Studies on the effects of interphase heat exchange during thermal explosion in a combustible dusty gas with general Arrhenius reaction-rate laws. (English) Zbl 1235.80042

Adegbie, K. S.; Alao, F. I.

J. Appl. Math. 2012, Article ID 541348, 15 p. (2012).

Summary: A mathematical model for thermal explosion in a combustible dusty gas containing fuel droplets with general Arrhenius reaction-rate laws, convective and radiative heat losses, and interphase heat exchange between gas and inert solid particles is investigated. The objective of the study is to examine the effects of interphase heat exchange between the gas and solid particles on (i) ignition of reacting gas, (ii) accumulation of heat by the solid particles during combustion process (iii) evaporation of the liquid fuel droplets, and (iv) consumption of reacting gas concentration. The equations governing the physical model with realistic assumptions are stated and nondimensionalised leading to an intractable system of first-order coupled nonlinear differential equations, which is not amenable to exact methods of solution. Therefore, we present numerical solutions as well as different qualitative effects of varying interphase heat exchange parameter. Graphs and Table feature prominently to explain the results obtained.

MSC:

80M25	Other numerical methods (thermodynamics) (MSC2010)
80A25	Combustion

References:

[1]	Ya. B. Zeldovich, G. I. Barenblatt, V. B. Librovich, and G. M. Makhviladze, The Mathematical Theory of Combustion and Explosions, Consultants Bureau, New York, NY, USA, 1985.
[2]	N. N. Semenov, Some Problems of Chemical Kinetics and Reactivity, vol. 1&2, Pergamon Press, New York, NY, USA, 1958.
[3]	K. K. Kuo, Principle of Combustion, John Wiley & Sons, New York, NY, USA, 1986.
[4]	J. J. Gray, Linear Differential Equations and Group Theory from Riemann to Poincaré, Birkhäuser, Boston, Mass, USA, Second edition, 2000. · Zbl 0949.01001
[5]	J. B. Heywood, Internal Combustion Engine Fundamentals, McGraw-Hill, Singapore, International edition, 1988.
[6]	R. Stone, Introduction to Internal Combustion Engines, Macmillan, New York, NY, USA, 1992.
[7]	J. Warnatz, U. Mass, and R. W. Dibble, Combustion: Physical and Chemical Fundamentals, Modelling and Stimulation, Experiments, Pollutants Formation, Springer, Berlin, Germany, 3rd edition, 2001. · Zbl 0976.80005
[8]	V. Gol’dshtein, A. Zlnoviev, V. Sobolev, and E. Shchepakina, “Criterion for thermal explosion with reactant consumption in a dusty gas,” Proceeding of Royal Society of London A, vol. 452, no. 1952, pp. 2103-2119, 1996. · Zbl 0874.34052 · doi:10.1098/rspa.1996.0111
[9]	I. Goldfarb, S. Sazhin, and A. Zinoviev, “Delayed thermal explosion in flammable gas containing fuel droplets: asymptotic analysis,” Journal of Engineering Mathematics, vol. 50, no. 4, pp. 399-414, 2004. · Zbl 1178.76377 · doi:10.1007/s10665-004-9388-5
[10]	I. Goldfarb, V. Gol’dshtein, and A. Zinoviev, “Delayed thermal explosion in porous media: method of invariant manifolds,” IMA Journal of Applied Mathematics, vol. 67, no. 3, pp. 263-280, 2002. · Zbl 1008.80004 · doi:10.1093/imamat/67.3.263
[11]	I. Goldfarb and A. Zinoviev, “A study of delayed spontaneous insulation fires,” Physics Letters, Section A, vol. 311, no. 6, pp. 491-500, 2003. · doi:10.1016/S0375-9601(03)00506-1
