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Springer Aerospace Technology Subramaniam Krishnan Jeenu Raghavan Chemical Rockets Performance Prediction and Internal Ballistics Design Springer Aerospace Technology The Springer Aerospace Technology series is devoted to the technology of aircraft and spacecraft including design, construction, control and the science The books present the fundamentals and applications in all fields related to aerospace engineering The topics include aircraft, missiles, space vehicles, aircraft engines, propulsion units and related subjects More information about this series at http://www.springer.com/series/8613 Subramaniam Krishnan • Jeenu Raghavan Chemical Rockets Performance Prediction and Internal Ballistics Design 123 Subramaniam Krishnan Professor of Aerospace Engineering (Retired) Indian Institute of Technology Madras Chennai, Tamil Nadu, India Jeenu Raghavan Solid Propulsion Research Entity VSSC, Indian Space Research Organisation Thiruvananthapuram, Kerala, India ISSN 1869-1730 ISSN 1869-1749 (electronic) Springer Aerospace Technology ISBN 978-3-030-26964-7 ISBN 978-3-030-26965-4 (eBook) https://doi.org/10.1007/978-3-030-26965-4 © Springer Nature Switzerland AG 2020 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface From its preliminary design stage to the hardware realization and also its improvement beyond, performance prediction is invariably required for any dynamic system Development of chemical propellant rocket engines and motors is not an exception to this The purpose of this book is to discuss the methods of performance prediction for chemical rocket propulsion in a manner more rigorous than what is usually taught at undergraduate level The two parts of this book, “Performance Calculation of Chemical Propellants by Energy Minimization” and “Performance Prediction and Internal Ballistics Design of Solid Propellant Rocket Motors,” are based on the lectures that the first author delivered to practicing engineers and scientists at many professional development programs Also, under the course topic known as “Selected Topics,” he taught the methods presented here for many years to senior undergraduate and graduate students The material presented in either part of this book could be covered in about 20 lecture hours The course participants are expected to have undergone a first course in aerospace propulsion at the undergraduate level The limitations that arise out of the assumptions used in the prediction methods are discussed in detail It is hoped that this will motivate young researchers to improve upon the discussed methods Many examples are presented to aid in understanding the practical applications As the purpose is to train the readers in internal ballistics design, the solutions to these examples are rather long and may not fit into the type to be solved in short time duration The first part of this book deals with the rocket performance calculations of propellants Towards the computation of equilibrium composition of combustion products, the method of equilibrium constant is a classical one, and the same is detailed in many textbooks But, this method is found to be less amenable for generalized treatment with more difficulty in testing for the presence of condensed species Whereas condensed species are frequently present in the combustion products of propellants, more so for solid propellants As an alternative to the method of equilibrium constant, the method of energy minimization, relatively a recent one developed in the 1960s by Sanford Gorden and his group of NASA Glenn Research Center at Lewis Field, is found to remove this shortcoming The v vi Preface synoptic presentation of the energy minimization method and the listing of the related computer code CEC71 was published by the NASA in 1971 as SP273 The code and its subsequent improvements CEA and FCEA have been widely used as “black boxes” by rocket industries and educational institutions for the last so many years However, a pedagogical presentation of energy minimization method has so far not been available The first part of this book attempts to fill this gap with particular reference to rocket propulsion systems This part has seven chapters with sufficiently large number of worked examples showing the basic procedures adopted “inside” the CEC71 and its variants Chapter details the procedure for propellant selection along with the introduction to the related thermodynamic fundamentals Furthermore, it explains that an isolated system attaining maximum entropy for equilibrium is equivalent to it reaching minimum energy The intensive property chemical potential, the potent energy of a species to react in a chemical product mixture, is detailed in Chap For an operating condition given by a pair of constant state functions (temperature and pressure T &p, enthalpy and pressure h&p, entropy and pressure s&p, etc.), the equilibrium is reached for a product mixture when the sum of all such potential energies (chemical potentials) reaches a minimum Although many routes are available to evaluate the chemical potential, it is convenient to evaluate it based on Gibbs energy, and it is shown that the chemical potential of a species is its unit molar Gibbs energy with respect to product-mixture’s temperature and pressure For equilibrium, in addition to the energy minimization, the simultaneous satisfaction of the conservation of elements between the propellant and its products or between two operating conditions is mandatory, and this is dealt in Chap Adopting Lagrange multipliers, the governing equations, encompassing in tandem the energy minimization and the conservation of elements, are derived in Chap Since rocket propulsion units are analyzed under parametric values of pressure, applicable Newton-Raphson iteration equations are