Repro-modelling

The reduction techniques which take advantage of this separation in scale are described below. They include the quasi-steady-state approximation (QSSA), the computational singular perturbation method (CSP), the slow manifold approach (intrinsic low-dimensional manifold, ILDM), repro-modelling and lumping in systems with time-scale separation. They are different in their approach but are all based on the assumption that there are certain modes in the equations which work on a much faster scale than others and, therefore, may be decoupled. We first describe the methods used to identify the range of time-scales present in a system of odes. 

The first stage in the formulation of a repro-model is the generation of suitable data. Usually, several thousand typical data points are considered and a detailed model is used for the calculation of the concentrations as a function of time. The evaluation of a large number of data points requires quite intensive computation, since, large sets of odes must be solved for all conditions. However, this effort has to be expended only once, and when the data have been fitted to polynomials, the repeated kinetic calculations, which are generally carried out during every CFD calculation, are avoided. 

Variables in the algebraic model can be identical to those in the approximated system of differential equations [228]. Another possibility is to approximate only a subset of the original variables, including the important variables and those which have the greatest influence on the important variables of the model. For example, a repro-model can be constructed which describes only the evolution of the slow variables [229]. This means an indirect utilization of the very different time-scales characteristic of combustion systems. In most applications of time-scale separation the basic problem is to derive expressions which describe the time derivative of slow variables as a function of the same slow variables. The algebraic equations, resulting from the repro-modelling, contain these dependencies and therefore constitute an implicit application of slow manifolds. 

There are possible alternative ways for the construction of algebraic models. If the look-up tables of the intrinsic low-dimensional manifold method are fitted by polynomials [234], the result is an algebraic model similar to a repro-model describing only slow variables. Polynomials can be fitted to the integrated solutions of few-step global mechanisms [233]. Such integrated solutions are found in the look-up tables used in the Monte-Carlo method for the simulation of turbulent flames. 

Fig. 4.16. Carbon monoxide-oxygen phase plane. The solid lines give the trajectories corresponding to a starting temperature of 1000 K and fuel-to-air ratios 0.5, 1.0 and 1.5, respectively, up to a simulation time of 0.01 s. The upper ends of the lines belong to the initial unburned mixture. For the preparation of the repro-model the initial composition of the mixture was uniformly distributed between = 0.5 and 1.5. Dots represent the values of CO and O2 concentrations used for the generation of the repro-model. Only a part of the 30,000 data points are plotted on the figure for clarity.

Fig. 4.17. An example of a fully optimized computer-generated multivariate Horner equation. The repro-model obtained consists of a set of such FORTRAN functions. This function calculates the temperature 10 s later, given the temperature and mole fractions...

Fig. 4.19. (a) Concentration-time and temperature-time curves for a stoichiometric mixture. Solid lines represent the solution obtained from the original detailed mechanism. The dotted lines were calculated using the three-variable repro-model. Calculation of the curves with the repro-model was 11,700 times faster (b) Concentration-time and temperature-time curves for the same initial conditions as in (a). Here the dots were calculated using the two variable repro-model, and the calculation of the curves with the repro-model was 24,000... 

T. TurSnyi, Application of Repro-Modelling for the Reduction of Combustion Mechanisms, 25th Symp. (Int.) Comb. (1994) pp. 948-955. 

W.S. Meisel and D.C. Collins, Repro-Modeling An Approach to Efficient Model Utilization and Interpretation, IEEE Trans. SMC-3 (1973) 349-358. 

Network and Repro-Modelling in Turbulent Combustion, Proceedings of the IEEE International Conference on Neural Networks 1 (1995) 911-916. 

In view of the relatively few parameters (temperature and pressure history, equivalence ratios) over which predictions are required, it is reasonable to consider whether simpler extrapolation methods are possible. Repro modelling, described in Chapter 4, avoids the computationally expensive integration of differential equations derived from chemical models, by solving them separately and representing the result parametrically. However, chemical models of low complexity have been more commonly used directly. Two types are distinguished reduced mechanisms and simplified models, which differ in the methodology used to generate them. 

Biiki, A., Perger, T., Turanyi, T., Maas, U. Repro-modelling based generation of intrinsic low-dimensional manifolds. J. Math. Chem. 31, 345-362 (2002)... 

Meisel and Collins were among the first authors who used this principle and called it the repro-modelling approach. Meisel and Collins suggested that within a large complex model, it is worth identifying very time-consuming subtasks which are used frequently, where the results depend only on the values of a few variables (Meisel and Collins 1973). It is immediately obvious that the presence of... 

Fig. 7.11 Simulation of a skeletal model of the Belousov-Zhabotinsky reaction based on the solution of the kinetic system of ODEs (solid line) and using a repro-model (dots), (a) Concentration-time curves (b) the solution in phase space. Reprinted from Turanyi (1994) with permission from Elsevier...

Fig. 7.12 The upper part of the detonation wave travelled further, whilst the lower part reflected back from the obstacle. A part of the wave also reflected back from the ceiling. The density maps were calculated using (a) a detailed mechanism including 9 species and (b) a repro-model. The latter calculation was one hundred times faster. Reprinted from Clifford et al. (1998) with permission from Elsevier...

Fig. 7.13 Comparison between model simulations based on ordinary differential equations describing the reduced scheme (solid) and fitted polynomial repro-model (dashed) for oscillatory ignition of CO-H2 mixtures alp = 25 Torr and 0.5 % H2 and initial temperatures (a) 720 K, (b) 735 K, (c) 750 K, (d) 770 K. Reprinted from Brad et al. (2007) with permission from Elsevier...

Lowe, R.M., Tomlin, A.S. The application of repro-modelling to a tropospheric chemical model. Environ. Model. Software 15, 611-618 (2000b)... 

Meisel, W.S., Collins, D.C. Repro-modeling an approach to efficient model utilization and interpretation. IEEE Trans. SMC-3/4, 349-358 (1973)... 

Turanyi, T. Application of repro-modelling for the reduction of combustion mechanisms. Proc. Combust. Inst. 25, 948-955 (1995)... 

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