Research Papers: Fuel Combustion

Invariant Forms of Conservation Equations for Reactive Fields and Hydro-Thermo-Diffusive Theory of Laminar Flames

[+] Author and Article Information
Siavash H. Sohrab

Department of Mechanical Engineering,
Northwestern University,
2145 Sheridan Road,
Evanston, IL 60208-3111
e-mail: s-sohrab@northwestern.edu

1Corresponding author.

Contributed by the Advanced Energy Systems Division of ASME for publication in the JOURNAL OF ENERGY RESOURCES TECHNOLOGY. Manuscript received May 28, 2014; final manuscript received May 29, 2014; published online August 27, 2014. Editor: Hameed Metghalchi.

J. Energy Resour. Technol 137(1), 012203 (Aug 27, 2014) (10 pages) Paper No: JERT-14-1168; doi: 10.1115/1.4028070 History: Received May 28, 2014; Revised May 29, 2014

A scale-invariant model of statistical mechanics is described leading to invariant Enskog equation of change that is applied to derive invariant forms of conservation equations for mass, thermal energy, linear momentum, and angular momentum in chemically reactive fields. Modified hydro-thermo-diffusive theories of laminar premixed flames for (1) rigid-body and (2) Brownian-motion flame propagation models are presented and are shown to be mathematically equivalent. The predicted temperature profile, thermal thickness, and propagation speed of laminar methane–air premixed flame are found to be in good agreement with existing experimental observations.

