Solid Propellant Rocket Motors - Professor … Starting from the preliminary design stage to the development of designed hardware and also beyond, performance prediction of propulsion

Solid Propellant Rocket Motors:

Performance Prediction and Internal Ballistics

Design

S. Krishnan

Professor of Aerospace Engineering (Retired)

Indian Institute of Technology Madras

Chennai, India

November 2016

Contents

List of Figures xii

List of Tables xv

1 Introduction 1

1.1 Solid Rocket Motor Components . . . . . . . . . . . . . . . 1

1.2 Pressure-Time Trace . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Methods of Performance Prediction . . . . . . . . . . . . . . 5

2 Equilibrium Pressure Analysis 11

2.1 Assumptions in Equilibrium Pressure Analysis . . . . . . . 11

2.2 Mass Conservation Equation . . . . . . . . . . . . . . . . . 14

2.3 Operational Stability of Rocket Motor . . . . . . . . . . . . 18

2.4 Prediction of Pressure-Time Trace . . . . . . . . . . . . . . 21

2.4.1 Ignition Transient . . . . . . . . . . . . . . . . . . . 21

2.4.2 Equilibrium Operation . . . . . . . . . . . . . . . . . 25

2.4.3 Tail-off Transient after complete burnout . . . . . . 44

3 Incremental Analysis 61

3.1 Frozen Flow Versus Shifting Equilibrium Flow . . . . . . . . 61

3.2 Incremental-Analysis Procedure . . . . . . . . . . . . . . . . 67

3.3 Assumptions in Incremental Analysis . . . . . . . . . . . . . 70

3.4 Erosive Burning . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.5 Incremental-Analysis Equations . . . . . . . . . . . . . . . . 81

3.5.1 Unsteady Flow Equations . . . . . . . . . . . . . . . 81

3.5.2 Steady Flow Equations . . . . . . . . . . . . . . . . . 85

3.5.3 Solution of Steady Port-Flow . . . . . . . . . . . . . 86

3.5.4 Solution of Unsteady Port-Flow . . . . . . . . . . . . 95

i

ii CONTENTS

4 Computer Program 107

4.1 Computer Program . . . . . . . . . . . . . . . . . . . . . . . 107

4.1.1 Main Program Steadyfull . . . . . . . . . . . . . . . 107

4.1.2 Subprogram Propellant . . . . . . . . . . . . . . . . 112

4.1.3 Subprogram Prsrratio . . . . . . . . . . . . . . . . . 112

4.1.4 Subprogram Geometry . . . . . . . . . . . . . . . . . 112

4.1.5 Subprogram Segsteady . . . . . . . . . . . . . . . . 112

4.1.6 Subprogram Erosive . . . . . . . . . . . . . . . . . . 113

4.1.7 Subprogram Starttransienteql . . . . . . . . . . . . . 113

4.1.8 Subprogram Falci . . . . . . . . . . . . . . . . . . . . 113

4.1.9 Error Messages . . . . . . . . . . . . . . . . . . . . . 113

4.1.10 Outputs . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.1.11 Sample Problems . . . . . . . . . . . . . . . . . . . . 115

Problem01 . . . . . . . . . . . . . . . . . . . . . . . 115

Problem02 . . . . . . . . . . . . . . . . . . . . . . . 115

Problem03 . . . . . . . . . . . . . . . . . . . . . . . 115

Case10 . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Case11 . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Case13 . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.1.12 Sample Output: Case13 . . . . . . . . . . . . . . . . 117

Computer Output for Case13 . . . . . . . . . . . . . 118

4.2 Source Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

About the author

Subramaniam Krishnan received degrees in mechanical engineering from

the University of Madras (B.E., 1964) and the Indian Institute of Science

Bangalore (M.E., 1968), and in aerospace engineering from the Indian

Institute of Technology Madras (Ph.D., 1976). He was a research assistant

at Gas Turbine Research Establishment, Bangalore from 1964 to 1966.

He joined the Indian Institute of Technology Madras (IIT Madras) as a

Lecturer in 1968 and rose to the position of Professor in 1987. He retired

from IIT Madras in 2002. Professor Krishnan taught a wide variety courses

in aerospace propulsion from 1968 to 2013. His research interest lies in solid

propulsion: solid rockets, solid fuel ramjet, and hybrid rockets. He has

several publications to his credit in these areas. He was a visiting professor

at the Israel Institute of Technology (1999-2000), Kyungpook National

University, South Korea (2002-2005), Universiti Teknologi Malaysia (2008-

2011), and SRM University Madras (2005-2008 and 2011-2013). He is an

Honorary Fellow of the High Energy Materials Society of India, a Fellow

of the Aeronautical Society of India, and an Associate Fellow of American

Institute of Aeronautics and Astronautics.

iii

iv CONTENTS

Preface

Starting from the preliminary design stage to the development of designed

hardware and also beyond, performance prediction of propulsion systems

is invariably required. Development of solid propellant rocket motors is

not an exception to this. Practicing engineers and scientists in space and

defence R&D centres involved in solid rocket development have to predict

the motor performance during the development and beyond. The material

compiled here was discussed by the author in many short term courses

conducted for them. Also, under the course known as “Selected Topics”

the methods presented here were taught to graduate and senior under-

graduate students for about 20 class-hours including practicing sessions.

The course participants were expected to have undergone a first course in

aerospace propulsion at the under graduate level.

The limitations that arise out of the assumptions used in the prediction

methods are discussed in detail here. 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-durations.

In Chapter 1, a brief introduction to the components of solid propel-

lant 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 Chapter 2. 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

v

vi CONTENTS

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 Chapter 3 incremental analysis is discussed. Related to this analy-

sis, 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 procedure for steady port flow as

well as unsteady port flow are explained along with examples.

Adopting steady-flow incremental-analysis, a FORTRAN program has

been realized to predict the performance of solid propellant rocket mo-

tors 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.

The source code, typical examples along with their outputs, and an exe file

of the code are stored in the attached USB-device. 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.

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 solid rocket technology.

Subramaniam Krishnan

Chennai (Madras)

Nomenclature

A = area, m2

a = pre-exponent factor in the burning rate equation, r = apn

CF = thrust coefficient

C0F = characteristic thrust coefficient

cp = specific heat at constant pressure, J/kg-K

cs = specific heat of solid propellant, J/kg-K

c∗ = characteristic velocity, m/s

D = characteristic dimension; hydraulic diameter, m

d = diameter, m

F = thrust, N

G = mass flux, kg/m2 − s

h = convective heat transfer coefficient, J/s−m2 −K; height, m

K = ratio of grain burning area to nozzle throat area

k = proportionality constant in Lenoir-Robillard erosive burning model,

vii

viii CONTENTS

m3K/J

L = length, m

l = length, m

M = Mach number

m = mass flow rate, kg/s

m = molar mass with respect to gaseous species, kg/kg-mole

mc = molar mass with respect to gaseous as well as condensed species,

kg/kg-mole

NG = number of gaseous species

NS = total number of gaseous and condensed species

n = burning rate index in the burning rate equation ; number of kg-moles

of species

Pr = Prandtl number

p = pressure, Pa

R = specific gas constant, J/kg-K

Re = Reynolds number

Ru = universal gas constant, 8314.51J/kg-mole-K

r = radius, m

r = burning rate, m/s

CONTENTS ix

T = temperature, K

s = burning perimeter, m; slant length of cone, m; entropy, J/(kg-K)

t = time, s

u = velocity, m/s

V = volume, m3

v = specific volume, m3/kg

x = mole fraction

y = burned distance, m

Subscripts

0 = stagnation condition; zero crossflow velocity

a = ambient condition

b = burning; burnout

c = combustion chamber

E = equilibrium operation

e = nozzle- exit plane; erosive burning condition

f = final

h = head end

x CONTENTS

I = ignition transient

i = initial

j = jth station in incremental analysis; jth species

N = nozzle entry

m = mean value

n = nth step or nth iteration; nozzle end

p = propellant; port

R = provision of nozzle rupture disc or nozzle plug

s = sliver; surface

T = tail-off

t = throat section of nozzle

Greek letters

α = erosive constant in Lenoir-Robillard erosive burning model, m2.8/(kg0.8s0.2)

β = dimensionless exponential constant in Lenoir-Robillard erosive burn-

ing model

Γ =√γ[2/(γ + 1)](γ+1)/[2(γ−1)] function of specific-heats ratio

γ = specific-heats ratio

γs = isentropic exponent

CONTENTS xi

ε = erosive burning ratio

θ = grain-port taper angle, deg.

λ = thermal conductivity of combustion products, J/(s-m-K)

µ = viscosity of combustion products, kg/m-s

ρ = density, kg/m3

xii CONTENTS

List of Figures

1.1 A schematic sketch of a solid propellant rocket motor. 1

1.2 A typical pressure versus time of a rocket motor. . . 4

1.3 Incremental control volume. . . . . . . . . . . . . . . . 8

2.1 Operational stability of a solid propellant rocket mo-

tor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Arrangements of nozzle closures. . . . . . . . . . . . . 23

2.3 Typical pressure-time traces of solid propellant mo-

tors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Effect of calculations with different time steps adopt-

ing quasi steady-state assumption (schematic). . . . 26

2.5 Plot of equilibrium pressure for propellants with two

different values of burning rate index n. cylindrical

grain length = 140mm; dpi = 10mm; dt = 5mm;

r = 1.8 ∗ 10−5pnm/s; p[Pa]. . . . . . . . . . . . . . . . . . . 27

2.6 Comparison of ignition transients with and without

a nozzle closure. . . . . . . . . . . . . . . . . . . . . . . . 34

2.7 Schematic diagram of the rocket motor of Example

2. 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.8 Schematic sketch of the solid propellant thruster of

Example 2. 3. . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.9 Tail-off transient of the rocket thruster of Example

2. 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.10 Schematic sketch of the rocket motor of Example 2.

5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.11 Pressure-time trace of the rocket motor of Example

2. 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

xiii

xiv LIST OF FIGURES

3.1 Incremental stations fixed with respect to a rocket

chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2 Deflection (exaggerated) due to static pressure fall

along the port of a pure cylindrical grain of low

structural strength. . . . . . . . . . . . . . . . . . . . . . 73

3.3 Typical incremental segment with flow conditions. . 81

List of Tables

2.1 Ignition transient with nozzle closure from an initial

pressure of 1 bar to 75 bar, Example 2. 1 . . . . . . . 31

2.2 Ignition transient after the nozzle closure is relieved

at 75 bar, Example 2. 1 . . . . . . . . . . . . . . . . . . 32

2.3 Ignition transient without nozzle closure from the

minimum choking pressure of 1.838 bar, Example 2.

1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 Iterative calculations for the initial equilibrium pres-

sure for Example 2. 3 . . . . . . . . . . . . . . . . . . . 43

2.5 Iterative calculations for the burnout equilibrium

pressure for Example 2. 3 . . . . . . . . . . . . . . . . . 44

2.6 Tail-off transient-calculation results, Example 2. 4 . 46

2.7 Grain characteristics of cylindrical segment . . . . . . 48

2.8 Ignition transient, Example 2. 5 . . . . . . . . . . . . . 53

2.9 Equilibrium operation during the first phase burn-

ing: pressure versus burned-distance, Example 2. 5 55

2.10 Equilibrium operation during second phase burning

(sliver burning) pressure versus burned-distance, Ex-

ample 2. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.11 Ignition transient: pressure-time trace . . . . . . . . . 59

2.12 First phase of equilibrium operation: pressure-time

trace, Example 2. 5 . . . . . . . . . . . . . . . . . . . . . 59

2.13 Second phase of equilibrium operation (tail-off tran-

sient): pressure-time trace . . . . . . . . . . . . . . . . 60

3.1 Results of a typical shifting equilibrium flow calcu-

lation using program CEC71 . . . . . . . . . . . . . . . 64

xv

xvi LIST OF TABLES

3.2 Typical iteration results for the unsteady flow through

grain port segmental volume, Example 3. 4 . . . . . 102

3.3 Typical iteration results for the steady flow through

grain port segmental volume, Example 3. 4 . . . . . 105

Chapter 1

Introduction

1.1 Solid Rocket Motor Components

A schematic sketch of a solid propellant rocket motor is given in Figure 1.1.

Solid propellant rocket motors are generally of cylindrical configuration

with propellant grains having central perforations. The major components

of the motor are: (i) the igniter, which is to initiate combustion of the

solid propellant grain, (ii) the grain, which is of a structurally-strong

solid-chemical that burns on ignition to produce hot combustion products,

mostly of gaseous species by volume, (iii) the nozzle that accelerates the

high pressure and high temperature combustion products to high velocities

to produce thrust, and (iv) the motor casing that accommodates the

grain and the igniter.

Figure 1.1: A schematic sketch of a solid propellant rocket motor.

Solid propellant motors can be classified in many ways. Based on

the casing geometry solid propellant motors can be grouped as cylindrical

1

2 CHAPTER 1. INTRODUCTION

or spherical ones. Cylindrical motors are the commonly adopted ones

while spherical configurations are adopted in special situations such as in

upper stage propulsion. Also based on the grain-burning pattern we can

characterize the motors as radial burning or end burning ones. We are

considering in Figure 1.1 a cylindrical motor with a centrally perforated

radial-burning grain. Most grain designs aim to protect the inner wall of

the motor casing against the high temperature combustion products. Thus,

a precast pure-cylindrical grain is slid to fit into the motor casing in the case

of free standing grains. Or, the grain, cylindrical or other configuration,

is cast and cured into the casing and hence bonded to the inner-wall in

the case of case-bonded grains. In both these types there is a central

perforation or the port cavity through which the combustion products

flow from the head end towards the nozzle end. The combustion-

chamber cavity of the rocket motor includes the port volume from

head end to nozzle end, the head end free volume, if any, and the

plenum chamber, and also usually the nozzle convergence.

The motor casing and the nozzle are structural components designed to

withstand high operating-pressures and steep pressure-rise rates. In addi-

tion, these components are to be protected from high temperature combus-

tion products by adopting suitable high-temperature resistant insulators

and liners. The nozzle inner-surfaces are exposed to high temperature

combustion products for the entire time of motor operation. For the cas-

ing walls, there are certain areas exposed to high temperature products for

the entire time of operation while the remaining areas are exposed only

after burnout. With reference to the schematic sketch shown in Figure

1.1, the areas exposed to the high temperature combustion products for

the entire time of operation are the areas around the igniter at the head

end and the portion in the plenum area at the nozzle end. In contrast, the

cylindrical part of the casing wall with the propellant grain is exposed to

high temperature products only after the grain burnout. Accordingly the

thickness of the insulation is chosen as per the severity of exposure to the

high temperature combustion products. The liner, usually thin, acts as an

interface between the insulator and the propellant grain. By the design of

a grain, certain areas of the grain are to be prevented from burning and

hence those areas are applied with inert chemicals known as inhibitors,

Figure 1.1.

1.2. PRESSURE-TIME TRACE 3

The theoretical-performance predication of rocket-motor basically an-

swers the question: What is the expected thrust F versus time t of the de-

signed motor? Under the assumption of one dimensional isentropic nozzle-

flow, we have learned that the thrust is given by,

F = p0NAtCF = p0NAt

{C0F +

Ae

At

(pe

p0N−

pa

p0N

)}(1.1)

F =

p0NAt

Γ

√√√√√√√√√√2γ

γ − 1

1−

(pe

p0N

)γ − 1

γ

+Ae

At

(pe

p0N−

pa

p0N

)

(1.2)

We note from Eqs. 1.1 and 1.2, that for a fixed-configuration rocket-

motor (nozzle throat area At and nozzle exit area Ae are constant), so long

as its nozzle flows full without separation, the thrust F depends on the

nozzle entry total pressure p0N and the flight altitude that has its ambient

pressure of pa . All other quantities are essentially constant. It follows

therefore, to predict the performance of the rocket motor at a specified

altitude of ambient pressure pa , we should first predict the value of nozzle

entry pressure p0N with respect to the motor operating-time.