derived from the governing equations Following this, in order to lessen the iteration load, reduced iteration equations are obtained for computations Chapter ends with four worked examples, calculating equilibrium composition for all the three possible rocket operating conditions given by the pairs of constant state functions: T &p, h&p, and s&p After the determination of equilibrium composition, to proceed further in rocket performance calculations, we require the values of three key derivatives ∂ ln v ∂ ln T p , ∂ ln v ∂ ln p T , and the specific heat at constant pressure cp The determination of these is dealt in Chap These key derivatives assume different values for the reacting composition and the frozen composition and hence lead to the different values of rocket performance parameters Linked to this, against the conventional ratio of specific heats γ , the concept of the isentropic exponent γs for reacting composition is considered along with worked examples Preface vii The primary inputs required for the equilibrium-composition calculations are the energies contained in propellant ingredients and additives and the molar standardstate enthalpies Selected thermodynamic data of these are given in Chap Calculation of the molar specific heat at constant pressure c¯p , the standard state enthalpy, and the standard state entropy of product species through fourth-order polynomials is explained in this chapter In Chap 7, internal gas dynamics of liquid rocket engines and solid rocket motors is discussed along with equilibrium flow and frozen flow in nozzles Rocket performance parameters characteristic velocity c∗ , thrust coefficient CF , and specific impulse Isp under different operating conditions are discussed Examples to calculate rocket performance under equilibrium flow and frozen flow are solved The application of the three key derivatives in evaluating the rocket performance under equilibrium flow is explained The second part of this book deals with the performance prediction and internal ballistics of solid propellant rocket motors The subject matter is dealt in four chapters (Chaps 8–11) including a computer code In Chap 8, a brief introduction to the components of solid propellant rocket motor is given Next, the two basic methods of performance prediction, namely, equilibrium pressure analysis and incremental analysis, are introduced, and their applicability conditions are explained Equilibrium pressure analysis is detailed in Chap The required mass conservation equation and its variations during ignition transient, equilibrium operation, and tail-off transient are derived The importance of having the burning rate index less than unity for operational stability of the rocket motor is discussed Governing equations for the burning area progression for tapered cylindrical grains housed in a cylindrical casings are derived through an example In Chap 10, incremental analysis is discussed Related to this analysis, discussions on frozen flow versus shifting equilibrium flow and erosive burning are presented For the unsteady port flow, mass and momentum conservation equations are derived To get the governing equations for the steady port flow, the two unsteady equations are readily simplified by dropping the unsteady terms Solution procedures for steady port flow as well as unsteady port flow are explained along with examples Adopting the steady-flow incremental analysis, a FORTRAN program has been realized to predict the performance of solid propellant rocket motors having tapered cylindrical grains All the three phases of operation, namely, ignition transient, equilibrium operation, and tail-off transient, are included For easy readability and quick understanding of the program logic, the print version of the source code with detailed comments is given in Chap 11 The source code, typical examples along with their outputs, and an exe file of the code are stored in the digital link By running the code and also by developing the code for other grain configurations, the readers will get hands-on experience in the performance prediction and internal ballistics design of solid propellant motors viii Preface It is hoped that the book will fit the needs of the faculty for instruction and be useful to the young practicing engineers and scientists in the field of chemical rocket propulsion Chennai, India Thiruvananthapuram, India June 2019 Subramaniam Krishnan Jeenu Raghavan Contents Part I Performance Calculation of Chemical Propellants by Energy Minimization Introduction 1.1 Propellant Selection 1.2 Equation of State 1.3 Equilibrium 1.3.1 Entropy Principle 1.3.2 Equilibrium Conditions 1.4 Problems 3 7 13 16 Chemical Potential 2.1 Reacting Systems Versus Non-reacting Systems 2.2 Chemical Potential: An Intensive Property 2.3 Chemical Potential and Equilibrium 2.4 Evaluation of Chemical Potential 2.5 Recap on Chemical Potential and Equilibrium 2.6 Problems 19 19 22 24 25 28 35 Mass Balance 3.1 Problems 39 49 Iteration Equations 51 4.1 Lagrange Multipliers 54 4.2 Newton–Raphson Method 55 4.3 Constant Pressure Systems 56 4.4 Reduced Iteration-Equations 62 4.5 Problems 101 ix ... Springer Aerospace Technology ISBN 97 8-3 -0 3 0-2 696 4-7 ISBN 97 8-3 -0 3 0-2 696 5-4 (eBook) https://doi.org/10.1007/97 8-3 -0 3 0-2 696 5-4 © Springer Nature Switzerland AG 2020 This work is subject to copyright... engine © Springer Nature Switzerland AG 2020 S Krishnan, J Raghavan, Chemical Rockets, Springer Aerospace Technology, https://doi.org/10.1007/97 8-3 -0 3 0-2 696 5-4 _1 Introduction There are two methods... constant-temperature and -pressure Many reactions occur under the condition of constant-temperature and -pressure and hence this condition is very important Furthermore, it can be shown that if a chemical
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