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Topics: Flames
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de Broglie, L., 1926, “Interference and Corpuscular Light,” Nature, 118(2969), pp. 441–442. [CrossRef]
de Broglie, L., 1927, “Sur la possibilité de relier les phénomènes d'interférence et de diffraction à la théorie des quanta de lumière,” C.R. Acad. Sci. Paris, 183, pp. 447–448.
de Broglie, L., 1927, “La structure atomique de la matière et du rayonnement et la mécanique Ondulatoire,” C. R. Acad. Sci. Paris, 184, pp. 273–274.
de Broglie, L., 1927, “Sur le rôle des ondes continues en mécanique ondulatoire,” C. R. Acad. Sci. Paris, 185, pp. 380–382.
de Broglie, L., 1960, Non-Linear Wave Mechanics: A Causal Interpretation, Elsevier, New York.
de Broglie, L., 1970, “The Reinterpretation of Wave Mechanics,” Found. Phys., 1(5), pp. 5–15. [CrossRef]
Madelung, E., 1926, “Quantentheorie in Hydrodynamischer Form,” Z. Phys., 40(3–4), pp. 322–326. [CrossRef]
Schrödinger, E., 1931, “Über die Umkehrung der Naturgesetze,” Sitz.-Ber. d. Preuss. Akad. d. Wiss., Phys.-Math. Kl., 9, pp. 144–153.
Fürth, R., 1933, “Über Einige Beziehungen zwischen klassischer Staristik und Quantenmechanik,” Z. Phys., 81(3–4), pp. 143–162. [CrossRef]
Bohm, D., 1952, “A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables. I,” Phys. Rev., 85(2), pp. 166–179. [CrossRef]
Takabayasi, T., 1952, “On the Foundation of Quantum Mechanics Associated With Classical Pictures,” Prog. Theor. Phys., 8(2), pp. 143–182. [CrossRef]
Bohm, D., and Vigier, J. P., 1954, “Model of the Causal Interpretation of Quantum Theory in Terms of a Fluid With Irregular Fluctuations,” Phys. Rev., 96(1), pp. 208–217. [CrossRef]
Nelson, E., 1966, “Derivation of the Schrödinger Equation From Newtonian Mechanics,” Phys. Rev., 150(4), pp. 1079–1085. [CrossRef]
Nelson, E., 1985, Quantum Fluctuations, Princeton University, Princeton, NJ.
de la Peña, L., 1969, “New Foundation of Stochastic Theory of Quantum Mechanics,” J. Math. Phys., 10(9), pp. 1620–1630. [CrossRef]
de la Peña, L., and Cetto, A. M., 1982, “Does Quantum Mechanics Accept a Stochastic Support?,” Found. Phys., 12(10), pp. 1017–1037. [CrossRef]
Barut, A. O., 1988, “Schrödinger's Interpretation of Ψ as a Continuous Charge Distribution,” Ann. Phys., 7(4–5), pp. 31–36. [CrossRef]
Barut, A. O., and Bracken, A. J., 1981, “Zitterbewegung and the Internal Geometry of the Electron,” Phys. Rev. D, 23(10), pp. 2454–2463. [CrossRef]
Vigier, J. P., 1980, “De Broglie Waves on Dirac Aether: A Testable Experimental Assumption,” Lett. Nuovo Cimento, 29(14), pp. 467–475. [CrossRef]
Gueret, Ph., and Vigier, J. P., 1982, “De Broglie's Wave Particle Duality in the Stochastic Interpretation of Quantum Mechanics: A Testable Physical Assumption,” Found. Phys., 12(11), pp. 1057–1083. [CrossRef]
Cufaro Petroni, C., and Vigier, J. P., 1983, “Dirac's Aether in Relativistic Quantum Mechanics,” Found. Phys., 13(2), pp. 253–286. [CrossRef]
Vigier, J. P., 1995, “Derivation of Inertia Forces From the Einstein-de Broglie-Bohm (E.d.B.B) Causal Stochastic Interpretation of Quantum Mechanics,” Found. Phys., 25(10), pp. 1461–1494. [CrossRef]
Arecchi, F. T., and Harrison, R. G., 1987, Instabilities and Chaos in Quantum Optics, Springer-Verlag, Berlin.
Maxwell, J. C., 1867, “On the Dynamic Theory of Gases,” Philos. Trans. R. Soc. London, 157, pp. 49–88. [CrossRef]