1.2 Pressure-Time Trace

A schematic pressure-time trace of a solid rocket motor is given in Figure

1.2. The pressure time trace can be broadly divided into (i) ignition

transient or start transient, (ii) equilibrium operation, and (iii) tail-

off transient.

The prediction of ignition transient considers the phenomena that occur

in the chamber cavity from the initiation of igniter (“countdown at 0”) to

the attainment of the first equilibrium pressure. For this prediction the

ignition transient can be subdivided into three intervals, namely, ignition

delay, flame spreading, and chamber filling. The ignition delay is the


Figure 1.2: A typical pressure versus time of a rocket motor.

time elapsed from the initiation of igniter to the appearance of the first

discernible flame on the grain surface. The flame spreading interval is that

between the instant of first flame appearance and the instant at which the

entire grain surface attains complete ignition. And, the chamber filling

interval is that between the instants of the complete ignition and the first

equilibrium pressure.

During the ignition transient the initially-produced burned-products,

from the igniter as well as from the grain surface, mostly fill the cavity to

build up the chamber pressure at a rapid rate while some products exit

through the rocket nozzle. Soon after a successful flame spread is estab-

lished, the igniter burns out. On the continuance of the chamber filling,

the filling rate into the cavity decreases with time with a simultaneous in-

crease in the rate of ejection of mass flow through the nozzle throat. This

culminates with the attainment of the first equilibrium pressure, at which

the rate of change of chamber pressure is substantially reduced to negligi-

ble levels. At this pressure, consequently, the rate of production of burned

products is almost equal to the rate of ejection through the nozzle, and

only a very small portion of the products goes to fill the volume vacated

by the burning propellant grain as the rate of mass accumulation within

the cavity.

The mass balance condition that the rate of production of burned prod-

ucts is almost equal to the rate of ejection through the nozzle continues

1.3. METHODS OF PERFORMANCE PREDICTION 5

for the duration of the equilibrium operation. During this operation, the

burning grain-surface recedes exposing the areas as per the designed grain

configuration. For example for the grain shown in Figure 1.1, the grain

burning area increases with the burned distance. This equilibrium opera-

tion goes on until the flame front first touches the liner of the casing with

some portion of the propellant grain still left to be consumed. This in-

stant is known as burnout and the left-out part of propellant grain after

burnout is known as sliver. The ratio of the sliver mass at burnout to the

total-propellant mass is known as the sliver fraction.

The tail-off transient starts from the burnout. During the tail-off, the

sliver burns with its burning area reducing at a very fast rate. This tran-

sient ends when the entire sliver is consumed. This zero sliver condition

is known as complete burnout. This is followed by the cavity pressure

dropping to the ambient pressure. Ordinarily, however, the cavity pressure

drops to ambient pressure at the complete burnout.

The sole purpose of a rocket motor in a rocket vehicle is the delivery

of the desired thrust-variation with respect to time. In most instants, this

is served by the equilibrium operation and therefore the designer usually

aims to have the ignition transient and the tail-off transient as short and

as smooth as possible.

1.3 Methods of Performance Prediction

Before looking at the methods of performance prediction to be presented,

let us consider the realities of the flow in the port cavity of a solid rocket

motor.

1. The regression rate or the burning rate of propellant is found to be

principally affected by local static pressure. In addition, the burn-

ing rate can have dependence on centrifugal acceleration introduced

through rocket-motor spin and other fluid dynamic properties such

as mass flux ρu and Mach number M .

2. Chemical reactions occur at the regressing grain-surface within a

thin combustion zone (of the order of 100µm or less under rocket

operating pressures) to introduce hot combustion products into the

port.


3. The mixture of combustion products contains a number of gaseous

species in addition to some species in liquid and solid phases. Con-

centrations of these species may change along the port flow depending

on the local static pressure and temperature, and the velocity.

4. Admitting that the flow in the port cavity can be three dimensional,

in most situations it is two dimensional (axisymmetric). In addition,

the port-flow is generally unsteady and turbulent.

5. Heat is transferred from the high-temperature port-flow to the burn-

ing propellant-surface by way of convection and radiation.

6. Furthermore, the burning surface that blows hot combustion prod-

ucts offers some frictional resistance for the port flow.

7. Structural deformation of the grain and the casing can occur due to

the high rates of pressure variations during the ignition and tail-off

transients as well as due to the spatial variation of static pressure in

the grain port cavity during the equilibrium operation.

8. Gravitational forces may become significant to affect the port as

well as nozzle flow-dynamics when the rocket vehicle is under high

acceleration or deceleration.

Modelling the above realities to various degrees of accuracy, and em-

ploying methods of computational fluid dynamics and finite-element struc-

tural analysis, complex computer codes do exist in industries and research

establishments. However, in the present treatment we will learn two basic

methods, which can be used to predict solid rocket performance within a

reasonable accuracy. These methods are invariably adopted for the first-

cut sizing of the motors to be developed.

The two basic methods by which we can predict the performance of a

solid rocket motor are (i) the equilibrium pressure analysis and (ii)

the incremental analysis. The equilibrium pressure analysis is relatively

a simple procedure. This analysis can be adopted for the low performance

motors, which have low maximum-velocities of combustion products in

the combustion chamber cavity. The velocities are so low such that the

differences between the total- and static-values of pressure anywhere in the

port cavity are not significant.


Let us find the parameter that characterizes a low performance mo-

tor for which equilibrium pressure analysis can be employed. Consider a

cylindrical motor with a centrally perforated radial-burning grain as shown

in Figure 1.1. Having got accumulated all along the central perforation,

namely the port, the total burned products exit the nozzle-end-port with a

maximum port-velocity and enter the plenum to get subsequently choked

at the nozzle throat. At this point let us define the ratio of nozzle-end-port

exit area to the nozzle throat area as port-to-throat area ratio. Since

the velocity is the one that determines the difference between the total and

static values of pressure and temperature, a high value of port-to-throat

area ratio will have a low port flow velocity. Deliberately ignoring the total

pressure loss that can occur due to sudden area variation from the port

exit to the plenum and hence assuming an isentropic flow from the port

exit to the nozzle throat and adopting the equation of continuity, we can

show that the port-to-throat ratio

Api

At=

1

Mi

[(2

γ + 1

)(1 +

γ − 1

2M2i

)] γ + 1

2 (γ − 1)(1.3)

where Api and Mi are the initial values of the port-exit area and the cor-

responding Mach number respectively. The ratio of specific heats for the

combustion products of typical solid propellants vary in the range of 1.15

to 1.30. For this range, the port exit Mach number of 0.2 or less has the

difference between total- and static-pressures less than 3 per cent. This in

turn corresponds to the port-to-throat ratios of 3 or more. The low perfor-

mance motors, therefore, are characterized by the high port-to-throat area

ratio of values greater than about 3 to 4. Low performance motors are

also characterized by low volumetric loading fraction of propellant

(Vpi/Vcf ) of less than around 0.75. Here Vpi is the initial propellant vol-

ume and Vcf is the empty chamber volume up to the nozzle throat section.

However, more importantly, rather than the low volumetric loading frac-

tion of propellant, it is the high port-to-throat ratio that characterizes the

low performance motors for which equilibrium pressure analysis could be

applied. In such motors, because of low port-velocities, a uniform constant

total-pressure can be assumed for the entire chamber cavity.


Since the equilibrium pressure analysis assumes a single uniform pres-

sure at a chosen time and the only variable is time, the corresponding

model followed is known as lumped chamber pressure model, p (t)

model, or zero dimensional (0−D) model.

Figure 1.3: Incremental control volume.

While the equilibrium pressure analysis is of zero-dimensional model,

the incremental analysis, the second method, adopts a one dimensional

model. As expected, it is more rigorous and involved when compared to

the first one, and is adopted for the high performance motors, characterized

by low port-to-throat area ratios of less than 3 to 4 and high volumetric

loading fractions of propellant. In such motors the average velocity of

combustion products inside the grain-port is high and the total- and static-

pressures are substantially different at most places along the port. And,

the two pressures significantly drop along the port from the head end to

the nozzle end. Under the incremental analysis, the port flow is evaluated

one dimensionally by considering a sufficiently large number of incremental

segments or incremental control-volumes along the port, Figure 1.3. Each

incremental control volume is bounded by the entry- and the exit-section of

the flow that are fixed in space, and the receding burning-surface of grain.

For a considered duration, there is an addition of mass of burned products

into the control volume from the burning surface at the rate dmj with a


simultaneous accumulation rate of burned products into the free volume

created by the receding burning surface (dm/dt)j . Under the assumption

of constant total enthalpy (thus energy conservation being automatically

satisfied), the flow through each incremental control-volume is analysed

applying equations of mass- and momentum-conservation, and ideal gas

equation of state.

Depending upon the complexity to be involved, the incremental anal-

ysis gets further classified as (i) the quasi steady flow model or p (x)

model and (ii) the temporal 1−D flow field model or p (x, t) model.

In the former, applicable with a moderate rate of chamber-pressure vari-

ation dp/dt, the mass-accumulation rate in the free volume (dm/dt)j cre-

ated by the propellant regression is considered negligible compared to the

mass-addition rate due to burning dmj . And, in the p (x, t) model the

mass accumulation-rate (dm/dt)j is significant with respect to the mass-

addition rate dmj . Mostly p (x) model is adopted during equilibrium op-

eration while the p (x, t) model is employed during ignition-transient and

tail-off-transient where the moduli of dp/dt are quite substantial.


Chapter 2

Equilibrium Pressure

Analysis

2.1 Assumptions in Equilibrium Pressure Analy-

sis

As indicated previously, the main assumption of the equilibrium pressure

analysis is that at a chosen instant there is one uniform equilibrium-total-

pressure p0c throughout the combustion-chamber cavity - hence the name

equilibrium pressure analysis. Thus p0c = p0N in Eqs. 1.1 and 1.2. Fur-

thermore, the velocities in the port volume and plenum are so small that

the differences between the static- and the total-properties (pressure, tem-

perature, and density) are negligible. The other assumptions made to

simplify the analysis are:

1. The port cavity of rocket motor is an adiabatic system with a con-

stant stagnation enthalpy. The combustion products are of fixed

composition and constant specific heats, and hence the specific heats

ratio γ is constant. Consequently the temperature of the combus-

tion product-mixture is at one uniform total-temperature T0, equal

to the adiabatic flame temperature at a chosen average equilib-

rium chamber-pressure.

2. The combustion products satisfy the ideal gas equation of state. The

inert materials in the combustion-chamber cavity, namely the insula-

tor and the liner do not decompose during the motor operation and

11

12 CHAPTER 2. EQUILIBRIUM PRESSURE ANALYSIS

hence do not form part of the nozzle flow. At an instant, the mass

of inhibitor fragmentation is negligible with respect to the mass of

combustion products from the propellant grain.

3. The nozzle of the rocket motor is always choked. The ignition of the

entire grain surface is instantaneous with negligible mass of igniter.

After the complete burnout, the calculation of the tail-off transient

is reckoned from zero sliver.

4. The propellant grain and the motor casing are infinitely rigid and

hence do not deform during the motor operation.

Let us consider the implication of the first simplifying assumption.

Depending on the grain design the equilibrium chamber-pressure may sub-

stantially vary during equilibrium operation. For the given propellant com-

position the stagnation enthalpy, representing the total energy content in

the propellant, is constant. However, with a change in chamber pressure,

the stagnation temperature can vary − for example an increased cham-

ber pressure gives higher adiabatic flame temperature. Specific heats and

hence T0 being weak functions of pressure, the assumption of constant T0

and γ for equilibrium operation is adequate. Therefore, using a computer

program similar to CEC71 [Ref.[1]] or the recent updated version CEA

Program [Ref. [2]] that can be downloaded from the web site of NASA

Glenn Research Center

www.grc.nasa.gov/WWW/CEAWeb/ceaguiDownload-win.htm

the values of T0 , molar mass m, and γ can be determined for an average

equilibrium chamber-pressure and held constant.

However, this assumption of constant property-values for transients

may be questionable in certain cases. During the ignition transient, the

temperature increases from a near-atmospheric value to T0. The value of

γ may change as the composition of the igniter combustion-products is

different from that of the products of the propellant combustion. Further-

more, on the initiation of igniter, the igniter combustion products mix with

the air contained in the port cavity to expel it first. During the tail-off

transient, reckoned from zero sliver of complete burnout, the temperature

drops from T0 to a value of a few hundred degrees warmer than atmo-

spheric value. And, the value of γ may slightly increase during tail-off.

Thus the first assumption of constant values of T0, m, and γ is a drastic


2.1. ASSUMPTIONS IN EQUILIBRIUM PRESSURE ANALYSIS 13

approximation as far as transients are concerned. However the effective

periods of ignition transient and tail-off transient being not appreciable

with respect to the period of equilibrium operation, the approximation is

acceptable in most cases.

Let us look at the second simplifying assumption. Most heteroge-

neous propellants (popularly known as composite propellants) in

use and also sometimes homogeneous propellants (popularly known as

double base propellants) contain metal powders as one of the fuel com-

ponents (usually of aluminium). On combustion, oxides of this fuel remain

in liquid- or solid-phase or sometimes in both the phases. These condensed

particles in the total combustion products can be about 5 percent by mole

fraction, about 25 percent by mass fraction, but quite negligible by volume

fraction, say 0.04 percent of the mixture volume. However, principally be-

cause of the negligible volume fraction, this multi-phase product-mixture

can be accommodated by adopting an equivalent molar mass m and the

ideal gas equation of state. During the regression of a propellant grain,

parts of inhibitor do get dislodged and form part of the nozzle flows but

here the flow due to the inhibitor fragmentation is assumed to be negligible

with respect to the flow rate of the propellant burned-products. The other

inert materials, the liner and the insulator, may “degas” and may become

significant particularly during tail-off. This degassing effect is considered

in rigorous prediction procedures, but not here.

Regarding the third assumption, we note that during most of its useful

operation, the rocket motor nozzle remains choked. However, unchoked-

nozzle flows exist at the beginning of ignition transient and at the end of

tail-off transient. While the unchoked duration at the beginning of ignition

transient is negligibly small relatively, this duration can be long during tail-

off transient with very small or insignificant thrust levels. These unchoked

conditions are considered in accurate prediction procedures. But in the

present treatment we consider only choked nozzle-flow condition. As far

as the zero-sliver assumption for the calculation of the tail-off transients

after complete burnout is concerned, in several grain designs considerable

quantity of sliver may be present at burnout. In such cases the calcula-

tion of equilibrium operation has to be continued after burnout until the

sliver is completely consumed. Thereafter, if necessary, the analysis of the

tail-off transient presented in this Chapter can be applied. Or in other


words, the application of the tail-off-transient analysis presented here has

the condition of zero sliver. In Example 2.5, we will see that the sliver

fraction is considerable and the calculation of equilibrium operation has to

be continued until the sliver is completely consumed, and thereafter the

calculation of tail-off transient with zero sliver becomes unnecessary.

Considering finally the fourth assumption, in low performance motors,

although the difference between the total and static pressures is negligible

spatially as well as temporally, there can be substantial temporal variation

in pressure depending upon the grain design. Furthermore, accelerating

field of significant strength may be imposed on the grain and the casing.

