^{1,a)}, Stephen B. Pope

^{1}, Alexander Vladimirsky

^{2}and John M. Guckenheimer

^{2}

### Abstract

This work addresses the construction and use of low-dimensional invariant manifolds to simplify complex chemical kinetics. Typically, chemical kinetic systems have a wide range of time scales. As a consequence, reaction trajectories rapidly approach a hierarchy of attracting manifolds of decreasing dimension in the full composition space. In previous research, several different methods have been proposed to identify these low-dimensional attracting manifolds. Here we propose a new method based on an invariant constrained equilibrium edge (ICE)manifold. This manifold (of dimension ) is generated by the reaction trajectories emanating from its -dimensional edge, on which the composition is in a constrained equilibrium state. A reasonable choice of the represented variables (e.g., “major” species) ensures that there exists a unique point on the ICEmanifold corresponding to each realizable value of the represented variables. The process of identifying this point is referred to as species reconstruction. A second contribution of this work is a local method of *species reconstruction*, called ICE-PIC, which is based on the ICEmanifold and uses preimage curves (PICs). The ICE-PIC method is *local* in the sense that species reconstruction can be performed without generating the whole of the manifold (or a significant portion thereof). The ICE-PIC method is the first approach that locally determines points on a low-dimensional invariant manifold, and its application to high-dimensional chemical systems is straightforward. The “inputs” to the method are the detailed kinetic mechanism and the chosen reduced representation (e.g., some major species). The ICE-PIC method is illustrated and demonstrated using an idealized system with six chemical species. It is then tested and compared to three other dimension-reduction methods for the test case of a one-dimensional premixed laminar flame of stoichiometric hydrogen/air, which is described by a detailed mechanism containing nine species and 21 reactions. It is shown that the error incurred by the ICE-PIC method with four represented species is small across the whole flame, even in the low temperature region.

This research is supported by the National Science Foundation through Grant No. CTS-0426787. Helpful comments were received from Stephen Vavasis and Paul Chew. The authors are grateful to Chris Pelkie and Steve Lantz at Cornell Theory Center for help with the graphics.

I. INTRODUCTION

II. THE INVARIANT CONSTRAINED EQUILIBRIUM EDGE (ICE)MANIFOLD

A. Homogeneous reacting system

B. Idealized system

C. Gibbs function, entropy, and chemical equilibrium

D. Reaction trajectories

E. Reduced composition

F. Attracting manifolds and species reconstruction

G. Constrained equilibrium manifold

H. Invariant constrained equilibrium edge (ICE)manifold

III. ICE-PIC: A METHOD TO DETERMINE POINTS ON THE ICEMANIFOLD USING THE CONSTRAINED EQUILIBRIUM PRE IMAGE CURVE

A. The preimage manifold

B. The constrained equilibrium preimage curve (CE-PIC)

C. The ICE-PIC method

IV. ICE-PIC METHOD FOR ADIABATIC SYSTEMS

V. COMPARATIVE TESTING OF SPECIES RECONSTRUCTION METHODOLOGIES

A. Computations of a premixed laminar flame

B. Species reconstruction using ICE-PIC

C. Species reconstruction using RCCE

D. Species reconstruction using ILDM

E. Species reconstruction using QSSA

F. Comparison of species reconstruction errors

VI. DISCUSSION AND CONCLUSION

### Key Topics

- Manifolds
- 156.0
- Ice
- 93.0
- Chemical kinetics
- 28.0
- Particle-in-cell method
- 24.0
- Flames
- 21.0

## Figures

Sketch, in the subspace, of the realizable region for the idealized system. Two hypothetical reaction trajectories are sketched, through the points and , starting from the boundary origin points and and ending at the single equilibrium point .

Sketch, in the subspace, of the realizable region for the idealized system. Two hypothetical reaction trajectories are sketched, through the points and , starting from the boundary origin points and and ending at the single equilibrium point .