Boltzmann, L., 1872, “Weitere Studien uber das Warmegleichgewicht unter Gasmoleculen,” itzungsberichte Akad.Wiss., Vienna, Part II, 66, pp. 275–370 [English translation in: Brush, G. S., 1965, Kinetic Theory, Vols. 1–3, Pergamon, New York].
Boltzmann, L., 1964, Lectures on Gas Theory, Dover, New York.
Reynolds, O., 1895, “On the Dynamical Theory of Incompressible Viscous Fluid and the Determination of the Criterion,” Philos. Trans. R. Soc. London, Ser. A, 186(1), pp. 23–164. [CrossRef]
Enskog, D., 1917, Kinetische Theorie der Vorgange in Massig Verdunnten Gasen, Almqvist and Wiksells Boktryckeri-A.B., Uppsala, [English translation in: Brush, G., S., 1965, Kinetic Theory, Vols. 1–3, Pergamon, New York].
Taylor, G. I., 1935, “Statistical Theory of Turbulence-Parts I-IV,” Proc. R. Soc. London, Ser. A, 151(873), pp. 421–478. [CrossRef]
von Kármán, T., and Howarth, L., 1938, “On the Statistical Theory of Isotropic Turbulence,” Proc. R. Soc. London, Ser. A, 164(917), pp. 192–215. [CrossRef]
Robertson, H. P., 1940, “The Invariant Theory of Isotropic Turbulence,” Proc. Cambridge Philos. Soc., 36, pp. 209–223. [CrossRef]
Kolmogoroff, A. N., 1941, “Local Structure on Turbulence in Incompressible Fluid,” C.R. Acad. Sci. U. R. S. S., 30, pp. 301–305.
Kolmogoroff, A. N., 1942, “Dissipation of Energy in Locally Isotropic Turbulence,” C.R. Acad. Sci. U. R. S. S., 32, pp. 19–21.
Kolmogoroff, A. N., 1962, “A Refinement of Previous Hypothesis Concerning the Local Structure of Turbulence in a Viscous Incompressible Fluid at High Reynolds Number,” J. Fluid Mech., 13(1), pp. 82–85. [CrossRef]
Obukhov, A. M., 1941, “On the Distribution of Energy in the Spectrum of Turbulent Flow,” C.R. Acad. Sci. U. R. S. S., 32, pp. 19–22.
Obukhov, A. M., 1962, “Some Specific Features of Atmospheric Turbulence,” J. Fluid Mech., 13, pp. 77–81. [CrossRef]
Chandrasekhar, S., 1943, “Stochastic Problems in Physics and Astronomy,” Rev. Mod. Phys., 151(1), pp. 1–89. [CrossRef]
Chandrasekhar, S., 1989, Stochastic, Statistical, and Hydrodynamic Problems in Physics and Astronomy, Vol. 3, University of Chicago, Chicago, pp. 199–206.
Heisenberg, W., 1948, “On the Theory of Statistical and Isotropic Turbulence,” Proc. R. Soc. London, Ser. A, 195(1042), pp. 402–406. [CrossRef]
Heisenberg, W., 1948, “Zur Statistischen Theorie der Turbulenz,” Z. Phys., 124(7–12), pp. 628–657. [CrossRef]
Batchelor, G. K., 1953, The Theory of Homogeneous Turbulence, Cambridge University, Cambridge.
Landau, L. D., and Lifshitz, E. M., 1959, Fluid Dynamics, Pergamon, New York.
Tennekes, H., and Lumley, J. L., 1972, A First Course in Turbulence, MIT Press, Boston, MA.
Sohrab, S. H., 1996, “Transport Phenomena and Conservation Equations in Multicomponent Chemically-Reactive Ideal Gas Mixtures,” Proceeding of the 31st ASME National Heat Transfer Conference, HTD, Vol. 328, pp. 37–60.
Sohrab, S. H., 1999, “A Scale Invariant Model of Statistical Mechanics and Modified Forms of the First and the Second Laws of Thermodynamics,” Rev. Gen. Therm., 38(10), pp. 845–853. [CrossRef]
Sohrab, S. H., 2014, “Boltzmann Entropy of Thermodynamics Versus Shannon Entropy of Information Theory,” Int. J. Mech., 8, pp. 73–84.
Sohrab, S. H., 2014 “Some Implications of a Scale Invariant Model of Statistical Mechanics to Classical and Relativistic Thermodynamics,” Int. J. Thermo. (accepted).
Sohrab, S. H., 2008, “Derivation of Invariant Forms of Conservation Equations From the Invariant Boltzmann Equation,” Theoretical and Experimental Aspects of Fluid Mechanics, S. H.Sohrab, H. C.Catrakis, and F. K.Benra, eds., WSEAS, pp. 27–35.
