed with the adsorption-oligomerization equilibria for formaldehyde, and the lumped kinetic parameters were estimated with non-linear regression. Generally, the model gave a very good agreement with experimental data obtained from aldolization of propion- and butyraldehyde. 

Eq. 6.2.6 was solved analytically to obtain the operation curve of the reactor (X vs t). Lumped kinetic parameters were determined by non-linear regression of experimental data using the numerical method of Newton-Raphson with first-order Taylor series expansion. Lumped parameters were smooth functions of temperature all parameters were adequately fitted to second order polynomials except for D that required a fourth order polynomial. The model can be used for reactor temperature optimization and can be extended to prolonged sequential batch operation provided that a sound model for enzyme inactivation is validated (Illanes et al. 2005b). 

Based on the complex-cage scheme, Eqs (23)-(32), combined wifli flie propagation, chain transfer, and termination reactions given by Eqs (5)—(18), the overall rate of polymeri2ation7 p can be written in terms of lumped kinetic parameters as... 

We will quantify the rate ratios of Figure 16.1. In Section 16.4 and the following, a molecular reaction energy diagram will be introduced that enables deduction of the lumped kinetic parameters of Scheme 16.1 from microkinetics simulations. 

Similar to Eq. (67), the first reaction (incorporating the enzyme phosphofructo-kinase) exhibits a Hill-type inhibition by its substrate ATP [126]. The overall ATP utilization v3 (ATP) is modeled by a saturable Michaelis Menten function. The system is specified by five kinetic parameters (with Gx lumped into Vm ), the Hill coefficient n, and the total concentration, 4 / = [ATP] + [ADP]. Note that the model is not intended to capture biological realism, rather it serves as a paradigmatic example to identify dynamic behavior in metabolic pathways. 

Kinetic parameters of fast pyrolysis were derived while assuming a single process for the decomposition of wood, including three parallel first-order decay reactions for the formation of the product classes. This is the so-called Shafizadeh scheme [56]. The three lumped product classes are permanent gas, liquids (biooil, tar), and char a classification that has become standard over the years. The produced vapors are subject to further degradation to gases, water and refractory tars. Charcoal, which is also being formed, catalyzes this reaction and therefore needs to be removed quickly [57]. 

The connection of the overall mass transfer coefficient of the lumped kinetic and the parameters of the general rate model is... 

In the washcoat, reaction rates are modeled via global reaction mechanisms. In such a global or macrokinetic reaction mechanism, several microkinetic adsorption, reaction and desorption steps are lumped together, reducing the overall number of kinetic parameters considerably. For some catalysts,... 

The hydraulic simulation tool AMESim (2006) has also been extended for exhaust aftertreatment simulation, by including routines developed together with IFP (2006). The software includes models for TWC, hydrocarbon (EIC) trap, NSRC, oxygen storage, DOC and DPF as well as pipes, etc. Catalysts are modeled via 0D approach, hence all transport effects are lumped into reaction kinetic parameters. These kinetic parameters can be adapted by the user. 

Another kinetic parameter in Eq. (2.11) is the maximum reaction rate rmax, which is proportional to the initial enzyme concentration. The main reason for combining two constants k3 and CEq, into one lumped parameter rmax is due to the difficulty of expressing the enzyme concentration in molar unit. To express the enzyme... 

Note that the results of our simulation via the pseudohomogeneous model tracks the actual plant very closely. However, since the effectiveness factors r]i were included in a lumped empirical fashion in the kinetic parameters, this model is not suitable for other reactors. A heterogeneous model, using intrinsic kinetics and a rigorous description of the diffusion and conduction, as well as the reactions in the catalyst pellet will be more reliable in general and can be used to extract intrinsic kinetic parameters from the industrial data. 

The chapter ends with a case study. Four different reduced kinetic models are derived from the detailed kinetic model of the phenol-formaldehyde reaction presented in the previous chapter, by lumping the components and the reactions. The best estimates of the relevant kinetic parameters (preexponential factors, activation energies, and heats of reaction) are computed by comparing those models with a wide set of simulated isothermal experimental data, obtained via the detailed model. Finally, the reduced models are validated and compared by using a different set of simulated nonisothermal data. 

First, the detailed model is used to simulate the behavior of the real system, and a set of simulated isothermal experimental data is generated including the total heat released by reaction. Then, these data are used to estimate the kinetic parameters of the reduced models and the heats of reaction of the lumped reactions. Finally, the reduced kinetic models are tested in a validation procedure which simulates the operation of a batch reactor and allows one to identify the best reduced model. 

Many working groups have modeled the performance of diesel particulate traps during the past few decades. Concentrated parameter models (CSTR assumption) have been applied for the evaluation of formal kinetic models and model parameters. The formal kinetic parameters lump the heat and mass transfer effects with the reaction kinetics of the complicated reaction network of diesel soot combustion. Those models and model parameters were used for the characterization of the performance of different filter geometries and filter materials, as well as of the performance of a variety of catalytically active coatings and fuel additives [58],... 

A simple model of lumped kinetics for supercritical water oxidation included in the partial differential equations for temperature and organic concentrations allows to qualitatively simulate the dynamic process behavior in a tubular reactor. Process parameters can be estimated from measured operational data. By using an integrated environment for data acquisition, simulation and parameter estimation it seems possible to perform an online update of the process parameters needed for prediction of process behavior. 

As shown in the preceding parts, kinetic parameters cannot be directly calculated when internal heat transfer limits pyrolysis. A model taking into account both kinetic scheme and heat- mass transfers becomes necessary, A one-dimension model has already been implemented and solved. It features a classical set of equations for heat and mass transfers in porous media, i.c. heat transfer through convection, conduction, radiation and mass transfer due to pressure gradient (Darcy s law) and binary diffusion. Different kinetic schemes from e literature arc and will be tested mass-loss as lumped first order reaction, formation of volatiles, tars and char from decomposition of cellulose, hcmicellulose and lignin [26], the Broido-Shafi2adeh model [30] and the one proposed by Di Blasi [31]. None of them can describe the composition of the volatiles and supplementary schemes have to be found. 

III the literature y f is u.sually denoted kj, a, kc,... We prefer the symbol Y instead of kj to remind the user that it is not a constant but lumps together diiTercnt kinetic parameters and operational variables. Thus, varies with a number of variables such as feedstock composition, catalyst surface type, and perhaps oil conversion (because it deiennines the gas-phase composition). Cqn I integrated, wi[h gives... 

Lumping and Mechanism Reduction It is often useful to reduce complex reaction networks to a smaller reaction set which still maintains the key features of the detailed reaction network but with a much smaller number of representative species, reactions, and kinetic parameters. Simple examples were already given above for reducing simple networks into global reactions through assumptions such as pseudo-steady state, rate-limiting step, and equilibrium reactions. 

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