E. Wintenberger

Analytical Model for the Impulse of Single-Cycle Pulse Detonation Tube

An analytical model for the impulse of a single-cycle pulse detonation tube has been developed and validated against experimental data. The model is based on the pressure history at the thrust surface of the detonation tube. The pressure history is modeled by a constant pressure portion, followed by a decay due to gas expansion out of the tube. The duration and amplitude of the constant pressure portion is determined by analyzing the gasdynamics of the self-similar flow behind a steadily moving detonation wave within the tube. The gas expansion process is modeled using dimensional analysis and empirical observations. The model predictions are validated against direct experimental measurements in terms of impulse per unit volume, specific impulse, and thrust. Comparisons are given with estimates of the specific impulse based on numerical simulations. Impulse per unit volume and specific impulse calculations are carried out for a wide range of fuel–oxygen–nitrogen mixtures (including aviation fuels) of varying initial pressure, equivalence ratio, and nitrogen dilution. The effect of the initial temperature is also investigated. The trends observed are explained using a simple scaling analysis showing the dependency of the impulse on initial conditions and energy release in the mixture.

Direct Experimental Impulse Measurements for Detonations and Deflagrations

Direct impulse measurements were carried out by using a ballistic pendulum arrangement for detonations and deflagrations in a tube closed at one end. Three tubes of different lengths and inner diameters were tested with stoichiometric propane– and ethylene–oxygen–nitrogen mixtures. Results were obtained as a function of initial pressure and percent diluent. The experimental results were compared to predictions from an analytical model and generally agreed to within 15% (Wintenberger, E., Austin, J., Cooper, M., Jackson, S., and Shepherd, J. E., “Analytical Model for the Impulse of a Single-Cycle Pulse Detonation Engine, AIAA Paper 2001–3811, July 2001). The effect of internal obstacles on the transition from deflagration to detonation was studied. Three different extensions were tested to investigate the effect of exit conditions on the ballistic impulse for stoichiometric ethylene–oxygen–nitrogen mixtures as a function of initial pressure and percent diluent.

Model for the Performance of Airbreathing Pulse-Detonation Engines

A simplified flowpath analysis of a single-tube airbreathing pulse detonation engine is described. The configuration consists of a steady supersonic inlet, a large plenum, a valve, and a straight detonation tube (no exit nozzle). The interaction of the filling process with the detonation is studied, and it is shown how the flow in the plenum is coupled with the flow in the detonation tube. This coupling results in total pressure losses and pressure oscillations in the plenum caused by the unsteadiness of the flow. Moreover, the filling process generates a moving flow into which the detonation has to initiate and propagate. An analytical model is developed for predicting the flow and estimating performance based on an open-system control volume analysis and gasdynamics. The existing single-cycle impulse model is extended to include the effect of filling velocity on detonation tube impulse. Based on this, the engine thrust is found to be the sum of the contributions of detonation tube impulse, momentum, and pressure terms. Performance calculations for pulse detonation engines operating with stoichiometric hydrogen–air and JP10–air are presented and compared to the performance of the ideal ramjet over a range of Mach numbers.

Thermodynamic Cycle Analysis for Propagating Detonations

Propagating detonations have recently been the focus of extensive work based on their use in pulse detonation engines [1]. The entropy minimum associated with Chapman–Jouguet (CJ) detonations [2] and its potential implications on the thermal efficiency of these systems [3] has been one of the main motivations for these efforts. The notion of applying thermodynamic cycles to detonation was considered first by Zel’dovich [4], who concluded that the efficiency of the detonation cycle is slightly larger than that of a cycle using constant-volume combustion. More recently, Heiser and Pratt [3] conducted a thermodynamic analysis of the detonation cycle for a perfect gas using a one-γ model of detonations. Other studies have used constant-volume combustion as a surrogate for the detonation process [5]. This work presents two main contributions. First, we present an alternative physical model for the detonation cycle handling propagating detonations in a purely thermodynamic fashion. The Fickett–Jacobs (FJ) cycle is a conceptual thermodynamic cycle that can be used to compute an upper bound to the amount of mechanical work that can be obtained from detonating a given mass of explosive. Second, we present computations of the cycle thermal efficiency for a number of fuel-oxygen and fuel-air mixtures using equilibrium chemistry, and we discuss the strong influence of dissociation reactions on the results.