For simplicity of the present analysis we assume that the motor casing and

the propellant grain are infinitely rigid so that there is no deflection of

these structural members during the motor operation.

2.2 Mass Conservation Equation

The mass conservation for the motor can be stated as follows.

At any instant of motor operation, the total

mass production rate by way of burning of the

propellant grain should be equal to the rate of

mass ejected though the rocket-nozzle and the

rate of accumulation of the combustion products

within the combustion chamber cavity.

That is,

mc = mt +dm

dt(2.1)

The total mass production rate by way of burning of the propellant

grain mc can be written as,

mc = ρpAbr (2.2)

where ρp is the propellant density, Ab is the grain burning area at the

instant, and r is the propellant burning rate. The rate of accumulation of

combustion products within the combustion-chamber cavity is,

2.2. MASS CONSERVATION EQUATION 15

dm

dt=

d

dt(ρ0cVc) (2.3)

where ρ0c is the stagnation density of the combustion products in the cav-

ity and Vc is the cavity volume. Noting that the mixture of the combustion

products satisfies the ideal gas equation of state with a constant tempera-

ture and a constant molar mass, and differentiating Eq. 2.3

dm

dt= Vc

dρ0c

dt+ ρ0c

dVc

dt=

Vc

RT0

dp0c

dt+

p0c

RT0Abr (2.4)

As per our assumptions, we are considering only nozzle-choked flow situ-

ations. Therefore the mass flow rate through the choked nozzle is given

by,

mt =p0cAt

c∗=p0cAtΓ√RT0

(2.5)

where c∗, the characteristic velocity, is defined as mt/p0NAt (note here

p0N = p0c) and can be theoretically shown to be equal to√RT0

/Γ. Sub-

stituting Eqs. 2.2, 2.4, and 2.5 into Eq. 2.1, we get,

ρpAbr =p0cAt

c∗+

Vc

RT0

dp0c

dt+

p0c

RT0Abr (2.6)

There are quite a few burning rate equations proposed for solid propellants.

The most widely used one is due to Saint-Robert and is given by,

r = apn (2.7)

where a is the pre-exponent factor, which is a function of initial or storage

temperature of the grain. n is the combustion index, and p is the static

pressure experienced by the burning surface. As per the assumption of

one uniform equilibrium total-pressure for the entire combustion-chamber

cavity, p = p0c. Therefore,

r = apn0c (2.8)


Substituting Eq. 2.8 into Eq. 2.6, we get,

Vc

RT0

dp0c

dt=

(ρp −

p0c

RT0

)Abap

n0c −

p0cAt

c∗(2.9)

Noting r = dy/dt,

dp0c

dt=dp0c

dy

dy

dt=dp0c

dyapn0c (2.10)

Therefore, Eq.2.9 can also be written as,

Vc

RT0

dp0c

dy=

(ρp −

p0c

RT0

)Ab −

p1−n0c At

c∗a(2.11)

Equations 2.9 and 2.11 are the fundamental mass conservation equations

for a solid propellant rocket motor in operation. In the fundamental

governing-equations, Eqs. 2.9 and 2.11, notice that p0c/(RT0) = ρ0c. Typ-

ical density of a solid propellant ρp varies from 1200 to 1800kg/m3 and the

density of gases in the combustion chamber cavity ρ0c is mostly around or

less than 10 kg /m3. Therefore, ρp being far greater than ρ0c, the latter

can be neglected. Equations 2.9 and 2.11 can then be simplified to,

Vc

RT0

dp0c

dt= ρpAbap

n0c −

p0cAt

c∗(2.12)

Vc

RT0

dp0c

dy= ρpAb −

p1−n0c At

c∗a(2.13)

At the design stage, a chosen solid-propellant grain is of geometrically

perfect shape or shapes and its burning area can be algebraically expressed

as a function of burned distance y. However, after the designed grain is

cast and cured, there may be some geometric imperfections due to the

fabrication tolerance on the grain mandrel and the grain shrinkage during

curing. Still the initial cavity -configuration of the grain can be deter-

mined through suitable surface measurements and the grain burning area

2.2. MASS CONSERVATION EQUATION 17

with respect to burned distance can be calculated using discretized surface

elements and their movements perpendicular to the respective tangent-

planes. All in all, we note here that the value of the grain burning area Ab

is known for a burned distance y, Eq. (2.14). Consequently, it follows that

the value of the combustion-chamber cavity Vc is also known with respect

to y, Eq. (2.15).

Ab = f (y) (2.14)

Vc = Vci +

∫Abdy (2.15)

Since the burning rate dy/dt by assumption is equal to apn0c, the burned

distance dy is simply,

∫dy =

∫apn0cdt (2.16)

Or,

∫dt =

∫dy

apn0c(2.17)

The nozzle throat region handles the highest mass flux and hence has

the maximum radial heat-flux. In order to manage this severe thermal-

loading condition highly heat-resistant materials are adopted for the noz-

zle throat regions. Despite this, the throat area may get enlarged during

the motor operation, Eqs. 2.9 and 2.11. This nozzle-throat erosion

rate depends on the nozzle throat material, and temperature and compo-

sition of combustion products. The throat-erosion rate is known through

experience from previous experimental results and is generally taken as a

linear function of the motor-operating time, in the absence of any detailed

experimental analysis.

The fundamental mass-conservation Eq. 2.9 has p0c as the dependent

variable and the time t as the independent variable - hence the name

p (t) model. Given the propellant properties (ρp, a, n, &c∗), the propellant

grain configuration and the nozzle shape, Eq.2.9 can be numerically solved

noting Ab and Vc are known functions of y, Eqs. 2.14 and 2.15, and At is


known with respect to time through experience on nozzle throat-erosion

rate. The burned distance y can be numerically evaluated from Eq. 2.16.

Similarly the variance of the fundamental mass-conservation Eq. 2.11 can

be numerically solved. We must note here that the time coordinate t can

always be converted to burned-distance coordinate y by using Eq. 2.16.

Conversely, y can be transformed into t by using Eq. 2.17.

In the performance prediction procedures, use of these different forms

of mass conservation equation will be detailed in Examples 2.1 to 2.5.

2.3 Operational Stability of Rocket Motor

Operational stability of a rocket motor is of paramount importance and it

can be deduced from the fundamental mass conservation equation given by

Eq. 2.1. To understand this, let us resort to the widely used burning rate

equation r = apn0c and inquire what will be the equilibrium pressure for a

constant burning area Ab and whether that attained equilibrium pressure

be stable if the motor is subjected to certain possible perturbations.

The stability is found to be influenced by the value of burning-rate

index n. As explained previously, the equilibrium pressure is reached when

the rate of production of burned products is almost equal to the rate of

ejection through the nozzle, and only a very small portion of the products

goes to fill the volume vacated by the burning propellant grain. Therefore,

while considering the attainment of the equilibrium pressure we can assume

that the mass-accumulation rate is negligible. Equation 2.1 can then be

written as,

mc∼= mt (2.18)

With this, the related mass conservation equation Eq. 2.9simplifies to

ρpAbapn0c =

p0cAt

c∗(2.19)

Equation 2.19 simply reveals that, for the selected propellant (therefore

ρp,a,n, and c∗ are fixed), grain burning area, and nozzle throat area, the

equilibrium pressure is reached when the mass production rate is equal to

the mass ejection rate. Let us now consider the stability of the attained

2.3. OPERATIONAL STABILITY OF ROCKET MOTOR 19

equilibrium-pressure for different values of the burning rate index n. As

we deal with a single variable p0c with a parametric variation on n and

all other quantities being constant, we can plot curves pn0c versus p0c in

normalized units. The mass-production rate and the mass-ejection rate

are represented by pn0c and p0c respectively, Eq.2.19. In Figure 2.1, the pn0ccurves for different values of n are potted against p0c with p0c in normalized

units varying from 0.1 to 2. E, the point (1, 1) in the x-y plane, represents

the equilibrium.

Figure 2.1: Operational stability of a solid propellant rocket motor.

Solid propellants are classified depending on their burning-rate depen-

dence on pressure. The line A represents the mass ejection rate. Also it

denotes the mass production rate for the propellant of n value of unity. If

the burning rate of a propellant decreases with the increase in pressure,

that propellant is known as a mesa propellant and the propellant’s burn-

ing rate index n will be less than zero that is negative. As a representative

curve for a mesa propellant, the curve B is drawn for an n value of -0.3.

If the burning rate of a propellant is insensitive to pressure variation, that

propellant is known as a plateau propellant and the propellant’s burn-

ing rate index n will be equal to zero. The line C represents a plateau

propellant. The curves D and F are drawn for the values of n being 0.4

and 0.7 respectively and the two curves represent the usually adopted pro-

pellants (so called “normal” propellants) with n values less than unity but

greater than zero. The curve G is drawn for the value of n equal to 1.5


as a representative curve for the fictitious propellant of value greater than

unity. The curves B, C, D, F, and G of mass production rates, meet the

mass ejection rate line A at the equilibrium point E.

That the equilibria attained for all the values of less than unity are

stable can be explained as follows. During the equilibrium operation at a

chamber pressure of p0c, small perturbations, positive or negative, can be

triggered by many factors. For example, a dislodged solid-fragment of the

inhibitor may pass through the nozzle and cause a positive perturbation

to the equilibrium pressure p0c due to a sudden reduction in the effective

nozzle-throat area. Or, there may be a small positive or negative per-

turbation in the neighborhood of the assumed constant burning-area Ab.

This may be caused by the opening or the vanishing of a tiny air-bubble

undesirably entrapped during the grain casting. First let us consider a

positive perturbation in the chamber pressure at the equilibrium point E

from p0c to (p0c + ∆p0c). At this perturbed state, the mass production

rates for all curves of n values less than unity, B, C, D, and F, are less

than the mass ejection rate, thus pushing the perturbed state back to the

original equilibrium point E. Whereas, at the positively perturbed state,

the mass production rate for the curve G of n value greater than unity is

more than the mass ejection rate. This pushes the positively perturbed

operating state at pressure (p0c + ∆p0c) further upward. On the quick

progression of this inevitable upward movement of the operating pressure,

motor-explosion is imminent for n > 1. In a similar manner we can con-

sider a negative perturbation in the chamber pressure at the equilibrium

point E from p0c to (p0c −∆p0c) . At this negatively perturbed state, we

find that the mass production rates for all the curves of n values less than

unity are more than the mass ejection rate, thus pushing the perturbed

state forward to the original equilibrium point E. But, at the negatively

perturbed state, the mass production rate for the curve G of n value greater

than unity is less than the mass ejection rate. This pushes the negatively

perturbed state at pressure (p0c −∆p0c) further downward. The quick

regression of this downward movement of the operating pressure leads to

motor extinction for n > 1 . Recall that the line A represents the mass

ejection rate as well as the mass production rate for the value of unity.

All points on the line A can be taken as points of transitory equilibria

that include the point E. Any positive or negative perturbation pushes

2.4. PREDICTION OF PRESSURE-TIME TRACE 21

the operating point sliding on the line A upward or downward leading to

explosion or extinction.

A very important point on the stability response of rocket motors with

respect to the value of burning rate index should be noted at this juncture.

Lower the value of n with respect to unity quicker is the stability response

in a solid rocket motor. From the examples that we considered, the ability

to return to the equilibrium point is stronger for n equal to 0.4 than for n

equal to 0.7. Generally we find that the combustion index of homogeneous

propellants is higher (say about 0.7) than that of heterogeneous propellants

(say about 0.4). In that respect the operational stability of solid rocket

motors of heterogeneous propellants is superior to those of homogeneous

propellants.

For the operational stability of a solid propellant

rocket motor, the burning rate index of the pro-

pellant has to be less than unity and lower the

value of the burning rate index, better is the sta-

bility.

2.4 Prediction of Pressure-Time Trace

In this Chapter we discuss the methods to calculate the pressure time

trace for a rocket motor under equilibrium pressure analysis. Recall that

this analysis assumes a single uniform pressure for the chamber cavity at

a chosen time instant. The pressure-time trace is of three parts, namely

ignition transient or start transient, equilibrium operation, and tail-off

transient.

2.4.1 Ignition Transient

As per the previously discussed assumptions, we may note in particular

that the nozzle is always choked and the ignition of the entire grain surface

is instantaneous with negligible igniter-mass. Generally, the duration of

ignition transient has to be small around 100ms or less for medium sized

motors. During this short period the burned distance or burned depth of

the grain is very small, say less than 1 mm. Therefore the chamber-cavity

volume and burning area can be assumed to be constant at their initial

values of Vci and Abi corresponding to zero burned distance. Alternatively


for the estimated small burned-distance, average values of Vc and Ab can

be determined and kept as constants as Vci and Abi. Proceeding now with

the constant values of Vci and Abi, and with the assumption of ρp � ρ0c

Eq. 2.9 becomes,

dt =Vci

Γ2c∗At

dp0cI

p0cI(ρpaKic∗p

n−10cI − 1

) (2.20)

where Ki is defined as the ratio of the grain burning area Abi and the nozzle

throat area, Abi/At. Note that during the ignition-transient calculation Ki

is a constant. By splitting the second fraction containing the variable p0cI ,

the ignition-transient time-increment,

dt =Vci

Γ2c∗At

−dp0cIp0cI

+ρpaKic

∗p(n−2)0cI dp0cI(

ρpaKic∗p(n−1)0cI − 1

)

This can be brought to the form,

dt =−Vci

(1− n) Γ2c∗At

(1− n) dp0cI

p0cI+d(ρpaKic

∗p(n−1)0cI − 1

)(ρpaKic∗p

(n−1)0cI − 1

)

Applying the limits after integration,

∆t =Vci

(1− n) Γ2c∗Atln

ρpaKic∗ − p(1−n)0cI1


(2.21)

wherep0cI1 is the ignition-transient chamber-pressure at time t and p0cI2 is

the one at t + ∆t. We can initiate the ignition-transient calculation from

the instant zero at which the nozzle gets first-choked. The corresponding

chamber-pressure,

p0cI1 = pa

(γ + 1

2

)γ/(γ−1)(2.22)

Considering a sufficiently small equal pressure-increment ∆p0cI , the next

pressure


p0cI2 = p0cI1 + ∆p0cI

For the pressure rise from p0cI1 to p0cI2 the time interval ∆tI1 can be

calculated using Eq.2.21. For the calculation of the next step,

p0cI3 = p0cI2 + ∆p0cI

The required ∆tI2 is calculated using again Eq. 2.21. This step-wise

calculation is continued with equal ∆p0cIs until we reach a time for which

the last-calculated pressure on the ignition transient p0cIn is sufficiently

close to the first equilibrium pressure, Fig. 1.2 - we will consider the

calculation of the first equilibrium pressure in the next sub-section. You

will learn then that the quantity ρpaKic∗ in Eq. 2.21 is the (1− n)th power

of the first equilibrium-pressure p0cE1 . Therefore, mathematically the time

required to reach the first equilibrium pressure is infinity. To be realistic,

generally the stepwise calculation is continued until the pressure p0cIn is

0.90 to 0.95 of the first equilibrium pressure. Appreciate here that the

procedure just described here to calculate the ignition transient is similar

to the one to calculate the boundary layer thickness in fluid mechanics.

Figure 2.2: Arrangements of nozzle closures.

