Macrokinetics

the kinetics of macroscopic processes, which describes the course of chemical conversions in their relation to the physical processes of the transfer of matter (mass), heat, and electric charge.

The term “macrokinetics” first came into use in the early 1940’s, in particular, in the works of the Soviet physical chemist D. A. Frank-Kamenetskii. Macrokinetics encompasses all phenomena that occur as a result of the effect of mass transfer and heat transfer on the rate of chemical conversion, whereas chemical kinetics examines only the rate of the chemical reaction itself. Outside of the laboratory, however, a chemical conversion is often accompanied by mass-transfer and heat-transfer processes, that are dependent on the hydrodynamic conditions of the motion of a gas, a liquid, or particles of a solid. The rates of of these processes often limit the overall rate of the process.

Several theories having practical importance have been constructed on the basis of macrokinetics, including theories of heterogeneous catalysis with porous catalysts, as well as theories of chemical reactors, electrochemical processes on electrodes (in particular, current generation in a fuel cell), fermenters (used in the microbiological industry), and combustion, dissolution, and lixiviation. Mathematical modeling has been used since the late 1950’s in solving the problems of macrokinetics.

In the macrokinetic method, a complex chemical-engineering process is divided into its chemical and physical components. After these have been studied separately, their effect on each other is determined mathematically with electronic computers. This approach is necessary because in most cases it is impossible to reproduce in the laboratory all aspects of a process as it occurs under industrial conditions, accompanied by a transfer of matter and heat. Attempts to solve the problems of macrokinetics on the basis of similarity theory and physical modeling have proved unsuccessful because of the incompatibility of the similarity conditions of the chemical and the physical components of various chemical-engineering processes.

To solve the problems of macrokinetics, both the principles of chemical conversion, undistorted by the influence of transfer processes, and the laws of mass and heat transfer must be known. The principles governing chemical conversion are expressed in kinetic equations that reflect the dependence of the rate of a chemical reaction on the composition of the reaction mixture, the temperature, the pressure, the properties of the catalyst (for catalytic processes), and other factors.

The practical problems with which macrokinetics must deal are as varied as chemical-engineering processes. However, a significant number of them are examined by various specialized disciplines: diffusion kinetics, the study of the influence of mass transfer on the rate of heterogeneous chemical reactions under conditions in which heat transfer may be disregarded; the theory of heterogeneous exothermic and endothermic processes that take place under conditions in which heat and mass transfer must simultaneously be taken into account; and the theory of combustion, the study of the role of mass and heat transfer in homogeneous exothermic reactions. Other specialized areas of macrokinetics include the macrokinetics of the processes of dissolution; the macrokinetics of electrochemical processes; and chemical hydrodynamics, which studies the hydrodynamic properties of gas or liquid flows by measuring the rate of well-studied chemical processes.

Diffusion kinetics. Any heterogeneous chemical process that takes place at the interfacial area between phases (for example, heterogeneous catalytic reactions, adsorption, electrochemical reactions on the surface of an electrode, chemical dissolution) consists of several stages: the transfer of the reactants to the surface on which the reaction takes place, the chemical reaction proper, and the removal of the products from the reaction surface. The overall rate of the process is determined by the rates of the individual stages. When the transfer of the reactant is the slow stage, it is assumed that the process takes place in the diffusion region, and it is described by diffusion kinetics. Diffusion kinetics has great value for the study of many processes in chemical engineering, especially heterogeneous catalytic processes. Industrial catalysts are porous kernels with a developed inner active surface whose area equals dozens or hundreds of sq m per g. A catalytic process consists of a number of stages: transfer of the reactants from the flow core through the boundary layer to the outer surface of the kernel; diffusion of the reactants within the kernel through the pores; chemical conversion on the active surface of the catalyst; and transfer of the products in the reverse direction. Depending on the relationships between the rates of these stages, a distinction is made between the external diffusion region, the internal diffusion region, and the kinetic region.

In the external diffusion region the rate of reaction is determined by the transfer of matter to the outer surface of the catalyst (or solid, reacting with a gas). The rate of mass transfer to a unit inner surface is proportional to the difference between the concentration in the gas flow core c_g and at the outer surface of the catalyst c_s and can be expressed in the form β(c_g -c_s), where β is the mass transfer coefficient, which describes the averaged mass transfer through the boundary layer and is dependent on the hydrodynamics of the flow.