Stagnation hugoniot analysis for steady combustion waves in propulsion systems

The combustion mode in a steady-flow propulsion system has a strong influence on the overall efficiency of the system. To evaluate the relative merits of different modes, we propose that it is most appropriate to keep the upstream stagnation state fixed and the wave stationary within the combustor. Because of the variable wave speed and upstream stagnation state, the conventional Hugoniot analysis of combustion waves is inappropriate for this purpose. To remedy this situation, we propose a new formulation of the analysis of stationary combustion waves for a fixed initial stagnation state, which we call the stagnation Hugoniot. For a given stagnation enthalpy, we find that stationary detonation waves generate a higher entropy rise than deflagration waves. The combustion process generating the lowest entropy increment is found to be constant-pressure combustion. These results clearly demonstrate that the minimum entropy property of detonations derived from the conventional Hugoniot analysis does not imply superior performance in all propulsion systems. This finding reconciles previous analysis of flowpath performance analysis of detonation-based ramjets with the thermodynamic cycle analysis of detonation-based propulsion systems. We conclude that the thermodynamic analysis of propulsion systems based on stationary detonation waves must be formulated differently than for propagating waves, and the two situations lead to very different results.

THERMODYNAMIC ANALYSIS OF COMBUSTION PROCESSES FOR PROPULSION SYSTEMS

A key issue in conceptual design and analysis of proposed propulsion systems is the role of the combustion mode in determining the overall e‐ciency of the system. Of particular recent interest are detonations and the e‐ciency of detonation-based propulsion systems as compared to more conventional systems based on low-speed ∞ames. Our goal is to understand, based on thermodynamics, the merits of detonative combustion relative to de∞agrative combustion characteristic of conventional ramjet and turbojet engines. After reviewing detonation thermodynamics, we analyze the merits of detonations for steady ∞ow systems and highlight the importance of the irreversible portion of the entropy rise in steady ∞ow analysis. The conventional analysis of steady combustion waves is reformulated to obtain solutions at a flxed stagnation enthalpy. The implications of this analysis are that detonations are less desirable than de∞agrations for a steady air-breathing combustion system since they entail a greater entropy rise at a given ∞ight condition. This leads us to consider the situation for unsteady, i.e., intermittent or pulsed, combustion systems which use various modes of operation. For unsteady detonation waves, we consider a notional cyclic process for a closed system (the Fickett-Jacobs cycle) in order to circumvent the di‐culties associated with analyzing a system with time-dependent and spatially inhomogeneous states. We use the thermodynamic principles for closed systems to compute the maximum amount of mechanical work produced by a cycle using an unsteady detonation process. This ideal mechanical work is used to compute a thermal e‐ciency for detonations. Although this e‐ciency cannot be precisely translated into propulsive e‐ciency, the results are useful in comparing detonations with other combustion modes. We flnd that the e‐ciency of cycles based on detonation and constant-volume combustion are very similar and superior to a constant-pressure combustion (Brayton) cycle when compared on the basis of pressure at the start of the combustion process.

Erratum for "Analytical Model for the Impulse of Single-Cycle Pulse Detonation Tube"

I N the original evaluation of our analytical model for the singlecycle impulse of a pulse detonation tube,1 we approximated the detonation product isentrope as having a frozen composition with a corresponding polytropic exponent γ f . As discussed in the accompanying comment by Radulescu and Hanson and our response,2 for many situations, it is more appropriate to use an equilibrium approximation to the isentrope. This implies a different value of the polytropic exponent γ = γe and a new computational procedure for computing the plateau pressure P3 and results in revised values for the predicted impulse. Although the general equations and the qualitative conclusions drawn in our paper are unchanged, the revised numerical values of the predicted impulse differ up to 9.5% for stoichiometric fuel– oxygen mixtures and less than 1.3% for fuel–air mixtures at standard conditions. In this Errata, we present a revised set of data along with a short description of the calculations. The choice of the isentropic exponent, issues associated with chemical equilibrium, and the relevance to impulse calculations are discussed in the associated comment by Radulescu and Hanson and in our response to them. The input parameters of our impulse model consist of the external pressure P0, the detonation velocity UCJ, the equilibrium speed of sound behind the detonation front c2, the Chapman–Jouguet (CJ) pressure P2, and an approximation to the equilibrium polytropic exponent γe for the adiabatic expansion of the detonation products. All parameters were computed using numerical equilibrium calculations3 performed with a realistic set of combustion products. Instead of the analytic computation used in our original paper, our revised properties at state 3 (behind the Taylor wave) are now calculated by numerically integrating the Riemann invariant along the equilibrium isentrope until the plateau region of no flow is reached, ∫ P2

