New, Accurate, Analytical Relations for Fractional Pressure Rise, Laminar Burning Velocity, and the Cubic Root Law in Constant Volume Combustion C.C.M. Luijten * , L.P.H. de Goey Eindhoven University of Technology, Section Combustion Technology, Department of Mechanical Engineering, PO Box 513, 5600 MB Eindhoven, The Netherlands Abstract In constant volume combustion, a simplified linear relation between pressure rise and burnt fraction x is often used. In the more accurate two- and multi-zone models, x is typically obtained by numerical solution of the volume and energy conservation laws. We show that it is possible to solve these laws analytically for x, resulting in a more accurate x(p) relation. Remarkably, our multi-zone result appears not to differ significantly from the two-zone result, indicating that the gradient in burnt temperature is not essential. Our new x(p) relation yields burning velocity values that differ significantly from the linear approach, and it also changes the expression for the explosion constant K G . * Corresponding author: c.c.m.luijten@tue.nl Proceedings of the European Combustion Meeting 2007 Introduction Experiments in which a combustible mixture is ignited in the centre of a constant volume vessel have a long history [1,2]. Their results are used for two main purposes. One of them is to collect data on the maximal pressure rise in confined explosions, often expressed in terms of the explosion constant K G , defined as (this is the so-called ‘Cubic Root Law’): 3 / 1 max V dt dp K G       ≡ . (1) The other main purpose of constant volume ‘bomb’ experiments is to obtain data on laminar burning velocities S L . Since pressure p and temperature T rise during the combustion event, data from one experiment can be used to collect S L values along a (typically isentropic) p,T-trajectory. Since the burning velocity appears as a parameter in the differential equation describing the pressure rise in the vessel, its experimental value is obtained by fitting the theoretical pressure history to the observed one. Next to burning velocity, a key role in the pressure equation is played by the burnt fraction x, expressing the mass fraction of the initial vessel content that has already burnt at time t. Since x fully determines the pressure rise, it is convenient to express x as a function of pressure, in order to use it directly in the differential equation (see section ‘State of the Art’). Although the functional form of x(p) has long been under development and debate, the form that is still most widely used dates back to Lewis and Von Elbe [3]. After a lengthy and not too enlightening derivation, they arrive at a linear relation reading i e i p p p p x - - = , (2) where p i and p e are the initial and end pressure values of the combustion event, respectively. Although many authors today agree that the derivation upon which (2) rests is questionable in some respects, the criticism was never made explicit. We have done so in a paper that was recently submitted to Combustion and Flame [4]. Although Lewis and Von Elbe are mostly cited in this field because of Eq. (2), the main result of the present paper – which differs from the linear approximation (2)! – is already implicitly contained in their 1934 work [2], as we have also demonstrated in Ref. [4]. This clearly proves that the pioneering quality of their contribution can hardly be overestimated. Specific Objectives In this paper we first briefly describe the state of the art in constant volume combustion modelling. After a summary of the general theory of constant volume combustion, we focus on different relations for x(p). Three main ‘schools’ can be distinguished: the Lewis and Von Elbe school, resulting in the linear relation (2); some intermediate results in (roughly) the third quarter of the last century; and finally the advent of zone-based modelling, in which all parameters during constant volume combustion are solved numerically by rigorous solution of the conservation laws of (specific) volume and energy. Then, we develop our analytical two-zone model, based on exactly the same starting points as the groups that have pioneered numerical two-zone modelling. Surveying the derivation of our two-zone model, there appears to be no reason prohibiting its extension to multiple zones. Although the resulting set of relations becomes more elaborate, we show that it is possible to provide analytical results for a multi-zone model, which also yield a realistic prediction for the temperature gradient in the burnt zone. We will compare our results to other analytical models available (including of course the linear one). For the example case of stoichiometric methane-air combustion, differences in obtained burning velocity values will be quantified. The sensitivity of our new model to some of its input parameters will be examined. Finally, we investigate the impact of our new result for x(p) on the value of the explosion constant K G in the Cubic Root Law, expressed by Eq. (1).