[12]	I. Goldfarb, V. Gol’dshtein, J. B. Greenberg, and G. Kuzmenko, “Thermal explosion in a droplet-gas cloud,” Combustion Theory and Modelling, vol. 4, no. 3, pp. 289-316, 2000. · Zbl 0995.76096 · doi:10.1088/1364-7830/4/3/304
[13]	I. Goldfard, V. Gol’dshtein, G. Kuzmenko, and S .S. Sazhin, “Thermal radiation effect on thermal explosion in a gas containing fuel droplets,” Combustion Theory and Modelling, vol. 3, pp. 769-787, 1999. · Zbl 0982.76568 · doi:10.1088/1364-7830/3/4/309
[14]	V. Gol’dshtein and V. Sobolev, “Integral manifolds in chemical kinetics and combustion,” in Singularity Theory and Some Problems of Functional Analysis, pp. 73-92, American Mathematical Society, Providence, RI, USA, 1992.
[15]	V. Gol’dshtein, I. Goldfard, I. Shreiber, and A. Zinoviev, “Oscillations in a combustible gas bubble,” Combustion Theory and Modelling, vol. 2, pp. 1-17, 1998. · Zbl 1040.80501 · doi:10.1088/1364-7830/2/1/001
[16]	I. Goldfarb, V. Gol’dshtein, D. Katz, and S. Sazhin, “Radiation effect on thermal explosion in gas containing evaporated fuel droplets,” International Journal of Thermal Sciences, vol. 46, pp. 358-370, 2007. · Zbl 1309.80006 · doi:10.1016/j.ijthermalsci.2006.06.014
[17]	V. Gol’dshtein and A. Zinoviev, “Thermal explosion in multiple phase media,” Nonlinear Analysis, vol. 30, no. 8, pp. 4771-4780, 1997. · Zbl 0896.34062 · doi:10.1016/S0362-546X(96)00356-2
[18]	J. Klammer, R. Klemens, V. P. Korobeinikov, V. V. Markov, I. S. Men’shov, and P. Wolanski, “On ignition and unsteady flows of dusty gases with combustion reactions,” Combustion Science and Technology, vol. 142, no. 1, pp. 81-90, 1999. · doi:10.1080/00102209908924189
[19]	A. Yu. Krainov and V. A. Shaurman, “On the limits of flame propagation in a dusty gas,” Combustion, Explosion and Shock Waves, vol. 33, no. 4, pp. 403-408, 1997. · doi:10.1007/BF02671832
[20]	G. Ben-Dor and O. Igra, “The relaxation zone behind normal shock waves in a reacting dusty gas. I Monatomic gases,” Journal of Plasma Physics, vol. 27, no. 3, pp. 377-395, 1982. · doi:10.1017/S0022377800010965
[21]	S. A. El-Sayed, “Adiabatic thermal explosion of a gas-solid mixture,” Combustion Science and Technology, vol. 176, no. 2, pp. 237-256, 2004. · doi:10.1080/00102200490256144
[22]	K. S. Adegbie, A mathematical model of thermal explosion in a combustible gas containing fuel droplets, Ph.D. thesis, Lautech, Ogbomoso, Nigeria, 2008.
[23]	K. S. Adegbie, “On delay-type behaviour of thermal explosion in a combustible gas mixture containing fuel droplets with arrhenius power-law model,” Journal of the Nigerian Association of Mathematical Physics, vol. 68, pp. 69-82, 2008.
[24]	International Union of Pure and Applied Chemistry (IUPAC) Goldbook Definition of Modified Arrhenius Equation, http://en.wikipedia.org/wiki/Arrhenius/equation.
[25]	K. S. Adegbie, “Some remarks on Arrhenius Power-law Model for thermal explosion in a combustible gas mixture containing fuel droplets,” Ife Journal of Science, vol. 10, no. 1, pp. 358-370, 2008.
[26]	R. O. Ayeni, O. Otolorin, O. T. Ayodele, and K. S. Adegbie, “Effect of frank-kamenetskii parameter on thermal explosion of flammable gas containing fuel droplets,” Ife Journal Of Science, vol. 9, no. 1, pp. 55-62, 2005.
[27]	D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics-Theory of Thermal Explosions, Plenum Press, New York, NY, USA, 1969.