Kinetic equations space-independent

The performed calculations demonstrate that a type of the asymptotic solution of a complete set of the kinetic equations is independent of the initial particle concentrations, iVa(0) and 7Vb(0). Variation of parameters a and (3 does not also result in new asymptotical regimes but just modifies there boundaries (in t and k). In the calculations presented below the parameters 7Va(0) = 7Vb(0) =0.1 and a = ft = 0.1 were chosen. The basic parameters of the diffusion-controlled Lotka-Volterra model are space dimension d and the ratio of diffusion coefficients k. The basic results of the developed stochastic model were presented in [21, 25-27],... 

Independent control parameters of the Lotka model are p and j3, describing reproduction of particles A and decay of B s, as well as the relative diffusion parameter k = DA/(DA+DB), and the space dimension d. Before discussing the solution of a complete set of the kinetic equations (8.3.20) to (8.3.24), let us formulate several statements. 

The independent variable in the deactivation kinetic equation at least 3 different types of kinetics of deactivation can be envisaged a-t, a-Cc and Cc-t. The mean residence time (or, in reality, the space-time) of the gas (tg) in the riser reactor can be calculated know ing the expansion factor (Ea) if cracking. But since the value of the slip... 

The answer to this question is mainly driven by the computational cost of solving the kinetic equation due to the large number of independent variables. In the simplest example of a 3D velocity-distribution function n t, x, v) the number of independent variables is 1 + 3 + 3 = 1. However, for polydisperse multiphase flows the number of mesoscale variables can be much larger than three. In comparison, the moment-transport equations involve four independent variables (physical space and time). Furthermore, the form of the moment-transport equations is such that they can be easily integrated into standard computational-fluid-dynamics (CFD) codes. Direct solvers for the kinetic equation are much more difficult to construct and require specialized numerical methods if accurate results are to be obtained (Filbet Russo, 2003). For example, with a direct solver it is necessary to discretize all of phase space since a priori the location of nonzero values of n is unknown, which can be very costly when phase space is not bounded. 

Since in an isotropic liquid system every space fixed axis is equivalent there should be no dependence of < Kim (u(o)) Tim(u(/)))> on m. This is a general property of isotropic systems and follows from the rotational invariance of equilibrium systems regardless of the kinetic equation (there is no unique axis of quantization). This independence of m is more general than our derivation indicates. We return to this in Sections 7.4 and 7.7. 

Let us briefly consider some aspects of the kinetic treatment based on the two-term approximation. When the equation system (12) is adapted to time- and space-independent plasmas, the simplified equation system (Shkarofsky et al, 1966 Winkler et al, 1982)... 

We consider a complex chemical system and focus on a set of S species M , u = 1,..., 5, which can carry one or more identical molecular fragments that are unchanged during the process in the following we refer to these species as carriers. For simplicity, in this section we limit ourselves to the case of isothermal, well-stirred, homogeneous systems, for which the concentrations c = c (t), u = 1,..., 5, of the chemicals M , m = 1,..., 5, are space independent and depend only on time. Later on we consider the more complicated case of reaction-diffusion systems. The deterministic kinetic equations of the process can be expressed in the following form ... 

Solution of space-independent kinetic equations. In this section the kinetic equations for a bare (or equivalent bare) homogeneous reactor are studied in the form... 

The total catalysis situation is reached when Xe — oo and Xe/y oo. a is not longer independent of the space coordinate. Since Xe is large, pure kinetic conditions are achieved and the q profile is squeezed into a thin reaction layer within which a is a constant equal to its value at the electrode surface, ay=q. Equation (6.77) may thus be simplified into... 

Example 1. Let us consider an example that exemplifies step 3 in the core-box modeling framework. The system to be studied consists of one substance, A, with concentration x = [A]. There are two types of interaction that affect the concentration negatively degradation and diffusion. Both processes are assumed to be irreversible and to follow simple mass action kinetics with rate constants p and P2, respectively. Further, there is a synthesis of A, which increases its concentration. This synthesis is assumed to be independent of x, and its rate is described by the constant parameter p3. Finally, it is possible to measure x, and the measurement noise is denoted d. The system is thus given in state space form by the following equations ... 