In the internal diffusion region the concentrations c_g and c_s are similar—that is, the transfer of matter to the outer surface does not lower the overall rate, and the concentration of reactants at the center of a catalyst kernel c_c approaches zero for nonreversible reactions or equilibrium for reversible reactions. The porous structure of the catalyst kernels is very complex and can only be described statistically. This complicates the determination of the effective diffusion coefficient D_eff. If the pores are so large that molecules of the diffusing substance collide with each other more often than with the walls of the pores, the effective diffusion coefficient is determined on the basis of the molecular D_m: D_eff = D_m€Π, where € is the porosity of the kernel and Π is a factor that takes into account pore structure. In narrow pores the molecules collide with the walls more often than with each other (Knudsen diffusion).

The reaction in the internal diffusion region takes place only on a certain portion of the inner surface. The main characteristic of the accessibility of the inner surface of the catalyst is the degree of its utilization η, which is equal to the ratio of the reaction rate in the kernel to the reaction rate calculated on the assumption that the concentration of the reactant is equal over the entire surface of the catalyst to its concentration on the outer surface of the kernel. For a first-order nonreversible reaction

where

in which V is the volume of the kernel, S_s is the outer surface of the kernel, and k is a constant for the rate of a first-order reaction, when the volume is equal to unity. The observed form of the kinetic equation in the internal diffusion region differs from the true form. The observed order of the reaction, based on the component whose diffusion defines the process, is the average between the real order and the first order but, based on all other components, is halved. The observed activation energy also becomes half the true value. The coefficients β and D_eff and the parameters € and II are determined experimentally.

If the transfer processes are sufficiently rapid in comparison with the rates of the chemical stages, and if the concentrations of the reactants in the flow core, at the outer surface, and in the center of the kernel are practically the same, then the mass transfer rates do not affect the overall rate of the reaction. This region is called the kinetic region.

Theory of heterogeneous exothermic processes. If a heterogeneous reaction has significant reaction energy, then the temperatures differ at the center of the kernel T_k, at the external surface of the kernel T_k, and in the gas-flow core T_g. For endothermic processes, T_g > T_s > T_k, and for exothermic processes, T_g < T_s < T_k.

Endothermic reactions always proceed under stable conditions. In exothermic reactions, several stable and unstable stationary conditions are possible. The transition from one set of thermal conditions to another takes place in a number of steps and is accompanied by the critical phenomena of combustion and attenuation. In particular, the combustion of a solid is associated with an abrupt transition of the reaction from the kinetic region to the external diffusion region. The temperature of the surface exceeds that of the gas in the flow core by the amount of adiabatic heating of the reaction mixture.

The reverse transition also takes place gradually and corresponds to the critical conditions of attenuation. In the transition region between the external diffusion region and the kinetic region, there are unstable stationary conditions that cannot be realized without forced stabilization by a special automatic control system. The presence of critical phenomena is determined by the adiabatic heating parameter of the reaction mixture for complete conversion (Δθ_ad = (Tθs - Tθg)E/RT²g, where E is the activation energy and R is the gas constant) and by the ratio of the reaction rate constant (calculated per unit external surface of the kernel kθs) to the mass-transfer coefficient β. If Δθ_ad > 4 and k_s/β > 0.135, then critical phenomena are possible. In a kernel of catalyst, critical phenomena may be observed if ψ_r = Δθ_ad (D_eff/a_eff) > 4.5 (where a_eff is the effective thermal conductivity) and ψ² < 0.08. Values of ψ_T > 4.5 are seldom encountered for industrial catalytic processes, but are achieved only for strongly exothermic reactions with high initial reactant concentrations.

Combustion. Combustion is a chemical reaction under intensive self-acceleration caused by the accumulation in the reaction mixture of heat or the active products of a chain reaction with branched chains. Analysis of the combustion process is also made on the basis of the data of chemical kinetics, thermal conduction, and reactant diffusion. The capacity for spatial propagation as a result of the transfer of heat or active particles is characteristic of combustion.

Macrokinetics of dissolution processes. Dissolution macrokinetics examines one of the most important processes of chemical engineering. Chemical dissolution is a complex heterogeneous process that consists of several stages: the transfer of the solvent to the surface on which the reaction takes place, the chemical reaction proper, and the removal of the products from the reaction surface. As in diffusion kinetics, the overall rate of dissolution is determined by the rates of the individual stages, and kinetic and diffusion regions are possible, depending on the relationships between the rates.