Nozzle Closure: Quite frequently rocket-motor chamber-cavities are

sealed with rupture discs or plugs at nozzle throats, Figure 2.2. After


ignition, at the attainment of the designed rupture-pressure in the cham-

ber cavity the disc or the plug gets relieved and the flow of combustion

products through the nozzle gets initiated. This arrangement of sealing

the chamber cavity is done on two counts. Firstly, to certain extent it is

to protect the chamber cavity containing the propellant grain and igniter

from different climatic conditions during storage and transportation of the

rocket motor. On the second count, under special situations, it is to have

a faster rise in chamber-pressure after the initiation of the igniter and to

provide quickly a high chamber-pressure to achieve a positive ignition of

the propellant grain. Furthermore, certain propellants, such as potassium-

perchlorate-based ones, have high minimum-pressures for ignition. For

such propellants, the rupture-disc or the nozzle plug becomes an essential

component of the motor.

For a rocket motor with a rupture-disc or a nozzle plug, the mass flow

rate through the nozzle is zero until the closure is relieved. Recall that we

have assumed that the ignition of the entire grain surface is instantaneous

with negligible mass of igniter. Therefore, considering ρp � ρ0c Eq. 2.9

can be written as

Vci

RT0

dp0cR

dt= ρpAbiap

n0cR

where,p0cR is the transitory cavity pressure for the motor before the relief

of the nozzle closure. This equation can be readily brought to the form,

dt =Vci

Γ2c∗At

p−n0cRdp0cR

ρpaKic∗

On integration,

∆t =Vci

(1− n) Γ2c∗At

(p(1−n)0cR2

− p(1−n)0cR1

)ρpaKic∗

(2.23)

Equation 2.23 can be used to calculate the pressure time trace with step-

wise increment of pressure from the initial cavity-pressure up to the release

pressure of the nozzle closure. Thereafter Eq. 2.21 can be adopted to take

the cavity pressure close to the first equilibrium pressure p0cE1 .


Figure 2.3: Typical pressure-time traces of solid propellant motors.

2.4.2 Equilibrium Operation

In this section we consider the performance prediction of equilibrium opera-

tion. Depending on propulsion requirements, different types of propellant

grains are designed. With respect to burned distance, propellant grains

may be designed to have near-constant burning area, increase in burning

area, decrease in burning area, or combination of the former three, Figure

2.3. The propellant grains of the first three types are known as near-

neutral grain, progressive grain, and regressive grain respectively

and these grains correspondingly give near-constant, continuous-increase,

and continuous-decrease in chamber pressures (and hence thrust) with re-

spect to time for a fixed area of nozzle throat.

Compared to its value during ignition transient or tail-off transient, the

modulus of the pressure gradient with respect to time is negligibly small

during equilibrium operation — more so for a near-neutral grain. We can,

therefore, assume a quasi steady-state operation during equilibrium

operation. Let us denote the chamber pressure during equilibrium oper-

ation as p0cE . Under the quasi steady-state operation, for a small time

interval ∆t, ∆p0cE/∆t = 0. This assumption simply means that for a

small incremental time of operation, the pressure is constant. If the curve

A in Figure Fig. 2.4 is the actual variation of p0cE versus time t, by the

quasi steady-state operation we try to calculate the variation of p0cE by

taking small time increments ∆ts during which ∆p0cE/∆t = 0. Evidently,


Figure 2.4: Effect of calculations with different time steps adoptingquasi steady-state assumption (schematic).

the accuracy of this assumption depends on the size of ∆t that we choose

— compare the curve C with larger time intervals with the curve B with

smaller time intervals. The error that occurs with the step wise increase in

time is cumulative. That is, the error that occurs in the first time step gets

added to the error that will occur in the next time step and so on. If the

calculations are done using a computer program, attaining the required

accuracy by choosing a suitably small ∆t is not an issue. Therefore, under

quasi steady-state assumption and assuming ρp � ρ0cE , Eq. 2.12 becomes,

ρpAbapn0cE =

p0cEAt

c∗(2.24)

Equation 2.24is readily simplified to the form

p0cE = (ρpaKc∗)

1

(1− n) (2.25)

Equation 2.25 prompts us once again that the value of n from its de-

sirable value of less than unity should never approach close to unity. For a

selected propellant (ρp, a, and c∗ being constant) and a fixed nozzle config-

uration (At being constant), as the burning area Ab varies as per the grain

design the chamber equilibrium-pressure p0cE gets amplified by the value


of the factor 1/(1− n) and hence this factor is known as amplification

factor. With a value of n close to unity the factor 1/(1− n) assumes a

very high value giving a wild fluctuations in the equilibrium pressure for

small perturbations in the burning area Ab. To explain further the impor-

tance of the value of n, in Figure 2.5 the variations of chamber pressure for

a rocket motor of a progressive grain are shown for two propellants with

the values of n equal to 0.40 and 0.44. Note that the chamber equilibrium

pressure markedly increases for a meager 10 per cent increase in the value

of n.

Figure 2.5: Plot of equilibrium pressure for propellants with twodifferent values of burning rate index n. cylindrical grain length= 140mm; dpi = 10mm; dt = 5mm; r = 1.8 ∗ 10−5pnm/s; p[Pa].

That the value of K cannot be very high is another important design

principle that we learn from Eq. 2.25. With a very high initial-value

of K, apart from the high equilibrium pressure, the pressurization rate

is expected to be very high during ignition transient. This may give a

“hard start” with a high vehicle-acceleration and also an ignition peak

endangering the structural integrity of propellant grain as well as the motor

casing. Generally the values of K is kept between 100 and 2000.

On the point of view of propellant-grain design, we may note that a

large thrust may be required initially to accelerate a rocket vehicle to a

required velocity and once that velocity is achieved, the thrust required is

less. Under this condition for the chosen rocket motor, during the initial

“boost” phase the grain burning area is kept high and thereafter through


the “sustainer” phase the grain burning area can be less. Related to this,

the structural-design factors specified for a rocket vehicle that will influ-

ence the grain design are the maximum allowable vehicle-acceleration and

the aerodynamic pressure 12ρau

2 = 12γpaM

2 experienced by the rocket

vehicle in flight. By the design practices being followed, allowable vehicle-

acceleration and aerodynamic-pressure are substantially lower for satellite

launch vehicles than those for small missiles and sounding rockets. Accel-

eration and aerodynamic pressure that are permissible for large satellite

launch vehicles are substantially less than 1.5g and 100kPa respectively.

Given the propellant properties (ρp, a, n, &c∗) and the nozzle throat

area At, the equilibrium operation can be determined through step-wise

increase of time intervals. The first equilibrium pressure can be calculated

from Eq. 2.25 from the initial grain burning area (burned distance yE1 = 0)

at its initial value of Ki = KE1 . For the chosen time interval ∆t and the

calculated equilibrium pressure p0cE1 that is assumed constant for the time

interval, under quasi steady-state assumption the burned distance∆yE1 is

given by,

∆yE1 = ∆t(apn0cE1

)(2.26)

For the burned distance yE2 (yE2 = yE1 + ∆yE1 ) the corresponding grain

burning area AbE2 is calculated, which gives KE2 . The next equilibrium

pressure p0cE2 is calculated using again Eq. 2.25. This procedure is re-

peated until burnout.

In the above calculation-procedure through step wise increase of time

intervals, as previously noted, the error is cumulative. In order to avoid this

cumulative error, the equilibrium operation can be calculated also through

a slightly different method for a geometrically perfect grain configuration.

Note that for a geometrically perfect grain and a constant nozzle throat

area, the burning area Ab and hence K and the chamber-cavity volume

Vc are known algebraic functions of burned distance y. Recall dp0c/dt =

rdp0c/dy. Therefore Eq. 2.11 with the approximation that ρp � ρ0cE can

be written as

aVc

Γ2c∗At

dp0cE

dy= ρpaKc

∗ − p(1−n)0cE (2.27)


As a first order approximation Eq. 2.25 can be differentiated with respect

to the burned distance y to give,

dp0cE

dy=

p0cE

(1− n)

K′

K(2.28)

where K′

is the derivative of K with respect to the burned distance y.

Substituting Eq. 2.28 into Eq. 2.27 we get

a

(1− n) Γ2c∗At

VcK′

Kp0cE + p

(1−n)0cE − ρpaKc∗ = 0 (2.29)

Equation 2.29 is an algebraic one. Although p0cE in Eq. 2.29 is implicit,

through a suitable procedure p0cE can be solved for any burned distance

y. The time coordinate at any y can be simultaneously obtained by nu-

merically integrating dy/apn0cE from burned distance 0 to y — refer Eq.

2.17.

Example 2. 1

A certain missile adopts a solid propellant rocket motor using a potassium

perchlorate based heterogeneous propellant of a high burning rate and a high

minimum-pressure for ignition. Initial grain burning area is 690 cm2. Initial vol-

ume of chamber cavity is 420 cm3. The propellant characteristics are the following.

Propellant density is 1770 kg/m3. Experimental characteristic velocity is 1130

m/s. Ratio of specific heats γ is 1.27. Burning rate r = 1.8836× 10−5p0.745cm/s;

p is expressed in Pa. The nozzle throat diameter is 22mm. Because of the high

minimum-pressure for ignition, the motor has a nozzle closure to be relieved at

75bar. (a) Determine the ignition transient pressure trace from 1 bar to 95 per-

cent of the first equilibrium pressure. (b) Also compare the ignition transient

pressure trace if ignition were possible at a pressure a little above minimum chok-

ing pressure. Assume that the motor is being tested under standard sea level

conditions.

Solution Ignition transient has to be calculated first with a nozzle closure and

next after the rupture of the nozzle closure. Also the ignition transient, assuming

that ignition is possible at the minimum choking pressure, has to be calculated.

This is for the purpose of comparing the ignition transient with nozzle closure and

without nozzle closure. Necessary motor-dimensions and propellant properties are

given.

Assumptions It is assumed that the entire grain area is ignited instanta-

neously. Igniter mass is negligible. The chamber cavity is an adiabatic system


with a uniform total pressure throughout and the difference between the static-

and total-pressure is negligible.

Analysis

(a)

The ratio of initial burning area to throat area is given by

Ki =690.0× 10−4

(π/4)× 0.0222=

690.0× 10−4

3.8013× 10−4= 181.516

The first equilibrium pressure is given by,

p0cE1= (ρpaKic

∗)1/(1−n)

=(1770× 1.8836× 10−7 × 181.516× 1130

)1/(1−0.745)

= 157.0× 105Pa

The chamber pressure corresponding to 95% of the first equilibrium pressure

is 149.15 bar. The chamber pressure rise with nozzle closure is given by Eq. 2.23,

∆t =Vci

(1− n) Γ2c∗At

(p(1−n)0cR2

− p(1−n)0cR1

)ρpaKic∗

(2.23)

Γ =√γ

(2

γ + 1

) γ + 1

2 (γ − 1)=√

1.27

(2

1.27 + 1

) 2.270.54

= 0.6618

Substituting the values in Eq. 2.23,

∆tR =420× 10−6

(1− 0.745)× 0.66182 × 1130× 3.8013× 10−4

×

(p(1−n)0cR2

− p(1−n)0cR1

)1770× 1.8836× 10−7 × 181.516× 1130

= 1.2802× 10−4(p(1−n)0cR2

− p(1−n)0cR1

)


Starting from an initial chamber pressure p0cR1 of 1 bar, we can calculate the

time required for the chosen values chamber pressures up to 75 bar. The results

of the calculations are given in Table 2.1.

Table 2.1: Ignition transient with nozzle closure from an initialpressure of 1 bar to 75 bar, Example 2. 1

p0cR2 (bar) ∆tR (ms) p0cR2 (bar) ∆tR (ms)

2 0.47 35 3.563 0.78 40 3.774 1.02 45 3.755 1.22 50 4.1310 1.93 55 4.2915 2.40 60 4.4420 2.77 65 4.5825 3.07 70 4.7130 3.33 75 4.84

We see from the results of Table 2.1 that the motor with a nozzle closure

attains the chamber pressure of 75 bar in 4.84ms. On the attainment of 75bar,

the nozzle closure is relived and the nozzle flow is initiated. We have to now

calculate the ignition transient from 75 bar to a pressure close to the first equilib-

rium pressure. Mathematically it takes infinite time to reach the first equilibrium

pressure, Eq.2.21. This is because when p0cI2 takes the value of p0cE1the de-

nominator within the logarithmic sign becomes zero; note ρpaKic∗ is nothing but

the (1− n)th power of the first equilibrium pressure p0cE1 , Eq. 2.25. In practice

the ignition transient is calculated up to the chamber pressure that is 95% of the

first equilibrium pressure. We have to now use Eq.2.21 to determine the ignition

transient from 75 bar to 149.15 bar that is 95% of the first equilibrium pressure.

∆t =Vci


(ρpaKic

∗ − p(1−n)0cI1


)(2.21)

Substituting the values,


∆tI =420× 10−6

(1− 0.745) 0.66182 × 1130× 3.8013× 10−4×

ln

(1770× 1.8836× 10−7 × 181.516× 1130−

(75× 105

)0.2551770× 1.8836× 10−7 × 181.516× 1130− p(1−n)0cI2

)

= 0.021564− 8.7548× 10−3 ln(68.3841− p0.2550cI2

)Calculating the time intervals from the pressure of 75 bar (p0cI1), at which the

nozzle closure gets relieved, to different pressure levels up to 95% of the first

equilibrium pressure, which is 149.15 bar, we get the time intervals as given in

Table 2.2

Table 2.2: Ignition transient after the nozzle closure is relieved at75 bar, Example 2. 1

p0cI2 (bar)∆tI (ms) ∆tRn +∆tI (ms)

p0cI2 (bar)∆tI (ms) ∆tRn +∆tI (ms)

80 0.73 5.57 125 9.74 14.5885 1.49 6.33 130 11.35 16.1990 2.28 7.12 135 13.26 18.1095 3.12 7.96 140 15.63 20.47100 4.01 8.85 145 18.79 23.63105 4.96 9.80 149.15 22.60 27.44110 5.98 10.82 150 23.62 28.46115 7.10 11.94 155.43 36.82 41.66120 8.34 13.18 156.21 42.91 47.75

The time interval ∆tI corresponds to the time taken for the chamber pressure

to rise to a pressure from 75 bar. In the previous calculation we have determined

the time interval ∆tRn as 4.84ms required for the pressure to rise from 1 bar to

75 bar. The total-time interval for the pressure to rise from 1 bar to a pressure

after the nozzle closure is relieved is ∆tRn + ∆tI where ∆tRn is 4.84ms, the time

elapsed at the relief of nozzle closure. These values are given in the third and the

sixth columns of Table 2.2.

(b)

Now we have to determine the ignition transient if there is no nozzle closure

and if ignition were possible at the minimum choking pressure. The minimum

choking pressure is given by


p0cI1 = 1.01325× 105

(γ + 1

2

)γ/(γ−1)

= 1.01325× 105

(2.27

2

)4.704

= 1.838× 105Pa

From this minimum choking pressure we can calculate the time interval for

various chamber pressures along the ignition transient. Substituting the values in

Eq.2.21,

∆tI =420× 10−6

(1− 0.745) 0.66182 × 1130× 3.8013× 10−4×

ln

(1770× 1.8836× 10−7 × 181.516× 1130−

(1.838× 105

)0.2551770× 1.8836× 10−7 × 181.516× 1130− p(1−n)0cI2

)

= 0.03359− 8.7548× 10−3 ln(68.3841− p0.2550cI2

)Calculating the time intervals from the minimum choking pressure of 1.838 bar

(p0cI1) to different pressure levels up to 95% of the first equilibrium pressure,

which is 149.15 bar we get the time intervals as given in Table 2.3.