Sohrab, S. H., 2014, “Invariant forms of Conservation Equations and Some Examples of their Exact Solutions,” ASME J. Energy Res. Technol. 136, pp. 1–9.
Sohrab, S. H., 2007, “Invariant Planck Energy Distribution Law and its Connection to the Maxwell-Boltzmann Distribution Function,” WSEAS Trans. Math., 6(2), pp. 254–262.
Sohrab, S. H., 2010, “Quantum Theory of Fields From Planck to Cosmic Scales,” WSEAS Trans. Math., 9(8), pp. 734–756.
Sohrab, S. H., 2012, “On a Scale Invariant Model of Statistical Mechanics, Kinetic Theory of Ideal Gas, and Riemann Hypothesis,” AIAA 50th Aerospace Meeting including the New Horizons Forum and Aerospace Exposition, Nashville, TN, Jan. 9–12.
de Groot, R. S., and Mazur, P., 1962, Nonequilibrium Thermodynamics, North-Holland, Amsterdam.
Schlichting, H., 1968, Boundary-Layer Theory, McGraw Hill, New York.
Williams, F. A., 1985, Combustion Theory, 2nd ed., Addison Wesley, Menlo Park, CA.
Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B., 1954, Molecular Theory of Gases and Liquids, Wiley, New York.
Chapman, S., and Cowling, T. G., 1953, The Mathematical Theory of Non-Uniform Gases, Cambridge University, Cambridge.
Sohrab, S. H., 2009, “Some Implications of a Scale Invariant Model of Statistical Mechanics to Transport Phenomena,” Recent Advances in Systems, N.Mastorakis, V.Mladenov, Z.Bojkovic, S.Kartalopoulos, A.Varonides, and M.Jha, eds., WSEAS Press, Athens, pp. 557–568.
Darrigol, O., 2002, “Between Hydrodynamics and Elasticity Theory: The First Five Births of the Navier–Stokes Equation,” Arch. Hist. Exact Sci., 56, pp. 95–150. [CrossRef]
Sohrab, S. H., 2014, “On a Scale Invariant Model of Statistical Mechanics and Derivation of Invariant Forms of Conservation Equations From Invariant Boltzmann and Enskog Equations,” The 2014 International Conference on Mechanics, Fluid Mechanics, Heat and Mass Transfer, Feb. 22–24, Interlaken, Switzerland.
Mallard, E., and le Chatelier, H. L., 1883, “Combustion des mélanges gazeuse explosifs,” Ann. Mines, 4, pp. 379–568.
Mikhel'son, V. A., 1889, “On the Normal Ignition Rate of Fulminating Gas Mixtures,” Thesis, University of Moscow, Vol. 1, Novyi Agronom, Moscow, 1930.
Taffanel, M., 1913, “Sur la combustion des mélanges gazeux et les vitesses de réaction,” C.R. Acad. Sci., Paris, 157, pp. 714–717.
Taffanel, M., 1914, “Sur la combustion des mélanges gazeux et les vitesses de réaction,” C.R. Acad. Sci., Paris, 158, pp. 340–343.
Jouguet, E., 1913, “Sur l'Onde Explosive,” C.R. Acad. Sci., Paris, 156, pp. 872–875.
Jouguet, E., 1904, “Sur la propagation des déflagration dans les mélanges,” C.R. Acad. Sci., Paris, 179, pp. 121–124.
Nusselt, W., 1915, “The Rate of Ignition of Combustible Gas Mixtures,” Z. Ver. Dtsch. Ing., 59, pp. 872–878.
Daniell, P. J., 1930, “The Theory of Flame Motion,” Proc. R. Soc. London, Ser. A, 126, pp. 393–405. [CrossRef]
Lewis, B., and von Elbe, G., 1934, “On the Theory of Flame Propagation,” J. Chem. Phys., 2, pp. 537–546. [CrossRef]
Jost, W., and Müffling, L. V., 1938, Z. Phys. Chem. A, 181, p. 208.
Kolmogoroff, A., Petrovsky, I., and Piscounoff, N., 1937, Bulletin de l'université d' état à Moscou, Série internationale, Sect. A, Vol. 1 [English translation in P.Pelcé, ed., 1988, Dynamics of Curved Fronts, Academic Press, New York.
Zeldovich, J. B., and Frank-Kamenetski, D. A., 1938, Acta Physicochimica U.R.S.S., Vol. IX, No.2, 341, Moscow [English translation in P.Pelcé, ed., 1988, Dynamics of Curved Fronts, Academic Press, New York].