Introduction to "To the Question of Energy Use of Detonation Combustion" by Ya. B. Zel'dovich

Ya. B. Zel’dovich (1914–1987) made numerous contributions [1] to the theory of detonation, beginning with his very well known and widely translated article [2] on detonation structure that first introduced the standard Zel’dovich-von Neumann-Doring (ZND) model of shock-induced combustion. Even at that early stage of detonation research, Zel’dovich was also considering the application of detonations to propulsion and power engineering. He published these ideas in another paper [3] that has been virtually unknown in the West and has apparently remained untranslated until now. We are indebted to Sergey Frolov of the N.N. Semenov Institute of Chemical Physics for first bringing this article to our attention. We believe that the focus of this paper, which is the application of detonation waves to power generation and propulsion, is very relevant to the current activity on pulse detonation engines. In particular, Zel’dovich was apparently the first researcher to consider the questions of the relative efficiency of various combustion modes, the role of entropy production in jet propulsion, and the distinction between unsteady and steady modes of detonation in power engineering and propulsion applications. Even 60 years later, we believe that his results are relevant and can be of value in modern discussions on thermodynamic cycle analysis of detonation waves for propulsion [4]. For these reasons, we have arranged for the paper to be translated and suggested that it be published by the Journal of Propulsion and Power.

Reply to Comment on "Analytical Model for the Impulse of Single Cycle Pulse Detonation Tube" by M. I. Radulescu and R. K. Hanson

Response We used the polytropic approximation P ∼ ρ to model the isentrope in the detonation products in our original study in order to simplify the computation and develop analytic formulas for the impulse. Radulescu and Hanson (R&H) point out that for the stoichiometric ethylene-oxygen case, an equilibrium approximation to the isentrope is more realistic than the frozen approximation that is implied by our choice of polytropic exponent. We appreciate their observation, noting that we were aware of the significance of chemical reaction in the products, and our use of the frozen rather than equilibrium isentrope was an oversight on our part. We have recomputed our results (see the tables and plots in the Errata accompanying this response) for all cases finding that the choice of frozen versus equilibrium isentrope makes less than a 10% difference in the impulse in the most extreme cases, at most a 1.3% difference for fuel-air cases, and changes none of the qualitative conclusions of our study. Their comment raises three issues that we did not address in our original study. 1) Is the polytropic approximation reliable for equilibrium detonation products? 2) To what extent are the detonation products in equilibrium within the Taylor wave? 3) What is the appropriate choice for the polytropic exponent γ in our model? The polytropic approximation has been used extensively for studying the nonsteady flow in equilibrium detonation products and comparing computed blast and expansion waves with experimental data. The thermochemical basis of this approximation has been examined assuming “shifting” equilibrium in the products to compute the dependence of internal energy and molar mass on temperature and density for adiabatic flow. These studies demonstrate that there is a limited, but for our purposes useful, range of thermodynamic states over which the approximation of polytropic behavior is quantitatively reliable. The modest variations in density and temperature within the Taylor wave are favorable in this regard but caution is indicated if a wider range of parameters will be considered. In addition, the relationships between sound speed, pressure variation, and the polytropic exponents on the equilibrium isentrope are rather subtle and more involved than indicated in the ∗Post-doctoral Scholar, Aeronautics, Caltech MC 205-45, Pasadena, CA 91125, Member AIAA †Graduate Student, Mechanical Engineering, Caltech MC 205-45, Pasadena, CA 91125, Member AIAA ‡Graduate Student, Applied Physics, Caltech MC 205-45, Pasadena, CA 91125 §Professor, Aeronautics, Caltech MC 105-50, Pasadena, CA 91125, Member AIAA