Table 2.3: Ignition transient without nozzle closure from the min-imum choking pressure of 1.838 bar, Example 2. 1

p0cI2 (bar)∆tI (ms) p0cI2 (bar)∆tI (ms) p0cI2 (bar)∆tI (ms)

2 0.10 60 9.97 120 20.385 1.31 65 10.64 125 21.7810 2.60 70 11.33 130 23.3915 3.59 75 12.04 135 25.3020 4.45 80 12.77 140 27.6725 5.22 85 13.53 145 30.8330 5.95 90 14.32 149.15 34.6435 6.64 95 15.16 150 35.6640 7.32 100 16.04 155.43 48.8645 7.98 105 16.99 156.22 54.9550 8.64 110 18.0255 9.30 115 19.14

Discussion These kinds of calculations are easily carried out using the spread-

sheet procedure, say Excel. From Tables 2.2 and 2.3, we note that with a nozzle


Figure 2.6: Comparison of ignition transients with and without anozzle closure.

closure the time required to reach 95% of the first equilibrium pressure is 27.44ms

against the time of 34.64ms without nozzle closure. Similarly longer time is re-

quired without nozzle closure to reach the specified pressure of 75bar. The two

ignition transient traces are plotted in Figure 2.6. Also note that as we approach

the first equilibrium pressure the ignition transient curve becomes asymptotic and

takes a longer time interval for the specified increment in pressure. To demon-

strate this we have calculated the time required to reach 99% and 99.5 % of first

equilibrium pressure, namely 155.43 bar and 156.22bar respectively. Although

nozzle closures are provided for the previously explained advantages, if the de-

signed rupture pressure is high, at the instant of the rupture, a sudden “impact”

of high thrust will be acting on the rocket vehicle resulting in a high acceleration.

Suitable structural strength for the rocket vehicle and safety of the vehicle com-

ponents, particularly the electronic ones, against the high acceleration are to be

ensured.

Even with optimally designed igniters, during motor firing no sooner the noz-

zle closure ruptures a downward kink will be mostly observed in the pressure-time

trace. This is due to the ensuing sudden nozzle-outflow.

Example 2. 2

A small solid propellant rocket motor has a tapered tubular grain with its

grain-ends inhibited. The taper of the grain port is 2o. Head-end port-diameter

of the grain is equal to 10mm. The length and outer diameter of the grain are

600mm and 100mm respectively. The nozzle with 40o convergence at the entry

has a throat diameter of 16mm. The length of the cylindrical chamber is 620mm.

The propellant density is 1750 kg/m3. The experimental characteristic velocity is

1400 m/s. The burning rate is given by the equation r = 1.2×10−2p0.4mm/s with


the unit of p in Pa. Calculate (a) the initial and the burnout burning areas, (b) the

sliver volume and the sliver fraction, (c) the propellant mass and its volumetric

loading fraction and the port-to-throat area ratio, and (d) the initial and burnout

equilibrium-pressure.

Solution The grain and motor dimensions are given. The properties of the

propellant are given. Initial and burnout burning areas are to be determined.

Sliver volume and its fraction have to be determined. Loaded propellant mass,

volumetric loading fraction, port-to-throat area ratio, and initial and burnout

equilibrium pressures are the other quantities to be determined. The sketch of the

motor is shown in Figure 2.7. The calculations are mainly towards the geometric

properties of the grain and the nozzle convergence, which are of the shapes of

truncated cones.

Figure 2.7: Schematic diagram of the rocket motor of Example 2.2.

Assumptions Equilibrium pressure analysis is applicable. Every point on the

propellant surface moves perpendicular to the tangent plane at that point. Note

that this is an important property of propellant surface-movement and is applied

while analysing grain configurations. The propellant grain and the casing are

infinitely rigid and there is no deflection of these components.

Analysis

For SI units, the burning rate equation becomes,

r = 1.2× 10−5p0.4m/s

The grain port forms a truncated cone with a half cone angle of 2o with

a truncated height of 600mm. With 4o cone angle (2o taper), the initial port

diameter at nozzrle end can be calculated.

The initial port diameter at the nozzle end port diameter is given by


0.010 + 2× 0.6× Tan2o = 0.0519m

The height of the cone h1I with 0.0519 m base diameter and the height of the

cone h2I with 0.010m base diameter are given by,

h11 =0.0519

2× Tan2o= 0.7431m

h21 =0.01

2× Tan2o= 0.1432m

Similarly the slant lengths of the cones are

s11 =

√√√√(0.0519

2

)2

+ 0.74312 = 0.7436m

s21 =

√√√√(0.01

2

)2

+ 0.14322 = 0.1433m

(a)

The surface area of the cone is given by πrs. Therefore the initial burning

area is given by,

AbE1= πr11s11 − πr21s21

= π

(0.0519

2× 0.7431− 0.005× 0.1433

)= 0.05833m2

At burnout the base diameter of the port at the nozzle end is 0.1m and the head

end diameter is given by [(0.1-0.0519) +0.01] = 0.0581m. The corresponding

heights and slant lengths at burnout are calculated.

h1n =0.1

2× Tan2o= 1.4318m


h2n =0.0581

2× Tan2o= 0.8319m

s1n =√

0.052 + 1.43182 = 1.4327m

s21 =

√√√√(0.0581

2

)2

+ 0.83192 = 0.8324m

The burnout burning area is given by,

AbEn= πr1ns1n − πr2ns2n

= π

(0.05× 1.4327−

0.0581

2× 0.8324

)= 0.14908m2

(b)

The initial port volume is given by

1

3π

(0.0519

2

)2

× 0.7431− 0.0052 × 0.1432

= 5.2027× 10−4m3

Initial propellant volume is given by

π

4× 0.12 × 0.6− 5.2027× 10−4 = 4.1921× 10−3m3

The burnout port volume is given by

1

3π

0.052 × 1.4318−

(0.0581

2

)2

× 0.8319

= 3.0133× 10−3m3

The sliver volume is given by


π

4× 0.12 × 0.6− 3.0133× 10−3 = 1.699× 10−3m3

The sliver fraction is given by

1.699× 10−3

4.1921× 10−3= 0.4053

(c)

Propellant mass is given by

4.1921× 10−3 × 1750 = 7.336kg

Volume of nozzle convergence cone is given by

1

3π

(0.052 ×

0.05

Tan40o− 0.0082 ×

0.008

Tan40o

)= 1.5536× 10−4m3

Total volume of the empty chamber is given by

π

4× 0.12 × 0.62 + 1.5536× 10−4 = 5.0248× 10−3m3

Propellant volume loading fraction is given by

4.1921× 10−3

5.0248× 10−3= 0.834

Port to throat area ratio is generally considered with respect to nozzle end port

area. It is given by

0.05192

0.0162= 10.522

(d)

Initial equilibrium pressure is given by


p0cE1= (ρpaK1c

∗)1/(1−n)

=

1750× 1.2× 10−5 ×0.05833

π

4× 0.0162

× 1400

1/(1−0.4)

= 35.61× 105Pa

p0cE1= 35.61× 105Pa

Burnout equilibrium pressure is given by

p0cEn = (ρpaK1c∗)

1/(1−n)

=

1750× 1.2× 10−5 ×0.14908

π

4× 0.0162

× 1400

1/(1−0.4)

= 170.109× 105Pa

p0cEn= 170.109× 105Pa

Discussion Port taper is given firstly for the easy retrieval of the grain man-

drel after the cast grain is cured. However, the taper for that purpose is generally

small, about 0.5o or so. The second reason can be to cater for a specific mission,

where in a long tail-off with substantial-thrust values is a requirement. The third

reason can be to avoid erosive burning of the propellant. Erosive burning is

the dependence of the burning rate on the cross flow properties of the burned

products over the burning surface and this is in addition to the burning rate be-

ing dependent on pressure. The phenomenon of erosive burning will be detailed

in Chapter 3 under Section 3.4. The solid propellant motor of the Example 2. 2

is one of progressive burning grain. The burning area increases with the burned

distance. As per our definition, the sliver propellant is the left-out propellant in

the motor at the instant that the flame front first touches the casing wall, which

is burnout. At complete burnout the sliver is zero. In the present Example, the


sliver fraction is 0.4053. More than 40% of the propellant is left-out at burnout.

This part of the propellant grain will give a regressive part of the pressuretime

trace. In total, the propellant grain is progressive initially and then becomes re-

gressive. Determination of equilibrium pressure, using Eq.2.25, can be continued

even after burnout until the propellant is consumed. We have to consider the

intersection of the line representing the surface of the grain at burnout with the

line representing the casing wall at 100mm diameter. Detailed analysis of tapered

grain is dealt in Example 2. 5.

Example 2. 3

A small solid rocket thruster contains a straight tubular grain of inner diam-

eter 9mm and outer diameter of 20mm. The ends of the grain are inhibited. The

grain length is 150mm. The nozzle throat diameter is 5mm. The empty volume

of the chamber is 55 cm3. The properties of the solid propellant are the following.

ρp = 1860kg/m3. r = 1.25 × 10−5p0.4m/s with p in Pa. Characteristic velocity

c∗ is equal to 1590 m/s. γ = 1.25. (a) Calculate through equilibrium pressure

analysis the first- as well as burn-out-equilibrium pressures. (b) Compare your

results by adopting Eq. 2.25 with that using Eq. 2.29.

Figure 2.8: Schematic sketch of the solid propellant thruster ofExample 2. 3.

Solution Small solid rocket motors or liquid rocket engines are also known

as solid or liquid rocket thrusters. The thruster dimensions and the propellant

properties are given. The sketch of the solid rocket thruster is given in Figure 2.8.

The first equilibrium pressure as well as the burnout equilibrium pressure has to

be calculated through equilibrium analysis.

Assumptions Equation 2.25 assumes negligible density of combustion prod-

ucts and quasi-steady state. Equation 2.29 assumes negligible density of combus-

tion products but introduces a first order approximation without assuming quasi

steady state condition.

Analysis First let us check the applicability of the equilibrium pressure anal-

ysis. The port to throat area ratio is

Api

At=

92

52= 3.24


Therefore we can adopt equilibrium pressure analysis. Volumetric propellant load-

ing fraction,

Vp

Vcf=

π4 × 0.15×

(0.022 − 0.0092

)55× 10−6

= 0.683

(a)

Let us first calculate the equilibrium pressures adopting Eq. 2.25.

p0cE = (ρpaKc∗)

1

(1− n)

At the first equilibrium pressure the value of K is given by

Kinitial =π × 0.009× 0.15

(π/4) 0.0052= 216

At the burnout the value of K is given by

Kburnout =π × 0.020× 0.15

(π/4) 0.0052= 480

Therefore the first equilibrium pressure

p0cE1=(1860× 1.25× 10−5 × 216× 1590

)1/0.6= 31.90× 105Pa

Similarly the burnout equilibrium pressure

p0cEn =(1860× 1.25× 10−5 × 480× 1590

)1/0.6= 120.72× 105Pa

(b)

Let us now calculate the equilibrium pressures using Eq. 2.29.

a

(1− n) Γ2c∗At

VcK′

Kp0cE + p

(1−n)0cE − ρpaKc∗ = 0


Γ =√γ

(2

γ + 1

) γ + 1

2 (γ − 1)=√

1.25

(2

2.25

)2.25

0.5= 0.6581

At =π

4× 0.0052 = 1.9635× 10−5m2

Initial and burnout chamber volumes are

Vcinitial= 55× 10−6 −

π

4× 0.15×

(0.022 − 0.0092

)= 17.42× 10−6m3

Vcburnout= 55× 10−6m3

K =Ab

At=

8Lg (ri + y)

d2tK′

=8Lg

d2t

K′

K=

1

(ri + y)

(K′

K

)initial

=1

ri=

1

0.0045= 222.22m−1

(K′

K

)burnout

=1

ro=

1

0.01= 100.0m−1

On substituting the initial values into Eq. 2.29

1.25× 10−5

0.6× 0.65812 × 1590× 1.9635× 10−5

×17.42× 10−6 × 222.22p0cE1

+p0.60cE1− 1860× 1.25× 10−5 × 216× 1590 = 0


5.9646× 10−6p0cE1 + p0.60cE1− 7984.8 = 0

The above equation can be solved iteratively.

First let us start with the value of equilibrium pressure that we got using Eq.

2.25. The fourth trial value in Table 2.4 has been calculated through interpolation.

Table 2.4: Iterative calculations for the initial equilibrium pressurefor Example 2. 3

No Trialp0cE1(Pa)

5.9646 ×10−6p0cE1

p0.60cE1Erroragainst0

1 31.9× 105 19.0271 7984.99 19.21772 31.8× 105 18.9674 7969.96 4.12983 31.7× 105 18.9078 7954.92 -10.97694 31.773×105 18.9511 7965.86 7.50×10−3

Since the error obtained in the fourth trial is very small the value of p0cEI= 31.773

bar is taken as the solution.

On substituting the burnout values into Eq. 2.29

1.25× 10−5

0.6× 0.65812 × 1590× 1.9635× 10−5× 55× 10−6 × 100p0cEn

+p0.60cEn− 1860× 1.25× 10−5 × 480× 1590 = 0

8.4744× 10−6p0cEn + p0.60cEn− 17744.4 = 0

Adopting the iterative procedure, the above equation can be solved.

In Table 2.5, the trial value of p0cEnin the third step has been obtained

through extrapolation and the trial value of p0cEnin the fifth step has been

obtained through interpolation. Since the error in the fifth step is very small the

value of p0cEn= 119.57 bar is taken as the solution.

Discussion The first equilibrium pressure p0ce1 = 31.9 bar is obtained using

Eq. 2.25 and p0ce1 = 31.773 bar by using Eq. 2.29. The difference is about 0.4

per cent. The burnout equilibrium pressure p0cEn= 120.72 bar by using Eq. 2.25

and p0cEn = 119.57 bar by using Eq. 2.29. The difference is about 1 per cent.

The variance is due to the accounting of pressure gradient in Eq.2.29. To explain

further, Eq. 2.29can be written in a form given below.


Table 2.5: Iterative calculations for the burnout equilibrium pres-sure for Example 2. 3

No Trialp0cEn(Pa)

8.4744 ×10−6p0cEn

p0.60cEnErroragainst0

1 120.6× 105 102.2013 17734.07 91.8702 120.5× 105 102.1165 17725.24 82.9613 119.6× 105 101.3538 17645.69 2.64634 119.5× 105 101.269 17636.84 -6.29235 119.57×105 101.328 17643.04 -0.0753

p0cE =

(ρpaKc

∗ −a

(1− n) Γ2c∗At

VcK′

Kp0cE

) 1

(1− n)

Thus, the value of p0cE calculated using Eq. 2.25 differs from that calculated

using Eq. 2.29 depending on the value of the gradient K′. As seen in the present

example, for a progressive grain with its positive K′

value, the p0cE through Eq.

2.25 will be more than the p0cE through Eq. 2.29. Extending this observation

further, for a regressive grain with its negative K′

value, the p0cE through Eq.

2.25 will be less than the p0cE through Eq. 2.29. There will be no difference in

the values for a neutral grain because its K′

value is zero.

2.4.3 Tail-off Transient after complete burnout

Recall the difference between burnout and complete burnout. The first

contact of flame front with the chamber liner is burnout. The propellant

left at burnout is the sliver. Complete burnout occurs at zero sliver. Dur-

ing the tail-off, the sliver burns with its burning area reducing at a fast

rate. Generally, this transient ends when the entire sliver is consumed,

which is complete burnout, along with the chamber pressure asymptoti-

cally reaching ambient pressure. However, in certain types of grains the

complete burnout can occur instantaneously with a substantial chamber

pressure. For example, theoretically a pure cylindrical grain as in Example

2. 3 can result in complete burnout instantaneously with a high chamber

pressure that has to get reduced to ambient pressure. This type of tail-off


following complete burnout with a high chamber pressure is considered in

the following analysis.