Hirschfelder, J. O., and Curtiss, C. F., 1949, “The Theory of Flame Propagation,” J. Phys. Chem., 17, p. 4211076.
Hirschfelder, J. O., Curtiss, C. F., and Campbell, D. E., 1953, “The Theory of Flames and Detonations,” Proc. Combust. Inst., 4(1), pp. 190–211. [CrossRef]
Bush, W. B., and Fendell, F. E., 1970, “Asymptotic Analysis of Laminar Flame Propagation for General Lewis Numbers,” Combust. Sci. Technol., 1(6), pp. 421–425. [CrossRef]
Williams, F. A., 1971, “Theory of Combustion in Laminar Flows,” Annu. Rev. Fluid Mech., 3, pp. 171–188. [CrossRef]
Williams, F. A., 1973, “Quasi-Steady Gas Phase Flame Theory in Unsteady Burning of a Homogeneous Solid Propellant,” AIAA J., 11, pp. 1328–1330. [CrossRef]
Liñán, A., 1974, “The Asymptotic Structure of Counterflow Diffusion Flames for Large Activation Energies,” Acta Astronaut., 1(7–8), pp. 1007–1039. [CrossRef]
Joulin, G., and Clavin, P., 1979, “Linear Stability Analysis of Nonadiabatic Flames: Diffusional-Thermal Model,” Combust. Flame, 35, pp. 139–153. [CrossRef]
Buckmaster, J. D., and Ludford, G. S. S., 1982, Theory of Laminar Flames, Cambridge University, Cambridge.
Clavin, P., 1985, “Dynamic Behavior of Premixed Flame Fronts in Laminar and Turbulent Flows,” Prog. Energy Combust. Sci., 11(1), pp. 1–59. [CrossRef]
Zel'dovich, Y. B., Barenblatt, G. I., Librovich, V. B., and Makhviladze, G. M., 1985, Mathematical Theory of Combustion and Explosion, Plenum Publishing Company, New York.
Warnatz, J., Maas, U., and Dibble, R. W., 1996, Combustion, Springer, New York.
Sohrab, S. H., 1998, “Laminar Flame Theory Revisited-Stationary Coordinates for Systems Under Rigid-Body Versus Brownian Motions,” Combustion Fundamentals and Applications, Central States Section Meeting, The Combustion Institute, Lexington, KY, May 31–Jun. 2, pp. 110–115.
Sohrab, S. H., 2014, “Invariant Forms of Conservation Equations for Reactive Fields and Hydro-Thermo-Diffusive Structure of Laminar Flames,” Central States Section of the Combustion Institute, The Spring Technical Meeting, Tulsa, OK, Mar. 16–18.
Sohrab, S. H., 2004, “Scale-Invariant Forms of Conservation Equations in Reactive Fields and a Modified Hydro-Thermo-Diffusive Theory of Laminar Flames,” WSEAS Trans. Math., 3(4), pp. 755–763.
Sohrab, S. H., 2006, “A Modified Hydro-Thermo-Diffusive Theory of Laminar Premixed Flames,” WSEAS Trans. Fluid Mech., 1(5), pp. 337–345.
Lindow, R., 1968, “Eine verbesserde brennermethode zur bestimmung der laminaren flammengeschwindigkeiten von brenngas/luft-gemischen,” Brennst.-Wärme-Kraft, 20, pp. 8–14.
Reed, S. B., Mineur, J., and McNaughton, P., 1971, “The Effect on the Burning Velocity of Methane of Vitiation of Combustion Air,” J. Inst. Fuel, 44, pp. 149–155.
Günther, R., and Janisch, G., 1972, “Measurements of Burning Velocity in a Flat Flame Front,” Combust. Flame, 19(1), pp. 49–53. [CrossRef]
Andrews, G. E., and Bradley, D., 1972, “The Burning Velocity of Methane-Air Mixtures,” Combust. Flame, 19(2), pp. 275–288. [CrossRef]
Andrews, G. E., and Bradley, D., 1973, “The Burning Velocity of Methane-Air Mixtures,” Combust. Flame, 20(1), pp. 77–89. [CrossRef]
Yamaoka, I., and Tsuji, H., 1985, “Determination of Burning Velocity Using Counterflow Flames,” Proc. Combust. Inst., 20(1), pp. 1883–1892. [CrossRef]
Wu, C. K., and Law, C. K., 1985, “On the Determination of Laminar Flame Speeds From Stretched Flames,” Proc. Combust. Inst., 20(1), pp. 1941–1949. [CrossRef]
Elia, M., Ulinski, M., and Metghalchi, M., 2001, “Laminar Burning Velocity of Methane-Air-Diluent Mixtures,” ASME J. Eng. Gas Turbines Power, 123(1) pp. 190–196. [CrossRef]
