Reactor Kinetics with Temperature Dependence

It is convenient to express the energy content in terms of the reactor temperature T by introducing a heat capacity c. The following relation is assumed  

0 = some convenient reference temperature The immediate objective is to obtain a solution of the reactor-kinetics equations [(9.84) and (9.85)] which also satisfy the energy-balance relation (9.125). As in the previous problems in time dependence, we look for a solution which is separable in time and space. This treatment is limited, therefore, to bare systems and to reflected systems which meet the requirements set forth in Sec. 9.Id. On this basis we can use the relations (9.88) for the flux and the precursor concentrations. In addition, we select 

The substitution of these relations, along with (9.88), into (9.84), (9.85), and (9.125), yields 

The first two equations are essentially the set (9.89), the only difference being that we have included here the time dependence of the change in multiplication This dependence has been introduced since in the 

In the analyses which follow, attention is focused on the transients which arise when a given system initially at steady state t 0) is perturbed in some prescribed fashion. At i = 0, a disturbance is introduced which affects the energy balance of the system, and the problem will be to determine, in particular, the time behavior of the flux, temperature, and power for all subsequent time t 0). The steady-state values of the flux and precursor concentrations were previously indicated in (9.80) [see also (9.101)]. The steady-state requirements on the power removal can be obtained directly from (9.125) thus, 

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Multiple steady states are not associated exclusively with temperature-dependent reactor operations. For some types of more complex reaction kinetics, steady-state multiplicity can exist under isothermal conditions. For example, Matsura and Rato [T. Matsura and M. Rato, Chem. Eng. Sci., 22, 171 (1967)]... 

The component mass balance, when coupled with the heat balance equation and temperature dependence of the kinetic rate coefficient, via the Arrhenius relation, provide the dynamic model for the system. Batch reactor simulation examples are provided by BATCHD, COMPREAC, BATCOM, CASTOR, HYDROL and RELUY. 

The first-order non-isothermal (FONI) reactor. A continuous, well-stirred magmatic reservoir similar to those discussed above is supposed to be thermally insulated. A dissolved element i precipitates with a temperature-dependent rate of crystallization. Crystallization rate is assumed to obey first-order kinetics with Boltzmann temperature dependence such as... 

In Fig. 1, various elements involved with the development of detailed chemical kinetic mechanisms are illustrated. Generally, the objective of this effort is to predict macroscopic phenomena, e.g., species concentration profiles and heat release in a chemical reactor, from the knowledge of fundamental chemical and physical parameters, together with a mathematical model of the process. Some of the fundamental chemical parameters of interest are the thermochemistry of species, i.e., standard state heats of formation (A//f(To)), and absolute entropies (S(Tq)), and temperature-dependent specific heats (Cp(7)), and the rate parameter constants A, n, and E, for the associated elementary reactions (see Eq. (1)). As noted above, evaluated compilations exist for the determination of these parameters. Fundamental physical parameters of interest may be the Lennard-Jones parameters (e/ic, c), dipole moments (fi), polarizabilities (a), and rotational relaxation numbers (z ,) that are necessary for the calculation of transport parameters such as the viscosity (fx) and the thermal conductivity (k) of the mixture and species diffusion coefficients (Dij). These data, together with their associated uncertainties, are then used in modeling the macroscopic behavior of the chemically reacting system. The model is then subjected to sensitivity analysis to identify its elements that are most important in influencing predictions. 

Because the catalytic reaction A R is highly exothermic with rate highly temperature-dependent, a long tubular flow reactor immersed in a trough of water, as shown in Fig. P18.39, is used to obtain essentially isothermal kinetic data. Pure A at 0°C and 1 atm flows through this tube at 10 cm / sec, and the stream composition is analyzed at various locations. 

The Model. The physical system considered is an entrained bed reactor. Pulverized coal, carried by a gas stream, mixes with hot gas at the reactor entrance. As coal particles are carried their temperatures increase and devolatilization takes place. For practical purposes the particle size distribution was approximated by 10 discrete cuts. Since the devolatilization kinetics are highly temperature dependent and the temperature transient of a particle is affected by its size and mass separate account must be taken of each of the 15 reactions in each of the 10 different size particles. Without any detail on the derivation of the model, the equations can be summarized as ... 

Silanisation is a heterogeneous reaction. Silanes can be in the gas or liquid phase or in solution. The reaction is carried out at elevated temperatures, depending on the volatility of the silane and solvent, in a vessel under gentle stirring or in a fluidized bed reactor. To enhance the kinetics, catalysts are added. With chlorosilanes, organic bases are added as acid scavengers acids are employed in case of alkoxysilanes as reagents. By-products must be carefully removed by extraction with solvents. 

The reactivity of the clusters can then be studied by various experimental techniques, including fast flow reactor kinetics in the postvaporization expansion region of a laser evaporation source [21, 22], ion flow tube reactor kinetics of ionic clusters [23, 24], ion cyclotron resonance [25, 26], guided-ion-beam [27], and ion-trap experiments [28-30]. Which of these techniques is applied depends on the charge state of reactants (neutral, cationic, anionic), on whether the clusters are size-selected before the reaction zone, on single or multiple collisions of the clusters with the reactants, on the pressure of a buffer gas if present, and on the temperature and collision energy of the reactant molecules. 

The process is described by an one-dimensional, pseudohomogeneous, non-steady state dispersion model for an adiabatic fixed bed reactor. The kinetics are modelled by a re-versibll reaction system where each reaction step follows a power law with a reaction order of one in the gas and in the solid component. The temperature dependency of the reaction rate constant follows the Arrhenius law. The equilibrium constant is set to be independent of temperature. 

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 mass transfer coefficient increases only slightly with temperature, so above a certain temperature the reaction becomes mass transfer controlled. Further increases in temperature give almost no change in conversion. The transition to mass transfer control occurs at a lower temperature for very reactive species, such as H2 and CO, than for hydrocarbons, but the kinetics of oxidation are often not known. The design temperature and flow rate are based on lab tests or experience with similar materials. The reactor is usually operated in the mass transfer control regime, where the conversion depends on the rate of mass transfer and the gas flow rate. 

In a kinetic study the activation energy is generally not known a priori, or only with insufficient accuracy. The use of the equivalent reactor volume concept therefore leads to a trial-and-error procedure a value of is guessed and with this value and the measured temperature profile Vp is calculated by graphical or numerical integration. Then, for the rate model chosen, the kinetic constant is derived. This procedure is carried out at several temperature levels and from the temperature dependence of the rate coefficient, expressed by Arrhenius formula, a value of is obtained. If this value is not in accordance with that used in the calculation of Vp the whole procedure has to be repeated with a better approximation for . 

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