An outline of Butler s theory for the terms of the low surface state (transport-controlled) case is given in Section 10.3.5. Uosaki s 1977 theory of kinetics in the high surface state case was developed in greater detail by Khan (1984). Here, the beginning equation for the steady state (dnx/dt = 0) in the space charge region has a flux independent of distance, so that from the Nemst-Planck equation (4.226) and with dJIdx = 0, one obtains ... 

This equation states that the change in the free energy of the critical germ with the chemical potential per molecule of species / in the original phase (i.e., the mother liquor) equals the negative of the excess number An of molecules of type i in the nucleus over that present in the same volume of original space. The nucleation theorem is independent of the model and of the transition it holds true for classical nucleation theory, density functional theory, or cluster kinetic analysis and for gas-to-liquid or liquid-to-solid conversions. 

Therefore, the trial model function will in general be a nonlinear function of the independent variable, time. Various mathematical procedures are available for iterative x2 minimization of nonlinear functions. The widely used Marquardt procedure is robust and efficient. Not all the parameters in the model function need to be determined by iteration. Any kinetic model function such as Equation 3.9 consists of a mixture of linear parameters, the amplitudes of the absorbance changes, A and nonlinear parameters, the rate constants, kb For a given set of kb the linear parameters, A, can be determined without iteration (as in any linear regression) and they can, therefore, be eliminated from the parameter space in the nonlinear least-squares search. This increases reliability in determining the global minimum and reduces the required computing time considerably. 

By a routine mathematical technique, the equation can be split into a space dependent part and a time dependent part. These two parts of the Schroedinger equation are set equal to the same constant (the energy E) and solved separately. For our purposes, we are interested in the space dependent, time independent part, which describes a system that is not in a state of change. In Eq. (3.4), H is an operator called the Hamiltonian operator by analogy to the classical Hamiltonian function, which is the sum of potential and kinetic energies, and is equal to the total energy for a conservative system... 

In chemical kinetics the concept of the order of a reaction forms the basis of a kinematics which constitutes a frame for most of the molecular theories of chemical reactions. The fundamental magnitudes of this kinematics are the concentrations and the specific rate constants. In simple cases only the time enters as an independent variable, whereas in a diffusion process both time and space are involved. Diffusion processes are generally described in terms of diffusion coefficients, volume concentrations and thermodynamic potential or activity factors. Partial volume factors and friction coefficients associated with the components of the diffusing mixture are also essential in the description. A feature of the macro-dynamical theory is that it covers any region of concentration. Especially simple equations are connected with the differential diffusion process (diffusion with small concentration differences), for which the different coefficients or factors mentioned above are practically constant. 

Development of dynamic kinetics models is most effective when transient kinetic experiments combined with physico-chemical methods of investigation of catalyst surface. There are examples of models that describe well the dynamic processes on the catalyst, such as studies by Balzhinimaev et. al. [15], Sadhankar and Lynch [16], Jobson et. al, [17]. Once a model of dynamic processes on catalyst surface is devised, it can further be used for numerical optimization of the periodically forced reactor. Invariance of such model, where all equations and parameters are independent on time at every space scale such as pellet, catalyst bed or reactor at any time, simplifies the further scale up. 

The motion of the particles is governed by the potential energy U. If U is independent of the position of the center of mass, as is the case in field-free space or in the presence of homogeneous fields, the equations of motion (classical or quantum) factorize into separate equations for the center of mass and for the relative motion. The total linear momentump is conserved in a collision, so the center of mass moves uniformly (constant velocity V) and is unaffected by the collision. The kinetic energy T is conserved in elastic collisions, and L is conserved if the angular momenta of the particles themselves do not change in magnitude or direction. 

Bieniasz LK (1997) ELSIM - a problem-solving environment for electrochemical kinetic simulations. Version 3.0 - solution of governing equations associated with interfacial species, independent of spatial coordinates or in one-dimensional space geometry. Comput Chem 21 1-12... 

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