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About this book

The purpose of this book is to discuss, at the graduate level, the methods of performance prediction for chemical rocket propulsion. A pedagogical presentation of such methods has been unavailable thus far and this text, based upon lectures, fills this gap. The first part contains the energy-minimization to calculate the propellant-combustion composition and the subsequent computation of rocket performance.

While incremental analysis is for high performance solid motors, equilibrium-pressure analysis is for low performance ones. Both are detailed in the book's second part for the prediction of ignition and tail-off transients, and equilibrium operation.

Computer codes, adopting the incremental analysis along with erosive burning effect, are included. The material is encouraged to be used and presented at lectures. Senior undergraduate and graduate students in universities, as well as practicing engineers and scientists in rocket industries, form the readership.

Table of Contents

Frontmatter

Performance Calculation of Chemical Propellants by Energy Minimization

Frontmatter

Chapter 1. Introduction

Abstract
The procedure for propellant selection for specified chemical rockets is explained. Related thermodynamic fundamentals (equation of state, entropy principle, and chemical equilibrium conditions) are briefly discussed. That an isolated system attaining maximum entropy for equilibrium is equivalent to it reaching minimum energy is explained.
Subramaniam Krishnan, Jeenu Raghavan

Chapter 2. Chemical Potential

Abstract
The chemical potential, the potent energy of a species to react in a chemical product mixture, is introduced. The generalized definition for chemical potential is derived. Since the chemical potential does not change in its value when the method of its calculation differs, it is an intensive property. Chemical reactions occur under two distinct operating conditions, either at constant pressure or at constant volume. For the former the chemical potential is evaluated with respect to Gibbs energy and for the latter with respect to Helmholtz energy. For an operating condition given by a pair of constant state functions (temperature and pressure, enthalpy and pressure, and entropy and pressure, etc.), the chemical equilibrium is reached for a product mixture when the sum of chemical potentials of all species reaches a minimum. Reactions under given pressures are of interest in rocket propulsion systems. 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.
Subramaniam Krishnan, Jeenu Raghavan

Chapter 3. Mass Balance

Abstract
For equilibrium, in addition to the energy minimization, the simultaneous satisfaction of the conservation of elements between a propellant and its products or between two operating conditions is mandatory and this is dealt in this chapter. That is, the total kgatoms of an element between two operating conditions must be equal. For this mass conservation, necessary governing equation is given for a multi-element system involving many product compositions. Related worked examples for typical propellants and their products are added.
Subramaniam Krishnan, Jeenu Raghavan

Chapter 4. Iteration Equations

Abstract
Adopting Lagrange multipliers, the energy minimization and the conservation of elements are considered in tandem to derive the governing equations. 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. The 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: temperature and pressure, enthalpy and pressure, and entropy and pressure.
Subramaniam Krishnan, Jeenu Raghavan

Chapter 5. Thermodynamic Derivatives

Abstract
After the determination of equilibrium composition, to proceed further in rocket performance calculations we require the values of three key derivatives: change of volume with respect to temperature at constant pressure, change of volume with respect to pressure at constant temperature, and specific heat at constant pressure, which is the change of enthalpy with respect to temperature at constant pressure. The determination of these is dealt in this chapter. These key derivatives assume different values for the reacting composition and the frozen composition and hence lead to different values of rocket performance parameters. Linked to this, against the conventional ratio of specific heats γ, the concept of the isentropic exponent γ s for reacting composition is considered along with worked examples. Also a brief treatment on the use of Bridgman table is presented.
Subramaniam Krishnan, Jeenu Raghavan

Chapter 6. Thermodynamic Data

Abstract
The primary inputs required for the equilibrium-composition calculations are the energies contained in propellant-ingredients and -additives and the molar standard-state enthalpies. Thermodynamic data of many propellant-ingredients and -additives are given in this chapter. Adopting fourth order polynomials, the calculation of molar specific heat at constant pressure, the standard-state enthalpy, and the standard-state entropy of species is explained through worked examples. For selected condensed and gaseous species, the least square coefficients of the fourth order polynomials are presented.
Subramaniam Krishnan, Jeenu Raghavan

Chapter 7. Theoretical Rocket Performance

Abstract
Internal gas dynamics of liquid rocket engines and solid rocket motors with equilibrium flow and frozen flow is discussed. In evaluating the rocket performance under equilibrium flow, the application of the three key derivatives [ https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-26965-4_7/479745_1_En_7_IEq1_HTML.gif , https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-26965-4_7/479745_1_En_7_IEq2_HTML.gif , and c p] is explained. The performance parameters (characteristic velocity c , thrust coefficient C F, and specific impulse I sp) under the equilibrium flow and the frozen flow are discussed to show that the parameters assume different values for the two flows. To demonstrate the application of the knowledge gained, the design analysis on the internal ballistics and engine-envelope determination of proven engines, such as Space Shuttle Main Engine (SSME), is carried out. Also problems with answers are added on the design analyses of some proven engines: R-1E Marquardt, Mitsubishi LE-5B, and Vulcain.
Subramaniam Krishnan, Jeenu Raghavan

Performance Prediction and Internal Ballistics Design of Solid Propellant Rocket Motors

Frontmatter

Chapter 8. Introduction

Abstract
A brief introduction to the components of a typical solid propellant rocket motor is given. The division of the pressure-time trace of solid motor into ignition transient, equilibrium operation, and tail-off transient is explained. Next, the two basic methods of performance prediction, namely equilibrium-pressure analysis and incremental analysis, are introduced and their applicability conditions based on port-to-throat area ratios and volumetric loading fractions of propellant are discussed.
Subramaniam Krishnan, Jeenu Raghavan

Chapter 9. Equilibrium-Pressure Analysis

Abstract
The required mass conservation equation and its variations during ignition transient, equilibrium operation, and tail-off transient are derived. For the operational stability of a solid motor, the importance of having the burning rate index of the propellant less than unity is discussed. Calculations of ignition transients with and without nozzle closure, equilibrium operation, and tail-off transient are explained through worked examples. Governing equations for the burning area progression for tapered cylindrical grains housed in cylindrical casings are derived. The computation of the complete pressure-time trace of a typical motor having a tapered cylindrical grain is presented through a worked example.
Subramaniam Krishnan, Jeenu Raghavan

Chapter 10. Incremental Analysis

Abstract
Related to the incremental analysis, discussions on frozen flow versus shifting-equilibrium flow and erosive burning are presented. Lenoir–Robillard erosive burning rate model is discussed in detail along with an example. 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.
Subramaniam Krishnan, Jeenu Raghavan

Chapter 11. Computer Program

Abstract
Adopting the steady-flow incremental-analysis inclusive of erosive burning, 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 order to explain the program capability, the computed performance results of a typical solid rocket motor are presented. In addition, the program source code, the exe file, and examples together with their detailed outputs of six different motor configurations are made available through a suitable digital link.
Subramaniam Krishnan, Jeenu Raghavan

Backmatter

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