Sketch, in the active subspace, of the reduced realizable region (shaded), which is the perpendicular projection of the realizable region onto the reduced subspace. The reduced composition is shown, corresponding to the full composition . The sketch also shows the one-dimensional feasible regions and corresponding to the interior and boundary points and ; and the zero-dimensional feasible region corresponding to the boundary point .

Sketch, in the active subspace, of the reduced realizable region (shaded), which is the perpendicular projection of the realizable region onto the reduced subspace. The reduced composition is shown, corresponding to the full composition . The sketch also shows the one-dimensional feasible regions and corresponding to the interior and boundary points and ; and the zero-dimensional feasible region corresponding to the boundary point .

The computed constrained equilibrium manifold (grid manifold) with and O being the represented species for the idealized system. The dot is the chemical equilibrium composition of the system. The bold curves and lines form the constrained equilibrium edge, which is the intersection between the constrained equilibrium manifold and the boundary of the realizable region.

The computed constrained equilibrium manifold (grid manifold) with and O being the represented species for the idealized system. The dot is the chemical equilibrium composition of the system. The bold curves and lines form the constrained equilibrium edge, which is the intersection between the constrained equilibrium manifold and the boundary of the realizable region.

The invariant constrained-equilibrium edge (ICE) manifold for the idealized system, which is the union of the reaction trajectories from the constrained equilibrium edge (bold lines and curves).

The invariant constrained-equilibrium edge (ICE) manifold for the idealized system, which is the union of the reaction trajectories from the constrained equilibrium edge (bold lines and curves).

The projection of the ICE manifold onto the reduced composition space , shown by the projected trajectories originating on the edge (bold lines). The fact that the projected trajectories do not cross demonstrates that this ICE manifold is regular: to each point in there is a unique manifold point . (The directions of the and axes are chosen to facilitate comparisons with the previous 3D illustrations.)

The projection of the ICE manifold onto the reduced composition space , shown by the projected trajectories originating on the edge (bold lines). The fact that the projected trajectories do not cross demonstrates that this ICE manifold is regular: to each point in there is a unique manifold point . (The directions of the and axes are chosen to facilitate comparisons with the previous 3D illustrations.)

Sketch showing the feasible region and the preimage manifold corresponding to . The preimage manifold intersects the boundary of the realizable region along the curves and .

Sketch showing the feasible region and the preimage manifold corresponding to . The preimage manifold intersects the boundary of the realizable region along the curves and .

Sketch showing the intersection between the CEM and the preimage manifold for . The intersection is the constrained equilibrium preimage curve (CE-PIC).

Sketch showing the intersection between the CEM and the preimage manifold for . The intersection is the constrained equilibrium preimage curve (CE-PIC).

Sketch showing (for given and ) the feasible region and the constrained equilibrium preimage curve . The general point on the CE-PIC is denoted by , and the reaction trajectory from it, , intersects the feasible region at . The feasible end of the CE-PIC is and the boundary end is . The reaction trajectory from is in the ICE manifold, and it intersects the feasible region at after time .

Sketch showing (for given and ) the feasible region and the constrained equilibrium preimage curve . The general point on the CE-PIC is denoted by , and the reaction trajectory from it, , intersects the feasible region at . The feasible end of the CE-PIC is and the boundary end is . The reaction trajectory from is in the ICE manifold, and it intersects the feasible region at after time .

For a case with two represented variables ( and ) and one unrepresented variable , a sketch of the realizable region showing the feasible region and the constrained equilibrium point , corresponding to the given reduced composition ; the constrained equilibrium preimage curve from to its boundary end , which lies in a facet of (the triangle at the left, on which is zero); the constrained equilibrium edge in this facet; and the trajectory , which intersects the feasible region at . Based on the two CE-PIC points and , a predicted value of is obtained by extrapolation to the facet. A Newton iteration is performed yielding a succession of estimates of [all in ]. The initial guess has the same value of the represented variables as , and the iteration is based on refining with the aim of making the projected reaction trajectory pass through .

