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The invariant constrained equilibrium edge preimage curve method for the dimension reduction of chemical kinetics
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10.1063/1.2177243
/content/aip/journal/jcp/124/11/10.1063/1.2177243
http://aip.metastore.ingenta.com/content/aip/journal/jcp/124/11/10.1063/1.2177243

Figures

Image of FIG. 1.
FIG. 1.

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 .

Image of FIG. 2.
FIG. 2.

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 .

Image of FIG. 3.
FIG. 3.

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.

Image of FIG. 4.
FIG. 4.

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).

Image of FIG. 5.
FIG. 5.

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.)

Image of FIG. 6.
FIG. 6.

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 .

Image of FIG. 7.
FIG. 7.

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

Image of FIG. 8.
FIG. 8.

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 .

Image of FIG. 9.
FIG. 9.

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 .

Image of FIG. 10.
FIG. 10.

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.

Image of FIG. 11.
FIG. 11.

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

Image of FIG. 12.
FIG. 12.

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

Image of FIG. 13.
FIG. 13.

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.

Image of FIG. 14.
FIG. 14.

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.

Image of FIG. 15.
FIG. 15.

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

Image of FIG. 16.
FIG. 16.

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.

Image of FIG. 17.
FIG. 17.

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

Generic image for table
Table I.

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 .

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/content/aip/journal/jcp/124/11/10.1063/1.2177243
2006-03-20
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: The invariant constrained equilibrium edge preimage curve method for the dimension reduction of chemical kinetics
http://aip.metastore.ingenta.com/content/aip/journal/jcp/124/11/10.1063/1.2177243
10.1063/1.2177243
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