At the complete burnout, the propellant burning area Ab = 0, Vc is the

empty chamber volume Vcf , a constant. Therefore, Eq. 2.9, after some

manipulations can be written as,

dp0c

p0c=−Γ2Atc

∗

Vcfdt (2.30)

On integrating and applying the limits of complete-burnout time tb with

the corresponding complete-burnout pressure of zero sliver (the last equi-

librium pressure p0cEn) to any time under tail-off t with the corresponding

pressure under tail-off p0cT we get

t = tb −Vcf

Γ2Atc∗ln

(p0cT

p0cEn

)(2.31)

For different values of tail-off transient pressures p0cT s, time t can be cal-

culated. Generally the tail-off transient is calculated until the tail-off pres-

sure p0cT is 10 % of the burnout pressure or until the nozzle gets unchoked,

whichever is earlier.

Example 2. 4

Calculate the tail-off transient for the rocket thruster of Example 2. 3.

Solution Adopting Eq. 2.31, the problem can be easily solved. From Example

2. 3 and its solution, the related known-values are the following. The chamber

pressure at burnout p0cEn= 120.72 bar. Γ = 0.6581. At = 1.9635 × 10−5m2.

c∗ = 1590m/s.. The chamber volume at complete burnout Vcf = 55 × 10−6m3.

We do not know the complete-burnout time. Arbitrarily we will keep it at 2s.

Assumptions Properties of combustion products are assumed to be same

during tail-off transient. It is further assumed that there is no degassing from the

liners and insulators.

Analysis

Recall Eq. 2.31

t = tb −Vcf

Γ2Atc∗ln

(p0cT

p0cEn

)


Substituting the values into Eq.2.31,

t = 2−55× 10−6

0.65812 × 1.9635× 10−5 × 1560ln

(p0cT

p0cEn

)

= 2− 4.0677× 10−3 ln

(p0cT

p0cEn

)

Calculating for different values of p0cT /p0cEn we can draw the tail-off transient

curve from the arbitrarily chosen complete-burnout time of 2s. Table 2.6 gives

the calculated values for the selected time intervals. Figure 2.9 shows the tail-off

transient curve.

Table 2.6: Tail-off transient-calculation results, Example 2. 4

Time(ms) p0cT (bar) Time(ms) p0cT (bar) Time(ms) p0cT (bar)

2000(burnout)

120.72 2004.51 39.84 2009.79 10.86

2000.52 106.23 2005.04 35.01 2010.82 8.452001.01 94.16 2005.48 31.39 2011.44 7.242001.51 83.30 2005.98 27.77 2012.19 6.042002.01 73.64 2006.55 24.14 2013.09 4.832002.51 65.19 2006.98 21.73 2014.26 3.622003.07 56.74 2008.00 16.90 2015.91 2.412003.53 50.70 2008.98 13.28 2017.06 1.82 b

2004.04 44.67 2009.37 12.07 a

a 10% of burnout pressure.b Limit of nozzle choking for standard sea level ambient pressure.

Discussion Here we have not calculated the complete-burnout time but have

taken arbitrarily its value as 2s. Generally the tail-off transient is calculated up to

10% of the burnout pressure, in this case the complete-burnout pressure. However,

as explained previously, tail-off transient can be for a long period, particularly at

high altitudes and in space environment. To demonstrate this, the points of tail-

off transient are calculated beyond 10% of the burnout pressure until the nozzle

gets unchocked for standard sea level condition. These points are also given in

Table 2.6.

Example 2. 5

A composite propellant, containing ammonium perchlorate as an oxidizer,

hydroxyl terminated polybutadiene and aluminium as fuels, has the following


Figure 2.9: Tail-off transient of the rocket thruster of Example 2.4.

characteristics. Density = 1780 kg/m3. Ratio of specific heats γ = 1.17, exper-

imental characteristic velocity = 1560 m/s. Burning rate equation is given by

r = 2.814 × 10−5p0.35m/s — p is expressed in Pa. Using this propellant a case

bonded solid rocket motor is to be designed with a tapered cylindrical grain. The

first trial dimensions of the grain are the following. Head end port diameter =

50mm. Taper angle = 1.5o. Grain length = 2400mm. The grain outer-diameter =

400mm. The ends of the grain are inhibited. The nozzle convergence angle is 40o.

The head-end free volume and the plenum may be taken as 0.070m3. (a) If the

initial equilibrium pressure has to be 25 bar, calculate the nozzle throat diameter

in mm. (b) Determine the port-to-throat ratio and propellant volumetric loading

fraction. (c) Calculate the entire pressure-time trace.

Figure 2.10: Schematic sketch of the rocket motor of Example 2.5.

Solution Properties of the solid propellant to be used are given. Dimensions

of rocket motor are given. A tapered ends-inhibited case-bonded cylindrical grain

is to be used. Its outer diameter, head-end port diameter, and port-taper angle

are given. The nozzle-end port diameter has to be calculated. For the specified

first equilibrium pressure of 25 bar the throat diameter of the nozzle has to be


Table 2.7: Grain characteristics of cylindrical segment

Phase I: 0 ≤ y ≤ (rc − rhi − l tan θ) cos θ

Head-end portradius

rh = rhi + y/cos θ

Nozzleend portradius

rn = rhi + l tan θ + y/cos θ

Propellantvolume

Vp = πr2c l − [π/(3 tan θ)][(rhi + l tan θ + y/cos θ)3 − (rhi + y/cos θ)3

]Burnoutslivervolume

Vs = πr2c l − π/

(3 tan θ)[r3c − (rc − l tan θ)3

]

Burningarea

Ab = (πl/cos θ) [2 (rhi + y/cos θ) + l tan θ]

Burningarea atburnout

Abbo = (πl/cos θ) (2rc − l tan θ)

Phase II: (rc − rhi − l tan θ) cos θ ≤ y ≤ (rc − rhi) cos θ

Head-end portradius

rh = rhi + y/cos θ

Nozzleend portradius

rn = rc

Grainlength

l = (rc − rhi − y/cos θ)/tan θ

Slivervolume

Vp = (π/tan θ)

{r2c (rc − rhi − y/cos θ)−[r3c − (rhi + y/cos θ)3

]/3

}Burningarea

Ab = (π/sin θ)[r2c − (rhi + y/cos θ)2

]

determined. The port-to-throat ratio and the propellant volumetric loading frac-

tion have to be determined. The entire pressure-time trace for the rocket motor

has to be obtained.

Cylindrical grain is a quite frequently adopted configuration for all classes

of solid rocket motors. Small thrusters to very large boosters of satellite launch

vehicles, and long range missiles adopt this configuration. For example the solid

rocket booster (SRB) of space shuttle has two pure cylindrical segments, as given

in this Example, and one nozzle-end cylindrical segment. SRB's head end segment


is a bit complicated configuration known as finocyl. In view of the importance of

the cylindrical configuration we shall first derive the general governing-equations

for a cylindrical grain and then use those equations to solve the present numerical

problem. A schematic sketch of the rocket motor is given in Figure 2.10.

Assumptions (1) Equilibrium pressure analysis is applicable. (2) Igniter

mass is negligible. (3) At time zero, the complete grain is ignited and the nozzle is

just choked. (4) Ignition transient occurs with a negligible burned distance — the

burning area Ab and the initial chamber-cavity-volume Vci are constant. (5) The

flow due to the inhibitor fragments is negligible and the liner and insulator do not

decompose. (6) The rocket motor is operating in sea level standard atmosphere.

(7) During the motor operation, the grain is infinitely rigid and hence does not

deflect due to pressure variations.

Analysis

With reference to Figure 2.10, we have two phases of burning. First one is of

a progressive phase until burnout and the second one, towards complete burnout,

is a regressive phase of sliver burning. These two phases can be given by the

limits. The first phase is for the range of burned distance y,

0 ≤ y ≤ (rc − rhi − l tan θ) cos θ

And, the second phase is for the range of burned distance y,

(rc − rhi− l tan θ) cos θ ≤ y ≤ (rc − rhi

) cos θ

For these two phases of burning, equations can be derived for propellant volume,

sliver volume, and burning area. The derivation, based mainly on geometrical

properties, is straight forward one. The derived equations are given in Table 2.7.

(a)

Let us first calculate the nozzle-end initial diameter of the grain port. This is

given by

dni= 2× rni

= 2 (rhi+ l tan θ) = 2× (0.025 + 2.4 tan(1.5)) = 0.1757m

The first equilibrium pressure can be determined through Eq.2.25,

p0cE = (ρpaKc∗)

1

(1− n)


Substituting the values in Eq.2.25,

25× 105 =(1780× 2.814× 10−5 ×Ki × 1560

)1/(1−0.35)

Ki = 184.41

During the first phase of burning, the grain burning area is given by,

Ab = (πl/cos θ) [2 (rhi+ y/cos θ) + l tan θ]

The initial grain burning area is obtained with y = 0. Substituting the values,

Abi =π × 2.4

cos(1.5)[2× 0.025 + 2.4× tan(1.5)] = 0.8511m2

Ki =Abi

At⇒ 184.41 =

0.8511

At⇒

At = 4.615× 10−3m2 ⇒ Dt = 76.66mm

Throat diameter = 76.66mm (say 76.7mm)

(b)

The port to throat area ratio = (0.1757/0.07666)2

= 5.25.

As the port-to-throat area ratio is greater than 3 we can adopt equilibrium

pressure analysis.

The chamber empty volume comprises of (i) empty cylindrical volume includ-

ing plenum and head end free volume and (ii) nozzle convergence volume. Empty

cylindrical volume including plenum and head-end free volume is given by,

Vcf1 = πr2c l + 0.070 = π × 0.22 × 2.4 + 0.070 = 0.37159m3

Nozzle convergence volume is given by


Vcf2 =π

3× tan 40

(0.23 − 0.0383293

)= 9.9137× 10−3m3

Therefore the chamber empty volume

Vcf = 0.37159 + 9.9137× 10−3 = 0.3815m3

Initial propellant volume is given by,

Vp = πr2c l − [π/(3 tan θ)]

[(rhi + l tan θ + y/cos θ)

3 − (rhi+ y/cos θ)

3]

where y = 0. Substituting the values,

Vp = π × 0.22 × 2.4− [π/(3× tan (1.5))]

[(0.025 + 2.4× tan (1.5))

3 − (0.025)3]

= 0.2751m3

Therefore the propellant volumetric loading fraction is given by

0.2751/0.3815 = 0.721

This value being less than 75%, the motor further qualifies for a low performance

class and equilibrium pressure analysis is applicable.

(c)

Before determining the pressure-time trace during equilibrium operation, we

have to calculate the ignition transient. This transient can be determined using

Eq.2.21

∆t =Vci


(ρpaKic

∗ − p(1−n)0cI1


)


The initial free volume comprises of initial port volume, plenum, head-end free

volume, and nozzle-convergence volume. The plenum and head-end free vol-

ume is given as 0.070m3. We have calculated the nozzle convergence volume

as 9.9137x10−3m3. Now we have to calculate the initial port volume. Initial port

volume is given by the equation (refer Table 2.7)

Vporti =π

3 tan θ

[(rhi + l tan θ)

3 − (rhi)3]


Vporti =π

3 tan (1.5)

[(0.025 + 2.4× tan (1.5))

3 − 0.0253]

= 0.026485m3

Therefore the initial-total free volume

Vci = 0.026485 + 0.070 + 9.9137× 10−3 = 0.10640m3

For the given value of γ = 1.17,

Γ =√

1.17

(2

1.17 + 1

)(1.17+1)/[2(1.17−1)]

= 0.6426

We have to determine the ignition transient from the minimum chamber pres-

sure at which the nozzle chokes. For the sea level standard atmospheric pressure

of 1.01325 bar, the minimum chamber pressure for choking condition is given by,

p0cI1 = pa

(γ + 1

2

)γ/(γ−1)= 1.01325× 105 ×

(2.17

2

)1.17/0.17

= 177646.5Pa

Theoretically, as explained previously, it requires infinite time to reach the first

equilibrium pressure by adopting Eq. 2.21 and therefore, as practiced usually, we

will have to stop the ignition transient calculations when the ignition transient


pressure reaches 95% of the first equilibrium pressure, that is, p0cIn = 0.95×25 =

23.75bar.

Substituting the values in Eq. 2.21,

∆t =0.10640

(1− 0.35)× 0.64262 × 1560× 4.615× 10−3

× ln

(1780× 2.814× 10−5 × 184.41× 1560− 177646.50.65

1780× 2.814× 10−5 × 184.41× 1560− p0.650cI2

)

= 0.05506×[9.378088− ln

(14410− p0.650CI2

)]

Table 2.8: Ignition transient, Example 2. 5

Time(s) p0cIa Time(s) p0cI

a Time(s) p0cIa Time(s) p0cI

a

0.00E+0 1.78 4.25E−2 12 1.51E−1 23 1.90E−1 249.72E−4 2 4.75E−2 13 1.54E−1 23.1 1.96E−1 24.15.11E−3 3 5.29E−2 14 1.57E−1 23.2 2.02E−1 24.29.06E−3 4 5.87E−2 15 1.60E−1 23.3 2.09E−1 24.31.30E−2 5 6.50E−2 16 1.64E−1 23.4 2.18E−1 24.41.68E−2 6 7.21E−2 17 1.67E−1 23.5 2.28E−1 24.52.08E−2 7 7.99E−2 18 1.71E−1 23.6 2.40E−1 24.62.48E−2 8 8.89E−2 19 1.75E−1 23.7 2.56E−1 24.72.89E−2 9 9.94E−2 20 1.78E−1 23.8b 2.66E−1 24.8c

3.32E−2 10 1.12E−1 21 1.80E−1 23.83.77E−2 11 1.28E−1 22 1.84E−1 23.9

a p0cI expressed in bar.b Pressure corresponds to 95% of first equilibrium pressure.c Pressure corresponds to 99% of first equilibrium pressure.

Starting from the first choking pressure of 1.78 bar, for the increasing values

of chamber pressure we can calculate the time interval. The upper limit for the

pressure is 0.95 of the first equilibrium pressure, namely, 23.75 bar. The calculated

results are given in Table 2.8.

Burnout occurs at the nozzle end and this burned distance is given by,

ybo = (rc − rhi− l tan θ) cos θ



ybo1 = [0.2− 0.025− 2.4× tan(1.5)] cos(1.5) = 0.1121m

At the burnout the propellant volume, the sliver volume, is given by,

Vp = π × 0.22 × 2.4

− [π/(3× tan(1.5))]

(0.025 + 2.4× tan(1.5) + 0.1121/cos(1.5))3

−(0.025 + 0.1121/cos(1.5))3

= 0.08484m3

Therefore the sliver fraction at the first burnout is 0.08484/0.2751 =

0.3084.

During the first phase of the burning (0 ≤ y ≤ 0.1121m) we can calculate the

propellant burning area using the equation,

Ab = (πl/cos θ) [2 (rhi + y/cos θ) + l tan θ]


Ab = [2.4π/cos (1.5)] [2 (0.025 + y/cos (1.5)) + 2.4 tan (1.5)]

= 0.8511 + 15.09y

Using the above relation, we can calculate the burning areas for different burned

distances. Corresponding values of K are given by,

K =Ab

At=

0.8511 + 15.09y

4.615× 10−3= 184.41 + 3267.772y

Corresponding equilibrium pressures can be calculated using Eq. 2.25.

p0cE = (ρpaKc∗)

1

(1− n)


p0cE =(1780× 2.814× 10−5 × (184.410 + 3267.772y)× 1560

)1.5385= 816.918× (184.410 + 3267.772y)

1.5385

For different values of burned distances within the range of first phase, we can

calculate the equilibrium pressures and the same are given in Table 2.9.