Sohrab, S. H., Ye, Z. Y., and Law, C. K., 1986, “Theory of Interactive Combustion of Counterflow Premixed Flames,” Combust. Sci. Technol., 45, pp. 27–45. [CrossRef]
Smoot, D. L., Hecker, W. C., and Williams, G. A., 1976, “Prediction of Propagating Methane-Air Flames,” Combust. Flame, 26, pp. 323–342. [CrossRef]
Tsatsaronis, G., 1978, “Prediction of Propagating Laminar Flames in Methane, Oxygen, Nitrogen Mixtures,” Combust. Flame, 33, pp. 217–239. [CrossRef]
Chen, C. L., and Sohrab, S. H., 1995, “Upstream Interactions Between Planar Symmetric Laminar Methane Premixed Flames,” Combust. Flame, 101(3), pp. 360–370. [CrossRef]
Fisher, R. A., 1937, “The Wave of Advance of Advantageous Genes,” Ann. Eugen., 7(4), pp. 355–369. [CrossRef]
Sohrab, S. H., 2007, “A Modified Hydro-Thermo-Diffusive Theory of Shock Waves,” WSEAS Trans. Heat Mass Transfer, 1(5), pp. 606–611.
Kurz, O., and Sohrab, S. H., 1999, “Modified Hydro-Thermo-Diffusive Theory of Laminar Counterflow Premixed Flames,” First Joint Western- Central-Eastern States Section Meeting, The Combustion Institute, George Washington University, Washington DC, Mar. 14–17.
Liu, G. E., Ye, Z. Y., and Sohrab, S. H., 1986, “On Radiative Cooling and Temperature Profiles of Counterflow Premixed Flames,” Combust. Flame, 64(2), pp. 193–201. [CrossRef]
Sohrab, S. H., and Law, C. K., 1984, “Extinction of Premixed Flames by Stretch and Radiative Loss,” Int. J. Heat Mass Transfer, 27(2), pp. 291–300. [CrossRef]
Chen, Z. H., Lin, T. H., and Sohrab, S. H., 1988, “Combustion of Liquid Sprays in Stagnation-Point Flow,” Combust. Sci. Technol., 60(1–3), pp. 63–77. [CrossRef]
Friedman, R., 1953, “Measurement of the Temperature Profile in a Laminar Flame,” Proc. Combust. Inst., 4(1), pp. 259–263. [CrossRef]
Tsuji, H., and Yamaoka, I., 1982, “Structure and Extinction of Near Limit Flames in Stagnation Flow,” Proc. Combust. Inst., 19(1), pp. 1533–1540. [CrossRef]
Lewis, B., and von Elbe, G., 1987, Combustion Flame and Explosion of Gases, Academic, New York.
Eng, J. A., Zhu, D. L., and Law, C. K., 1995, “On the Structure, Stabilization, and Dual Response of Flat-Burner Flames,” Combust. Flame, 100(4), pp. 645–652. [CrossRef]
Turns, R. S., 1996, An Introduction to Combustion, McGraw Hill, New York, p. 221.


Grahic Jump Location
Fig. 1

Scale-invariant model of statistical mechanics. Equilibrium-β-Dynamics on the left-hand-side and nonequilibrium Laminar-β-Dynamics on the right-hand-side for scales β = g, p, h, f, e, c, m, a, s, k, and t as defined in Sec. 2. Characteristic lengths of (system, element, “atom”) are (Lβ , λβ, ℓβ) and λβ is the mean-free-path [45].

Grahic Jump Location
Fig. 2

Schematic of steady propagating laminar premixed flame in infinite domain according to (a) modified theory and (b) classical theory [85]

Grahic Jump Location
Fig. 3

Laminar premixed flame (a) propagating in laboratory; (b) stationary due to imposed convection; and (c) stationary with normalized velocity

Grahic Jump Location
Fig. 4

Flame thermal structure according to (a) modified theory of laminar flame and (b) classical theory of laminar flame

Grahic Jump Location
Fig. 5

(a) Finite value or reaction rate during τ-τi and (b) periodic onset of light during τ-τi

Grahic Jump Location
Fig. 6

Measured temperature profiles for methane–air premixed flames in stagnation point flow with ϕ = 0.8, w'zo= (a) 30 (b) 50 (c) 70 (d) 90 cm/s [102]



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