Computational Chemical Kinetics
Introduction
The most efficient approach to modeling a multicomponent chemical system is by numerical means. That is, when chemical species are calorically imperfect and reactions are rate-limited by temperature-dependent rate coefficients. A chemical system consists of a group of species in a given composition and state, which are governed by chemical kinetics and the laws of thermodynamics and material transport.
The subjects that will be introduced are the basic type of chemical kinetics, the essential building blocks of a chemical system and the making and selection of those blocks.
Classification of Chemistry
Components of a Chemical System 3
- Reaction Mechanisms
- Thermodynamic Properties
Regardless of the type of chemistry, frozen or chemically reactive, a set of species must be specified in order to build the system. The properties and composition of individual species then determine the overall behavior of the system. The second coefficient, a, which appears equally on both sides of the reaction, is the increased factor for the third body.
However, it is often true that most species have the same effect as a third body and result in the same rate coefficient. The rate coefficient k, kj or kr in each of the above reactions is the key to the finite rate process. The most commonly used rate coefficient model is given by the Arrhenius equation, which has the form
The probability that a species has a thermal energy greater than Ea is reflected in the Boltzmann factor. They are acquired experimentally at discrete temperatures and are listed as such in the databases.
Selection of Species, Reactions, Rates and Thermodynamic Properties 8
To select the relevant species, the reaction mechanisms, the physical conditions of the system and the desired accuracy of the chemical model must be considered. The other factor in a rate equation, species concentration, models the contribution of particle collision frequency to the rate of chemical production. In the case of feedback, their selection is based on the species pool, physical conditions, and desired model accuracy.
A complete reaction set includes all the known steps or mechanisms that relate to the components of the entire species set. Because of the non-linearity of the Arrhenius expression, two slightly different sets of kinetic data may produce significantly different sets of constants, although they can. U n- like the rate coefficient, it can be theoretically predicted according to the thermodynamic properties of the species involved.
Given the complete description, with initial conditions and boundary conditions specified, the evolution of the system can be solved. Finally, solutions of finite-rate problems are accomplished by numerical integration of the governing equations in time.
Model
It translates to the first inflection point of the size of a second stage variable. This is not necessarily the case in the last equation according to the single step approach. The main difference between the heat output and the composition of a species (for example, the mole fraction of OH used in section 4.5.3) is that the former represents the combined behavior of all species, while the latter only relates to individual behaviour.
The pressure profile is shown in Figure 5.3.2 as a function of the normalized time defined by ft (ie the number of cycles). Finally, the equivalence of the wall boundary condition in Equation 6.3.6 is written in the transformed plane as They are directly responsible for the behavior of the flow field as well as for the chemical processes in the nozzle.
In this formulation, the values of all the dependent variables at the boundaries are known. and 8Yk/8~ derived in chapter 6 need only be integrated along the internal streamlines 'lj;j for j is equal to 2 to N - 1. The Pj and OJ derivatives needed for the evaluation of the ~-derivatives are obtained with the same methods discussed in the simple wave solution - either by means of difference or the Pade's scheme. However, since the expansion rate is large enough, such as along the early part of the wall streamline (x ;S 0.42 m) or along the last part of the center streamline (x ,<: 0.4 m), the production rate of H is overtaking the decline and ' result in a net increase.
Formulation
- State Equation 17
- Energy Equation
Summary of Static Model
In the two cases, only a slight increase in start-up time and bending time is found. The destruction aspect in the reverse step has been ignored because it is assumed that only one of the steps can be dominant. The second requirement is essentially identical to one of the assumptions in the temperature correlation study.
Linearity in the behavior under pressure can be checked again by studying the duration of the rapid rise, which is defined from the start-up time to the inflection point. Nevertheless, the difference is small and the use of the times defined by the heat output provides a very good collapse of the curves. The change in the overall thermodynamic properties of the system during the reaction must therefore be small.