At the end of the second phase, the sliver is completely consumed. This

second phase is given by,

(rc − rhi − l tan θ) cos θ ≤ y ≤ (rc − rhi) cos θ

Table 2.9: Equilibrium operation during the first phase burning:pressure versus burned-distance, Example 2. 5

y (mm)Ab

(m2

)Vp

(m3

)p0cE (bar) y (mm)Ab

(m2

)Vp

(m3

)p0cE (bar)

0 0.85113 0.27511 25.00 58 1.72635 0.20036 74.212 0.88131 0.27338 26.38 60 1.75653 0.19688 76.214 0.91149 0.27158 27.78 62 1.78671 0.19333 78.246 0.94167 0.26973 29.21 64 1.81689 0.18973 80.288 0.97185 0.26782 30.66 66 1.84707 0.18607 82.3410 1.00203 0.26584 32.14 68 1.87725 0.18234 84.4212 1.03221 0.26381 33.64 70 1.90743 0.17856 86.5214 1.06239 0.26171 35.16 72 1.93761 0.17471 88.6316 1.09257 0.25956 36.71 74 1.96779 0.17081 90.7618 1.12275 0.25734 38.28 76 1.99797 0.16684 92.9120 1.15293 0.25507 39.88 78 2.02815 0.16282 95.0822 1.18311 0.25273 41.49 80 2.05833 0.15873 97.2724 1.21329 0.25033 43.13 82 2.08851 0.15458 99.4726 1.24347 0.24788 44.80 84 2.11869 0.15038 101.6928 1.27365 0.24536 46.48 86 2.14887 0.14611 103.9330 1.30383 0.24278 48.18 88 2.17905 0.14178 106.1832 1.33401 0.24015 49.91 90 2.20923 0.13739 108.4534 1.36419 0.23745 51.66 92 2.23941 0.13294 110.7436 1.39437 0.23469 53.43 94 2.26959 0.12843 113.0438 1.42455 0.23187 55.22 96 2.29977 0.12386 115.3740 1.45473 0.22899 57.03 98 2.32995 0.11923 117.7042 1.48491 0.22605 58.86 100 2.36013 0.11454 120.0644 1.51509 0.22305 60.71 102 2.39031 0.10979 122.4346 1.54527 0.21999 62.58 104 2.42049 0.10498 124.8148 1.57545 0.21687 64.47 106 2.45067 0.10011 127.2150 1.60563 0.21369 66.38 108 2.48085 0.09518 129.6352 1.63581 0.21045 68.31 110 2.51103 0.09019 132.0754 1.66599 0.20715 70.25 112 2.54121 0.08514 134.5256 1.69617 0.20378 72.22 112.12 2.54295 0.08484 134.66



0.11212m ≤ y ≤ 0.17494m

During the second phase, sliver burning area and sliver volume are given by,

Ab = (π/sin θ)[r2c − (rhi

+ y/cos θ)2]

Vp = (π/tan θ)

{r2c (rc − rhi − y/cos θ)

−[r3c − (rhi

+ y/cos θ)3]/

3

}


Ab = 4.8005− 120.0137× (0.025 + 1.0003y)2

Vp = 4.7989× (0.175− 1.0003y)

−39.9909×[8× 10−3 + (0.025 + 1.0003y)

3]

The K is given by,

K =Ab

At=

4.8005− 120.0137× (0.025 + 1.0003y)2

4.615× 10−3

= 1040.20− 26005.13× (0.025 + 1.0003y)2

Corresponding equilibrium pressures can be calculated using Eq. 2.25.

p0cE = (ρpaKc∗)

1

(1− n)


p0cE = 816.918×K1.5385

Substituting different values of burned distances we can calculate the values of

equilibrium pressures. The results of the calculations are given in Table 2.10.

In Table 2.11 chamber-pressure values against time are given for the ignition

transient — note that we have assumed zero burned distance during ignition

transient. In Tables 2.12 and 2.13 chamber-pressure values are given against time

for the burning durations of phases 1 and 2 respectively. While the values in Table

2.12 are before burnout, those in Table 2.13 are after burnout corresponding to

the sliver burning. The pressure-time trace of the rocket motor is given in Figure

2.11.

Discussion The generalized cylindrical grain, an important grain configura-

tion adopted in solid rocket motors of different applications, is analyzed. The

pressure-time trace comprising ignition transient and equilibrium operation has

been determined. Tail-off transient is the result of the regressivity of the sliver

burning and was calculated under equilibrium operation, Eq. 2.25. Equation

2.31 is applicable only at zero sliver and when the chamber pressure is sufficiently

above the minimum pressure required for the choking of nozzle. In the present

Example, therefore, Eq. 2.31 was not applicable.

Figure 2.11: Pressure-time trace of the rocket motor of Example2. 5.


Table 2.10: Equilibrium operation during second phase burning(sliver burning) pressure versus burned-distance, Example 2. 5

y (mm) Ab

(m2

)Vp

(m3

)p0cE (bar)

112.1154 2.5429496 0.0848432 134.6584114 2.4804595 0.0801095 129.6014116 2.4132099 0.0752157 124.2353118 2.3449994 0.0704573 118.8741120 2.2758283 0.0658363 113.5226122 2.2056963 0.0613546 108.1855124 2.1346036 0.0570142 102.8677126 2.0625501 0.0528169 97.57448128 1.9895358 0.0487646 92.31135130 1.9155608 0.0448594 87.08402132 1.840625 0.041103 81.89851134 1.7647285 0.0374975 76.76115136 1.6878712 0.0340447 71.67864138 1.6100531 0.0307467 66.65806140 1.5312742 0.0276052 61.70692142 1.4515346 0.0246222 56.83321144 1.3708342 0.0217997 52.04548146 1.289173 0.0191395 47.3529148 1.2065511 0.0166436 42.76534150 1.1229684 0.0143139 38.29354152 1.0384249 0.0121524 33.94919154 0.9529207 0.0101609 29.74518156 0.8664556 0.0083413 25.6958158 0.7790299 0.0066957 21.81711160 0.6906433 0.0052259 18.12739162 0.601296 0.0039338 14.64782164 0.5109879 0.0028213 11.4035166 0.4197191 0.0018905 8.42507168 0.3274895 0.0011431 5.751575170 0.2342991 0.0005811 3.436002172 0.1401479 0.0002065 1.558452174 0.045036 2.118E-05 0.271763174.94 1.53E-06 2.436E-14 3.62E-08


Table 2.11: Ignition transient: pressure-time tracet(s) y(m) p0E(bar) t(s) y(m) p0E(bar)

0.000 0 0.000 0.072 0 17.0000.001 0 2.000 0.080 0 18.0000.005 0 3.000 0.089 0 19.0000.009 0 4.000 0.099 0 20.0000.013 0 5.000 0.112 0 21.0000.017 0 6.000 0.128 0 22.0000.021 0 7.000 0.151 0 23.0000.025 0 8.000 0.154 0 23.1000.029 0 9.000 0.157 0 23.2000.033 0 10.000 0.160 0 23.3000.038 0 11.000 0.164 0 23.4000.042 0 12.000 0.167 0 23.5000.048 0 13.000 0.171 0 23.6000.053 0 14.000 0.175 0 23.7000.059 0 15.000 0.177 0 23.7500.065 0 16.000

Table 2.12: First phase of equilibrium operation: pressure-timetrace, Example 2. 5

t(s) y(mm) p0E(bar) t(s) y(mm) p0E(bar)

0.177 0.000 25 9.839 58.000 74.2080.583 2.000 26.377 10.117 60.000 76.2130.981 4.000 27.779 10.394 62.000 78.2371.373 6.000 29.207 10.667 64.000 80.2791.758 8.000 30.659 10.938 66.000 82.342.136 10.000 32.136 11.207 68.000 84.4192.508 12.000 33.637 11.473 70.000 86.5162.874 14.000 35.162 11.738 72.000 88.6313.235 16.000 36.711 12.000 74.000 90.7643.591 18.000 38.282 12.259 76.000 92.9143.941 20.000 39.877 12.517 78.000 95.0824.287 22.000 41.494 12.773 80.000 97.2684.627 24.000 43.133 13.026 82.000 99.474.964 26.000 44.795 13.278 84.000 101.695.295 28.000 46.479 13.528 86.000 103.9275.623 30.000 48.184 13.776 88.000 106.1815.947 32.000 49.91 14.022 90.000 108.4526.266 34.000 51.658 14.266 92.000 110.746.582 36.000 53.427 14.508 94.000 113.0446.894 38.000 55.216 14.749 96.000 115.3657.203 40.000 57.026 14.988 98.000 117.7037.508 42.000 58.856 15.225 100.000 120.0567.81 44.000 60.707 15.461 102.000 122.4268.109 46.000 62.577 15.695 104.000 124.8128.405 48.000 64.467 15.928 106.000 127.2158.697 50.000 66.377 16.159 108.000 129.6338.987 52.000 68.306 16.388 110.000 132.0679.274 54.000 70.254 16.616 112.000 134.5179.558 56.000 72.222 16.63 112.115 134.659


Table 2.13: Second phase of equilibrium operation (tail-off tran-sient): pressure-time trace

t(s) y(mm) p0E(bar) t(s) y(mm) p0E(bar)

16.63 112.115 134.658 21.153 146 47.35316.845 114 129.601 21.487 148 42.76517.077 116 124.235 21.833 150 38.29417.313 118 118.874 22.193 152 33.94917.552 120 113.523 22.57 154 29.74517.795 122 108.185 22.965 156 25.69618.043 124 102.868 23.382 158 21.81718.295 126 97.574 23.826 160 18.12718.552 128 92.311 24.301 162 14.64818.814 130 87.084 24.817 164 11.40319.081 132 81.899 25.385 166 8.42519.355 134 76.761 26.024 168 5.75219.635 136 71.679 26.771 170 3.43619.921 138 66.658 27.704 172 1.55820.216 140 61.707 29.107 174 0.27220.519 142 56.833 30.974 174.94 020.831 144 52.045

Chapter 3

Incremental Analysis

3.1 Frozen Flow Versus Shifting Equilibrium Flow

As mentioned previously, the incremental analysis is more rigorous and in-

volved than the equilibrium-pressure analysis and it is adopted for the high

performance motors, which are characterized by low port-to-throat area ra-

tios less than 3 to 4 and high volumetric loading fractions of propellant

(Vpi/Vcf ). In such motors the average velocity of combustion products in-

side the grain-port is high and the total- and static-pressures substantially

drop along the port because of mass addition. Furthermore, the static pres-

sure is significantly less than the total pressure. The total-pressure drop,

that is a loss, occurs essentially because of the continuous mass addition of

combustion products along the port. Recall that you have learned in gas

dynamics that in a pipe flow, the loss of total pressure occurs due to one or

more of the following: friction, heat addition, and mass addition. On the

other hand, the drops along the port in static-pressure and -temperature,

ideally not being losses, are due to flow acceleration consequent to the

mass addition.

Before learning the elements of incremental analysis, we should first

look into the two different assumptions generally adopted in analyzing the

flow of propellant combustion-products. The two assumptions are frozen-

flow assumption and shifting-equilibrium flow assumption. Solid

rocket propellants contain different chemicals, typically of the elements car-

bon, hydrogen, nitrogen, oxygen, and chlorine, and usually a metallic fuel –

mostly aluminum from the possible group of aluminum, boron, magnesium,

61

62 CHAPTER 3. INCREMENTAL ANALYSIS

and zirconium. For example a typical heterogeneous solid propellant con-

tains fine particles of the oxidizer ammonium perchlorate NH4ClO4 and

the fuel aluminum. The particles of the both are randomly distributed by

mixing with a liquid pre-polymer fuel, say hydroxyl-terminated polybuta-

diene (HTPB, a pre-polymer compound of hydrogen, carbon, and oxygen).

By adding a suitable chemical, the mix is subsequently cured into a solid

form. Furthermore, in order to “tailor” the processing-, burning-, as well

as structural-characteristics, small quantities of additive compounds of a

few additional elements may be added to the propellant mix during mixing.

Therefore, with several elements involved, a large number of species num-

bering to about 200 or more are to be considered to determine iteratively

the most significant species of the combustion equilibrium-composition and

the related properties such as molar mass m and ratio of specific heats γ.

Most of the species are in gaseous phase with a few of them being in liquid

and/or solid phase. For example, at a chosen flow condition of combustion

products from an aluminized propellant, particulates of alumina Al2O3 can

be in liquid as well as solid phase or in one of the phases.

With the mass addition along a constant-area port, the port flow, that

can be taken as an adiabatic one, has to accelerate with a drop in its

static-temperature as well as -pressure. Under this condition, if sufficient

time is available, some of the dissociated species in the combustion prod-

ucts such as CO, OH, H, and O may recombine to form stable species

of CO2, and H2O . Possibly, the fine liquid particulates of Al2O3 may

partially or fully get solidified. With such a scenario we visualize that, if

sufficient time is available for the recombination to take place, the equilib-

rium composition of the combustion products changes (“shifts”) from one

condition of static-temperature and -pressure to another with the change in

flow velocity. Realize further that for each such condition, the equilibrium

composition has to be calculated considering about 200 or more species

to identify iteratively the most significant ones. Therefore, in summary,

we can say that under the shifting-equilibrium-flow assumption the equi-

librium composition of propellant-combustion, with the conserved stag-

nation enthalpy, changes as per the flow conditions. Having understood

the features of shifting-equilibrium flow assumption, it follows obviously

that under frozen flow assumption, with the conserved stagnation enthalpy,

the equilibrium composition of propellant combustion is constant, that is

3.1. FROZEN FLOW VERSUS SHIFTING EQUILIBRIUM FLOW 63

“frozen”.

Shifting equilibrium-flow assumption and frozen-flow assumption the-

oretically represent the two extremes of flow behavior and rocket perfor-

mance. On the point of view of chemical kinetics, we can say that the shift-

ing equilibrium flow assumes relatively a negligible reaction-time against

the flow-residence time. By the same token, the frozen flow assumes a neg-

ligible flow-residence time against the reaction time. Actual performance

of a rocket motor is expected to be closer to the results of one of the as-

sumptions and it largely depends on the propellant composition, barring

other realities in the motor. The determination of equilibrium composi-

tion of propellant-combustion and other thermodynamic-properties have

to be carried out adopting complex computer-programs. Under shifting

equilibrium-flow assumption, it is extremely complex to develop a com-

puter program to calculate the compositions of combustion products for

changing flow conditions along the port and subsequently along the rocket

nozzle, and then to calculate the rocket performance. Programs such as

CEC71 and CEA [Refs.[1] [2]] can calculate the rocket performance adopt-

ing any one or combination of the two assumptions — for example the

combination of equilibrium flow up to the nozzle throat and frozen flow

thereafter. In industries, with the availability of such programs, adopting

high speed computers, rocket performance calculations are carried out and

analyzed exhaustively before finalizing the design.

In Table 3.1, the results of an isentropic shifting-equilibrium flow of

combustion products are presented. The flow is calculated at a cham-

ber total-pressure of 6.895 MPa (1000 psia) for a heterogeneous solid-

propellant of composition of ammonium perchlorate: HTPB: aluminium

= 69:12:19. The calculated results are from the program CEC71 [Ref. [1]].

