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Journal of Engineering Mathematics 44: 5782, 2002. 2002 Kluwer Academic Publishers. Printed in the Netherlands. Modelling gas motion in a rapid-compression machine M.G. MEERE 1 , B. GLEESON 1 and J.M. SIMMIE 2 Department of Mathematical Physics, NUI, Galway, Ireland 2 Department of Chemistry, NUI, Galway, Ireland Received 25 July 2001; accepted in revised form 8 May 2002 Abstract. In this paper, a model which describes the behaviour of the pressure, density and temperature of a gas mixture in a rapid compression machine is developed and analyzed. The model consists of a coupled system of nonlinear partial differential equations, and both formal asymptotic and numerical solutions are presented. Using asymptotic techniques, a simple discrete algorithm which tracks the time evolution of the pressure, temperature and density of the gas in the chamber core is derived. The results which this algorithm predict are in good agreement with experimental data. Key words: gasdynamics, rapid-compression machines, shock-waves, singular perturbation theory 1. Introduction 1.1. RAPID-COMPRESSION MACHINES A rapid-compression machine is a device used to study the auto-ignition of gas mixtures at high pressures and temperatures, with particular reference to auto-ignition in internal combus- tion engines; see 13. A typical combustion engine is a very dirty and complex environment, and this has prompted the development of rapid-compression machines which enable the scientific study of compression and ignition in engines in a cleaner and simpler setting. In Figure 1 we schematically illustrate a two-piston rapid-compression machine, such as the one in the department of Chemistry at NUI, Galway. However, single-piston machines, with a piston at one end and a stationary solid wall at the other, are more typical. The analysis developed in this paper is appropriate to both single- and two-piston machines. The operation of a rapid-compression machine is very simple - the pistons are simul- taneously driven in pneumatically, compressing the enclosed gas mixture, thereby causing the gas pressure, temperature and density to rise quickly. In Figures 1(a), 1(b) and 1(c) we schematically represent a rapid-compression machine prior to, during, and after compression, respectively. The ratio of the final volume to the initial volume of the compression chamber for the machine at NUI, Galway is about 1:12, this value being typical of other machines. At the end of the compression, the gas mixture will typically have been pushed into a pressure and temperature regime where auto-ignition can occur. In Figure 2, we display an experimental pressure profile for a H 2 /O 2 /N 2 /Ar mixture which has been taken from Brett et al. 4, with the kind permission of the authors. In this graph, the time t = 0 corresponds to the end of the compression time. We note that, for the greater part of the compression, the pressure in the chamber is rising quite gently, but that towards the end of the compression (that is, just before t = 0), there is a steep rise in the pressure. After compression, the pressure profile levels off as expected; the extremely steep rise at the end of 58 M.G. Meere et al. Figure 1. Schematic illustrating the operation of a rapid-compression machine; we have shown the configuration (a) prior to compression, (b) during compression and (c) after compression. Figure 2. An experimental pressure profile for a gas mixture H 2 /O 2 /N 2 /Ar = 2/1/2/3, as measured in the rapid-compression machine at NUI, Galway. It is taken from 4, and has an initial pressure of 005 MPa and an initial temperature of 344 K. Modelling gas motion in a rapid-compression machine 59 the profile corresponds to the ignition of the mixture. We note that the compression time and the time delay to ignition after compression are both O(10) ms. The pressure history is the only quantity which is measured in experiments. However, the temperature in the core after compression is the quantity which is of primary interest to chemists since reaction rates depend mainly on temperature for almost all systems, although there may also be some weaker pressure dependence. Measuring temperature accurately in the core can be problematic because of the presence of a thermal boundary layer; see the comments below on roll-up vortices. However, with the experimental pressure data in hand, the corresponding temperatures can be estimated using the isentropic relation ln(p/p i ) = integraldisplay T T i (s) s((s) 1) ds, (1) where (T i ,p i ) are the initial values for the core temperature and pressure, (T,p) are these quantities at some later time, and (s)is the specific heat ratio at temperature s. In exper- iments, the initial core temperature is typically O(300 K), while the core temperature after compression is usually O(1000 K). In this paper, we shall consider only the behaviour of the gas mixture during compression; the post-compression behaviour is not considered here, but this will form the subject for future work. Nevertheless, the model presented here does provide a reasonable description of the post-compression behaviour of a single species pure gas, or an inert gas mixture; see Section 3.5. 1.2. THE MODEL We suppose that the compression chamber is located along 0
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