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Research Papers: Gas Turbines: Combustion, Fuels, and Emissions

# Creating Reduced Kinetics Models That Satisfy the Entropy Inequality

[+] Author and Article Information
Nathan H. Jones

Department of Aerospace Engineering,
Texas A&M University,
College Station, TX 77843-3141
e-mail: nathan.h.jones@navy.mil

Paul G. A. Cizmas

Department of Aerospace Engineering,
Texas A&M University,
College Station, TX 77843-3141
e-mail: cizmas@tamu.edu

John C. Slattery

Department of Aerospace Engineering,
Texas A&M University,
College Station, TX 77843-3141
e-mail: slattery@tamu.edu

While not extensively studied [see for example Ref. [2]], thermal and pressure diffusion are negligible effects, unless the temperature and pressure gradients are extraordinarily large, much larger than those seen here. Forced diffusion is not applicable, since any ions are not subjected to an electric field.

The law of mass action states that the rate of disappearance of a chemical species is proportional to the products of the concentrations of the reacting chemical species, each concentration being raised to a power equal to the corresponding stoichiometric coefficient [15, p. 112].

1Present address: Naval Air Systems Command, Patuxent River, MD 20670.

2Corresponding author.

Contributed by the Combustion and Fuels Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received September 2, 2014; final manuscript received November 6, 2014; published online December 23, 2014. Editor: David Wisler.

J. Eng. Gas Turbines Power 137(7), 071504 (Jul 01, 2015) (9 pages) Paper No: GTP-14-1524; doi: 10.1115/1.4029172 History: Received September 02, 2014; Revised November 06, 2014; Online December 23, 2014

## Abstract

In simulating chemically reacting flows, the differential entropy inequality (the local form of the second law of thermodynamics) must be satisfied in addition to the differential mass, momentum, and energy balances. Previously, we have shown that entropy violations occur when using a global/reduced mechanism. Herein we show that entropy violations also occur when using a detailed/skeletal/reduced mechanism. Using a recent theorem of “Slattery et al. (2011, “Role of Differential Entropy Inequality in Chemically Reacting Flows,” Chem. Eng. Sci., 66(21), pp. 5236–5243),” we illustrate how to modify a reduced chemical kinetics model to automatically satisfy the differential entropy inequality. The numerical solution of a methane laminar flame was improved when using reduced chemical kinetics modified in this way. In addition, an ad hoc temperature limiter is no longer necessary.

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## References

Denbigh, K., 1963, The Principle of Chemical Equilibrium, Cambridge University, Cambridge, UK.
Slattery, J. C., 1999, Advanced Transport Phenomena, Cambridge University, Cambridge, UK.
Keck, J. C., 1990, “Rate-Controlled Constrained-Equilibrium Theory of Chemical Reactions in Complex Systems,” Progress Energy Combust. Sci., 16(2), pp. 125–154.
Ren, Z., and Pope, S. B., 2004, “Entropy Production and Element Conservation in the Quasi-Steady-State Approximation,” Combust. Flame, 137(1), pp. 251–254.
Slattery, J. C., Cizmas, P. G. A., Karpetis, A. N., and Chambers, S. B., 2011, “Role of Differential Entropy Inequality in Chemically Reacting Flows,” Chem. Eng. Sci., 66(21), pp. 5236–5243.
Westbrook, C. K., and Dryer, F. L., 1981, “Simplified Reaction Mechanisms for the Oxidation of Hydrocarbon Fuels in Flames,” Combust. Sci. Technol., 27(1–2), pp. 31–43.
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Jones, N. H., 2012, “The Importance of the Entropy Inequality on Numerical Simulation Using Reduced Methane-Air Reaction Mechanisms,” Master’s thesis, Texas A&M University, College Station, TX.
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## Figures

Fig. 1

Variation of temperature and molar concentrations of major species with distance

Fig. 2

Computational domain; all dimensions in mm. Due to symmetry, only half of it is used [5].

Fig. 3

(a) Coarse mesh used for grid convergence study and (b) detail of coarse mesh near burner

Fig. 4

Grid convergence study: (a) temperature variation versus radial position; (b) methane mass fraction versus radial position. 2025 K temperature limiter, 25 mm downstream from burner.

Fig. 5

Cells that violate entropy inequality and cells where temperature limiter is applied

Fig. 6

Variation of magnitude of second law violation versus iteration number

Fig. 7

Temperature contours for four temperature limiters; from left to right: 2025, 2300, 2600, and 2900 K

Fig. 8

(a) Temperature variation and (b) methane mass fraction versus radial position. 25 mm downstream from burner, 2025 K temperature limiter for Westbrook and Dryer [6] model and no limiter for new model.

Fig. 9

(a) Temperature variation and (b) methane mass fraction versus radial position. 50 mm downstream from burner, 2025 K temperature limiter for Westbrook and Dryer [6] model and no limiter for new model.

Fig. 10

(a) Temperature variation and (b) methane mass fraction versus radial position. 100 mm downstream from burner, 2025 K temperature limiter for Westbrook and Dryer [6] model and no limiter for new model.

Fig. 11

(a) Temperature and (b) methane mass fraction difference between the Westbrook and Dryer [6] model that did not satisfy the differential entropy inequality and the model proposed herein that satisfied the differential entropy inequality. Temperature differences between the two models were as high as 1750 K.

Fig. 12

## Errata

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