For a case with two represented variables ( and ) and one unrepresented variable , a sketch of the realizable region showing the feasible region and the constrained equilibrium point , corresponding to the given reduced composition ; the constrained equilibrium preimage curve from to its boundary end , which lies in a facet of (the triangle at the left, on which is zero); the constrained equilibrium edge in this facet; and the trajectory , which intersects the feasible region at . Based on the two CE-PIC points and , a predicted value of is obtained by extrapolation to the facet. A Newton iteration is performed yielding a succession of estimates of [all in ]. The initial guess has the same value of the represented variables as , and the iteration is based on refining with the aim of making the projected reaction trajectory pass through .

Top row: composition along the CE-PIC with ; the dot is the boundary end of the CE-PIC, , where . Bottom row: the feasible composition mapped from the CE-PIC for the same case; the dot is the reconstructed composition on the ICE manifold.

Top row: composition along the CE-PIC with ; the dot is the boundary end of the CE-PIC, , where . Bottom row: the feasible composition mapped from the CE-PIC for the same case; the dot is the reconstructed composition on the ICE manifold.

Temperature and species specific moles across the flame. Lines: composition obtained using PREMIX with detailed chemistry; dots: compositions reconstructed using the ICE-PIC method.

Temperature and species specific moles across the flame. Lines: composition obtained using PREMIX with detailed chemistry; dots: compositions reconstructed using the ICE-PIC method.

Species specific moles across the flame. Lines: composition obtained using PREMIX with detailed chemistry; dots: composition reconstructed using the ICE-PIC method.

Species specific moles across the flame. Lines: composition obtained using PREMIX with detailed chemistry; dots: composition reconstructed using the ICE-PIC method.

Species specific moles across the flame. Lines: from PREMIX; dots: from ICE-PIC; dot-dashed line: . The profiles are plotted against the temperature , which is an increasing function of distance through the flame.

Species specific moles across the flame. Lines: from PREMIX; dots: from ICE-PIC; dot-dashed line: . The profiles are plotted against the temperature , which is an increasing function of distance through the flame.

Species specific moles of . Solid line: obtained using PREMIX with detailed chemistry; dashed line: , reconstructed using ILDM. Note that passes through zero at about . The profiles are plotted against the temperature , which is an increasing function of distance through the flame.

Species specific moles of . Solid line: obtained using PREMIX with detailed chemistry; dashed line: , reconstructed using ILDM. Note that passes through zero at about . The profiles are plotted against the temperature , which is an increasing function of distance through the flame.

Normalized errors [Eq. (35)] in reconstructed compositions. The profiles are plotted against the temperature , which is an increasing function of distance through the flame.

Normalized errors [Eq. (35)] in reconstructed compositions. The profiles are plotted against the temperature , which is an increasing function of distance through the flame.

Reaction rates of across the flame based on PREMIX calculations and different reconstructed compositions. The profiles are plotted against the temperature , which is an increasing function of distance through the flame.

Reaction rates of across the flame based on PREMIX calculations and different reconstructed compositions. The profiles are plotted against the temperature , which is an increasing function of distance through the flame.

Normalized errors [Eq. (36)] in the reconstructed reaction rate vectors. The profiles are plotted against the temperature , which is an increasing function of distance through the flame.

Normalized errors [Eq. (36)] in the reconstructed reaction rate vectors. The profiles are plotted against the temperature , which is an increasing function of distance through the flame.

## Tables

Chemical mechanism of the ideal system. in mol/cm/s/K; in cal/mole; ; universal gas constant. represents a third body that could be any of the species H, , OH, O, , and . The collision efficiencies for the third bodies are , , , , , and .

Chemical mechanism of the ideal system. in mol/cm/s/K; in cal/mole; ; universal gas constant. represents a third body that could be any of the species H, , OH, O, , and . The collision efficiencies for the third bodies are , , , , , and .

Article metrics loading...

Full text loading...

Commenting has been disabled for this content