BURNING VELOCITY and the INFLUENCE of FLAME STRETCH Simon Crispin Taylor Bsc Cphys Minstp Submitted in Accordance with the Requi
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BURNING VELOCITY AND THE INFLUENCE OF FLAME STRETCH Simon Crispin Taylor BSc CPhys MInstP Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds Department of Fuel and Energy September 199 1 ABSTRACT A new technique is presented for determining burning velocities and stretch effects in laminar flames, and applied to a range of fuel/air mixtures. The speeds of expanding spherical flames, measured by high-speed schlieren cine-photography, are shown to vary with flame radius. A simple phenomenological model has been developed to analyse the data and obtain the one-dimensional flame speed by extrapolation to infinite radius. The validity of the simple model has been tested by using it to analyse the results of detailed simulations of expanding spherical flames. The true one-dimensional flame speeds in this case are known from planar flame modelling using the same kinetic scheme. The simple model predicted flame speeds within 2% of the true values for hydrogen/air mixtures over most of the stoichiometric range. This demonstrates that the extrapolation procedure is sound and will produce reliable results when applied to experimental data. Since the flame speeds derived from experiments are one-dimensional values, multiplying them by the density ratio gives one-dimensional burning velocities (s,'). Maximum burning velocities of hydrogen, methane, ethane, propane and ethylene mixtures with air were 2.85 ms-', 0.37 ms-', 0.41 ms-', 0.39 ms-' and 0.66 ms-' respectively. These are considerably smaller than most burner-derived values. The discrepancies can be explained by flow divergence and stretch effects perturbing burner measurements. The rate at which the measured flame speed approaches its limiting value depends on flame thickness and flame stretch. By subtracting the flame thickness term, the influence of flame stretch, expressed as the Markstein length, can be derived. Again values are given across the whole stoichiometric range of all fuels listed above, and form the most complete set of Markstein lengths reported to date. The Markstein lengths are negative in lean hydrogen and methane and in rich ethane and propane mixtures: this means that stretch increases the burning rate. They are positive in all other mixtures, showing that stretch decreases the burning rate. The results are in line with predictions based on Lewis number considerations. An alternative method of deriving one-dimensional burning velocities and Markstein lengths has been investigated. Burning velocities were measured at different stretch rates in flames in stagnation-point flow. Particle tracking was used to derive burning velocities referred to the hot side of the flame from the upstream values. The two burning velocities extrapolated to different one-dimensional values, both of which differed slightly from the expanding flame results. The suggested reason is that the upstream velocity gradient is not an accurate measure of the stretch experienced by the flame. Markstein lengths were consistent with those from the expanding flame method but the uncertainties were much larger. The method in its present form is therefore useful qualitatively but not quantitatively. To Frances, Michael and Jessica ACKNOWLEDGMENTS I would like to thank my supervisors, Professor Alan Williams of the University of Leeds and Dr. David Smith of the London Research Station, British Gas, for their support, advice and encouragement during the period of this research. Thanks are due to Dr. Chris Robinson for supplying the hydrocarbon oxidation scheme and for calculating pressure-dependent reaction rates, to Pierre Laugerette for printing Figures 6 and 12, and to the Photographic Section of Watson House Research Station, British Gas, for printing Figure 7. I would also like to thank present and former members of the Combustion Group at the London Research Station for many fruitful discussions on flames in general and on burning velocity and stretch in particular. Finally, my thanks go to my wife, Frances, and my children, Michael and Jessica, to whom this work is dedicated. Their help and support, particularly during the writing-up period, were much appreciated. Special mention must go to my children's ability to take my mind off flames so as to concentrate on really important matters like whether whales have teeth and which dress Lucy Teddy should wear. As a means of keeping a sense of proportion, this could not be bettered. CONTENTS Abstract ............................................................................................................................... ii Acknowledgments ................................................................................................................v List of Tables .......................................................................................................................x List of Figures ...................................................................................................................... x Nomenclature .................................................................................................................... xiv 1. INTRODUCTION ............................................................................................................1 1.1 A BRIEF HISTORY OF BURNING VELOCITY ............................................. 3 1.2 BURNING VELOCITY AND FLAME STRETCH .......................................... 9 1.3 THE ORGANIZATION OF THIS THESIS .....................................................12 2 . THEORY OF PREMIXED LAMINAR FLAMES ......................................................16 2.1 CONSERVATION EQUATIONS ...................................................................16 2.1.1 Derivation of the flame equations ..................................................... 17 2.1.2 Phenomenological analysis .................................................................. 23 2.1.2.1 Planar one-dimensional flame .................................................23 2.1.2.2 Stationary spherical flame........................................................ 26 2.1.2.3 Non-unity Lewis number ....................................................... 28 2.1.3 Asymptotics ............................................................................................31 2.1.3.1 Derivation of the model equations .......................................... 31 2.1.3.2 Asymptotic analysis ..................................................................37 2.2 FLAME STRETCH ............................................................................................ 46 2.2.1 Mathematical analysis ..........................................................................47 2.2.1.1 Extensional stretch ...................................................................52 2.2.1.2 Dilatational stretch .................................................................. 55 2.2.2 Phenomenological analysis ................................................................. 56 2.2.2.1 Planar flame in a stagnation-point flow ................................ 56 2.2.2.2 Expanding spherical flame .................................................... 60 2.2.3 Theories .................................................................................................66 2.2.4 The mean extensional stretch theorem ...............................................69 3 . EXPERIMENTAL METHODOLOGIES .......................................................................74 3.1 DEFINITIONS OF BURNING VELOCITY AND MARKSTEIN LENGTH......................................................................................................... 74 3.2 EXPANDING SPHERICAL FLAMES .......................................................... 78 3.2.1 Simple model of an expanding spherical flame ............................... 79 3.2.2 The effect of flame thickness ..................................... ... ..................... 83 3.2.3 Errors due to thermal radiation ...........................................................85 3.3 FLAMES IN STAGNATION-POINT FLOW .................................................. 92 3.3.1 Analysis of methodology ..................................................................... 93 3.3.2 Particle tracking errors ........................................................................ 97 3.4 BUTTON-SHAPED FLAMES ........................................................................ 102 4 . EXPERIMENTAL TECHNIQUES ...................................................................... 107 4.1 EXPANDING SPHERICAL FLAMES ........................................................... 107 4.1.1 Apparatus ............................................................................................ 107 4.1.2 Method ................................................................................................. 114 4.1.2.1 Calibration .............................................................................. 114 4.1.2.2 Gases ....................................................................................... 115 4.1.2.3 Experimental procedure ........................................................ 115 4.1.3 Data reduction ..................................................................................... 118 4.1.3.1 Derivation of Sb 0 and b ........................................................118 4.1.3.2 Experimental errors ................................................................120 4.2 STAGNATION FLOW FLAMES...................................................................