The y-axis represents the fractional reduction in start-up time given by ts/tso. The relationship is much clearer after the rapid rise or in the second half of the pressure cycle. Note also that the subscripts 'I' and 'u' are equivalent to the indices 'l' and 'N' in the discretized plane.
The step width increases downstream, but not to a large extent. This accounts for the overshoot representation in the variables along the wall immediately after the bend.
Finite-Rate Reaction in a I-D Constant Pressure Combustor
Setup
- Governing Equations
- Survey of Kinetic Data for a Hydrogen-Air System
- Estimation of Initial Conditions and Combustor Pressure 30
- Combustor input conditions 33
- Reduced kinetic data set 35
- Finite-Rate Reaction in a I-D Constant Pressure Combustor
This is a good approximation for the main part of the flow, but the effect of the boundary layer on the nozzle wall is lost.
Parametric Studies
The Reference Case
Although each calculation involves many variables as well as some derived quantities, not all are required in the analyses. The time histories for some fundamental variables and various quantities of interest for the combustion performance analysis are shown in figure 4.2.1a to figure 4.2.9b. According to the assumptions in the static model, the fundamental set of variables includes the composition and temperature for which the equations are derived in section 3.3.
Since the combustor flow rate is decoupled from the problem, knowledge of an I can help translate time scale to length scale which is a physical parameter in an engine. The density can provide an estimate on the channel area in the proposed one-dimensional combustor. However, those of the radicals are not plotted because their initial values are zero and their normalization curves will resemble the former, except for a multiplying factor.
On the larger timescale of Figure 4.2.9a, they appear very close; they can only be distinguished on the enlarged scale of Figure 4.2.9b. These features, along with other specific features, are described in detail in the following sections.
Variation of Equivalence Ratio 51
- Fast-rise behavior
- Conditions at chemical equilibrium 53
- Low temperature behavior 63
- Conditions at chemical equilibrium 67
- Start-up behavior
- Fast-rise behavior
- Scaling with pressure: Mole fraction
Its existence in the initiation phase is unlikely as the temperatures were found to be very different at the initiation time for the seven cases. Of course, the scaling in the start-up period and the period of rapid growth do not have to be the same. Therefore, introducing some radicals into the starting mixture can reduce the onset delay.
Although not one of the governing equations, the formulas for the inverse transformation are necessary so that the results can be represented on the physical plane.
Finite-rate Chemistry Calculations
Chemical activity along the center line (#21) is indicated by increasing pressure and temperature near the inlet. The response of hydrogen-air kinetics to an expansion is best described along the wall streamline as it undergoes the highest expansion rate. The evolution of individual species composition along the half-mouth flow lines (21) is shown in Figures 8.6.3 to 8.6.11.
This can be observed along the first three lines from the wall (lower part of figure 8.6.5) at the bottom of the mouth. H20: Along the initial part of the central line, H20 is produced; thus continuing the same processes as in the burner. Along the centerline where initially no expansion is felt, O 2 is consumed in the combustion process.
However, along the wall where the expansion takes place immediately, consumption soon turns into net production. Without the effect of expansion, as is the case initially along the midstream line, OH is slowly used up in combustion. The effect should be observed along the wall, as the behavior along streamlines close to the centerline is identical to that in case 1.
The half-nose line model is shown in Figure 8.7.1 to describe the rotational velocities along the wall and internal lines. The pressure and temperature distribution along the half-nozzle flow lines are shown in Figures 8.7.2 and 8.7.3. The variables examined in the case where Rt is equal to 100 Lx are again plotted along the half-nose streamlines.
0.00053 m, wall turn is complete; P and T reached their minimum points and became almost constant along the wall**. In the latter two cases this is not the case, as the compositional change along the wall slows down at about the same point where P and T become nearly constant. This is a wall score that leads to the completion of high levels back and forth.
- Species response to expansion
Discussion
Recommendation for Future Work
- Chemical Kinetics 208
Thermodynamic Equations for a Calorically Imperfect Gas
- The Absolute Entropy
- The frozen Speed of Sound
A One-Dimensional Diffuser Model