The program identifies 238 species to be the possible products of combus-

tion. For the specified mole fractions of significance greater than 0.00001

the program iteratively finds 18 species for all assigned conditions. The

mole fractions of these species are given in Table 3.1. The results show how

the properties and composition change in an isentropic shifting-equilibrium

flow. The first row of the table represents area ratios with respect to the

nozzle throat-area. The second cell of the first row of the table represents

the stagnation or total condition and hence A/At =∞. Next two cells,

representing subsonic flow at the port exit, are for the port-to-throat area


ratios of 4 and 1.5, which can be taken as representative cases of low-

performance and high-performance rocket motors respectively. In the fifth

cell of the first row A/At = 1, represents the throat section. The last

two cells represent the supersonic-flow conditions, which have the ratios of

exit area to throat area of the nozzle (Ae/At) at 1.6667 and 9.19. These

two nozzle area ratios correspond to the nozzle pressure ratios 6 and 70

respectively. The numbers in the 2nd to 6th rows are the corresponding

values of Mach number, temperature, nozzle pressure-ratio, molar mass,

and isentropic exponent along the port and nozzle [Isentropic exponent

γs is defined as γs ≡ γ/(∂ ln v/∂ ln p)T where γ is the specific heat ratio.

For more details on γs refer [3]]. The numbers in other rows represent the

mole fractions of the combustion product species.

Table 3.1: Results of a typical shifting equilibrium flow calculationusing program CEC71

A/At ∞ 4 1.5 1 1.6667a 9.1900a

M 0 0.151 0.44 1 2.026 3.163T (K) 3027 3021 2979 2789 2327 1561p0/p 1 1.0134 1.1193 1.7617 6 70mc(kg/kgmol)23.49 23.491 23.503 23.551 23.624 23.649γs 1.1724 1.1727 1.1744 1.1828 0.9993 1.2306

ALCL 0.00117 0.00115 0.00098 0.00043 0.00003 0ALCL2 0.00016 0.00015 0.00013 0.00006 0 0ALCL3 0.00016 0.00016 0.00015 0.00011 0.00003 0ALOH 0.00082 0.0008 0.00067 0.00027 0.00001 0ALOHCL 0.00019 0.00018 0.00015 0.00006 0 0ALOHCL2 0.0007 0.00069 0.00063 0.00041 0.0001 0AL(OH)2CL 0.00019 0.00019 0.00017 0.0001 0.00002 0CO 0.2877 0.28771 0.28777 0.28787 0.28653 0.2762CO2 0.01294 0.01295 0.01304 0.01356 0.01583 0.02649CL 0.00314 0.00311 0.0029 0.00199 0.00055 0.00001H 0.01142 0.0113 0.01046 0.00702 0.00184 0.00002HCL 0.13105 0.13114 0.13184 0.13445 0.13783 0.13888H2 0.32159 0.3217 0.32251 0.32571 0.33126 0.34288H2O 0.10783 0.10779 0.10751 0.10645 0.10405 0.09349N2 0.06897 0.06898 0.06901 0.06915 0.06937 0.06944OH 0.0014 0.00138 0.00123 0.0007 0.00011 0AL2O3(S) 0 0 0 0 0.01322 0.05259AL2O3(L) 0.05053 0.05057 0.05081 0.05164 0.03922 0

a ratio of nozzle exit area to throat area for supersonic flow condition.

Scrutinizing the mole fractions and other properties at stagnation con-

dition A/At =∞ and at A/At =4 in the second and third columns respec-

tively, we note that the properties are more or less the same, indicating

3.1. FROZEN FLOW VERSUS SHIFTING EQUILIBRIUM FLOW 65

the suitability of equilibrium-pressure analysis using frozen flow assump-

tion for low performance motors. Examining similarly the mole fractions

and other properties in the second and fourth columns, we observe that

the properties in the fourth column corresponding to A/At =1.5 are sub-

stantially different from the ones in the second column, implying the un-

suitability of equilibrium-pressure analysis using frozen flow assumption

for high performance motors.

Note how the mole fractions of liquid alumina Al2O3 (l) changes from

stagnation condition to flow conditions along the port and the nozzle.

Appreciate the possibility of Al2O3 coexisting in liquid as well as solid

phase in a flow. Up to the throat, Al2O3 exists in liquid phase and at the

nozzle pressure ratio of 6, partial solidification of alumina has taken place

and this solidification has been completed downstream.

Be alerted that the results in Table 3.1 are as per the isentropic shifting

equilibrium flow where the total pressure is held constant. But the flow

in a propellant-grain port of a high performance motor is one with mass

addition and consequent loss of total pressure with increase in entropy.

However, the trend of shift in mole fractions that are dependent on static

properties is similar to the one shown in Table 3.1.

Example 3. 1

From the data given in Table 3.1 calculate the mass- and volume-fraction of

alumina Al2O3 in the products of combustion under combustion chamber condi-

tions. Density of liquid alumina is 3800 kg/m3.

Solution In the second column of Table 3.1, the properties of combustion

products from a typical heterogeneous propellant containing aluminum are given

along with mole fractions of product species. In the case of mixtures containing

gaseous- and condensed-species two definitions for molar mass can be given. Molar

mass with respect to only gaseous species m is defined as follows.

m ≡

NS∑j=1

njmj

NG∑j=1

nj

where nj and mj are kg-moles and molar mass of the species j respectively. NG

are the total number of gaseous species and NG + 1, ..., NS are the condensed

species. Molar mass with respect to all the species, gaseous as well as condensed


species, mc is defined as follows.

mc ≡

NS∑j=1

njmj

NS∑j=1

nj

It follows then,

mc = m

1−NS∑

j=NG+1

xj

where xj is the mole fraction of species j relative to all species in the multiphase

mixture. For the total products including condensed species, the molar mass mc

is given as 23.490 kg/kg-mole.

Analysis

Mole fractions of the combustion products are given and all these mole frac-

tions add up to 1 kg-mole. Of this Al2O3 (l) is of 0.05053 kg-mole/kg-mole of

mixture containing gaseous as well as condensed species.

Molar mass of the mixture mc = 23.49 kg/kg-mole

In this one kg-mole of mixture, number of kg-mole of Al2O3 (l) = 0.05053kg-

mole/kg-mole

In this one kg mole mixture, number of kg-moles of gaseous species = (1-

0.05053) = 0.94947 kg-mole/kg-mole

Number of kg-moles ofAl2O3 (l) per kg of mixture = 0.05053/23.49 = 2.15113×10−3kg-

mole/kg

Number of kg-moles of gaseous species per kg of mixture = 0.94947/23.49 =

0.04042 kg-mole/kg

Noting that molar mass of Al2O3 is 101.96 kg/kg-mole, mass of Al2O3 (l) per

kg of mixture = 2.15113× 10−3×101.96 = 0.2193 kg/kg of mixture

Therefore mass fraction of condensed species = 0.2193 kg/kg of

mixture.

Volume of Al2O3 (l) = 0.2193/3800 = 5.7711×10−5 m3/kg of mixture

Ideal gas equation of state for the mixture is given by

pv = nRuT

3.2. INCREMENTAL-ANALYSIS PROCEDURE 67

where v is the specific volume (m3/kg), n is the number of kg-moles of gaseous

species per kg of mixture, and Ru is the universal gas constant = 8314.51 J/kg-

mole-K.

Therefore the volume of gaseous mixture,

v =nRuT

p=

0.04042× 8314.51× 3026.73

68.95× 105= 0.14795

m3

kg of mixture

Volume fraction of Al2O3 (l) with respect to the volume of the gaseous

species = 5.7711× 10−5/0.14795 = 3.90064× 10−4.

Discussion

1. In a typical heterogeneous propellant having a metal component, the con-

densed species in the combustion products is of mole fraction around 5 to

6 percent or a little more. Note however, its mass fraction exceeds 20 per-

cent. Realize that for every kg of combustion products being ejected by the

rocket nozzle the mass of condensed particulates exceeds 0.2 kg.

2. Environmental aspect on the ejection of abrasive particulates such asAl2O3(s)

apart, rocket scientists bother about the realities of the particulates flow

with respect to gas flow. In analyzing this multiphase flow they have to

look into the aspects of heat transfer between gaseous species and condensed

species and of velocity lag between them — recall that the rocket thrust

is equal to the nozzle exit flow rate multiplied by the effective nozzle-exit

velocity. These aspects are included in rigorous analyses to estimate the

performance of rocket motors.

3. Despite the high mass fraction, note that the volume fraction of the con-

densed species with respect to gaseous species is around 4× 10−4. This is

the reason for the statement in the Section 1.1 that a typical propellant on

burning produces mostly gaseous species by volume.

3.2 Incremental-Analysis Procedure

The realities of the flow in the port cavity of a solid rocket motor were

briefly discussed in Chapter 1 under Section 1.3. In applying the incremen-

tal analysis for high-performance motors, option is available to adopt the

less elaborate frozen flow assumption or the complex shifting-equilibrium-

flow assumption. The flow in the grain port and in the nozzle convergence


is subsonic and hence significantly slower than that in the nozzle diver-

gence, where it is supersonic. Therefore, the flow-residence time available

for the port flow is more than that for the supersonic flow in the nozzle di-

vergence. Notwithstanding the fact that the residence time for the burned

products entering the grain port reduces from the head end to nozzle end, a

suitable shifting-equilibrium-flow assumption up to the nozzle throat looks

appropriate. In rocket industries elaborate incremental-analysis programs

adopting shifting equilibrium flow assumption can be developed. How-

ever, for the purpose of easy and quick understanding of the incremental-

analysis we adopt here frozen-flow assumption. We will therefore discuss

here a “relatively simplified” incremental analysis. The analysis is for a

one-dimensional frozen port-flow that is adiabatic with mass addition and

zero surface-friction. Conservation equations for unsteady flow that are

more applicable for ignition- and tail-off-transients, will be first derived.

From these, the equations for steady flow, which are more applicable for

equilibrium operation, will be deduced.

As the word equilibrium is coming into many places, it is appropriate

that you note the differences between its usage with reference to rocket

operating conditions and assumptions adopted: (a) equilibrium opera-

tion (against ignition transient and tail-off transient — Section 1.2, (b)

equilibrium-pressure analysis (against incremental analysis — Section 1.3)

and (c) shifting-equilibrium flow (against frozen flow — Section 3.1)

Figure 3.1: Incremental stations fixed with respect to a rocketchamber.

Let us look at the procedure to be followed for the incremental analysis.

A sufficiently large number of incremental stations,

(1, 2, .....j, j + 1, ......, n)

spatially fixed with respect to the rocket chamber, are chosen along the

3.2. INCREMENTAL-ANALYSIS PROCEDURE 69

port as shown in Fig. 3.1. The port cavity between any two successive

incremental stations becomes the control volume, Fig. 1.3. Under the

assumption of constant stagnation enthalpy, the energy conservation is

automatically satisfied. The incremental analysis is as per the following

steps. These steps are common for the entire operation of the rocket motor:

ignition transient, equilibrium operation, and tail-off transient.

1. For a known port-envelope at a given instant, adopting a trial total-

pressure at the head end, the flow of combustion products through

the first incremental-segment between the station numbers 1 and 2 is

analysed applying the equations of mass- and momentum-conservation,

and the ideal gas equation of state. Through this analysis the flow

conditions at the exit station 2, namely total pressure, static pres-

sure, static temperature, and velocity, are calculated.

2. The evaluated flow conditions at the exit station of a segment be-

come the conditions at the entry station of the next segment for the

analysis to be continued.

3. This spatial march leads to the determination of the conditions of

the accumulated mass flow rate at the nth station, the exit of the

nozzle-end segment, Fig. 3.1.

4. After accounting for the pressure loss in the plenum, the nozzle-entry

total pressure p0N is evaluated.

5. With this total pressure p0N , the accumulated total mass flow rate

of combustion products at the nozzle entry should be able to pass

through the nozzle throat within an acceptable error band. If this

mass balance condition is satisfied go to step 6; if not, a new trial

total-pressure at the head end is chosen to repeat the calculations —

go to step 1.

6. On the fulfillment of the mass-balance, the spatial iterative process

is complete and the internal flow field of the rocket chamber is known

for the given instant.

7. Calculate the thrust for the instant. Evaluate the mass of propellant

ejected for the time increment and sum it up with the cumulative

mass of propellant ejected so far.


8. Under quasi steady state assumption, for a chosen small time-increment,

the burning propellant-surface is allowed to regress to a new port en-

velope.

9. If complete burnout has occurred go to step 10. If not go to step 1.

10. The adequacy on the number of spatially-fixed incremental stations

as well as the number of temporal increments is checked after the

complete burnout by the global mass conservation, namely, within

an acceptable error-band, the total cumulative mass ejected through

the nozzle [step 7] plus the mass of the combustion products in the

combustion chamber cavity should be equal to the total mass of pro-

pellant stored in the rocket chamber. If this global mass balance

is satisfied within an acceptable error band, the rocket performance

prediction is complete; if not choose an increased number of incre-

mental stations and finer temporal increments and go to step 1.

3.3 Assumptions in Incremental Analysis

The assumptions under incremental analysis adopting frozen flow assump-

tion are the following.

1. The port flow is one dimensional. The port wall is frictionless and

the mass addition of combustion products into port occurs with zero

axial momentum.

2. The port flow is adiabatic with a constant stagnation enthalpy. There

is no heat transfer from the port flow into the propellant surface. At

a given instant of operation and the corresponding port envelope,

the composition of the combustion products is calculated for a to-

tal pressure and is held constant, that is “frozen” and hence molar

mass is constant. Furthermore, the specific heats are assumed to be

constant and hence the ratio of specific heats γ is also a constant.

Consequently, the total temperature is constant and it is equal to

the adiabatic flame temperature determined for the total pressure.

3. The combustion products satisfy the ideal gas equation of state. The

flow due to the inhibitor fragmentation is negligible with respect to

the flow of combustion products from the propellant grain. The inert

3.3. ASSUMPTIONS IN INCREMENTAL ANALYSIS 71

materials in the combustion-chamber cavity, namely the insulator

and the liner do not decompose during the motor operation and

hence do not form part of the nozzle flow.

4. The nozzle of the rocket motor is always choked. The ignition of the

entire grain surface is instantaneous with negligible mass of igniter.

5. The propellant grain and the motor casing are infinitely rigid and

hence do not deform during the motor operation due to the spatial

and temporal variations in pressures.

Let us consider the rationales for the first simplifying assumption. This

is an assumption not found under the “zero-dimensional” equilibrium-

pressure analysis, detailed in Chapter 2 under Section 2.1. In the equilib-

rium pressure analysis, by neglecting the difference between the total and

static pressure and taking one uniform pressure for the entire combustion-

chamber cavity, we simulate the rocket-motor grain-cavity to an enlarging

“pressure-vessel” or settling-chamber with some mass entering (combus-

tion products from the burning propellant) and some other mass exiting

(combustion products exiting through the nozzle throat). But in the in-

cremental analysis we analyze the flow of combustion products within the

enlarging pressure-vessel one dimensionally. Under rocket operating con-

ditions, the burning propellant blows the combustion products essentially

normal to its surface with velocities in the range of 1-3 m/s. As the flame

distance from the solid surface is about 100µ or less, the blowing occurs

very close from the surface. Therefore the port flow having a boundary

layer with blowing can be assumed to be frictionless and to be receiving

the mass addition with zero axial momentum.

Let us elaborate the features involved in the second assumption. Heat

transfer from the high temperature port flow into the propellant grain does

take place. But compared to the total energy contained in the port flow,

the energy involved in the heat transfer to the grain surface is negligible.

Hence, we assume the port flow to be adiabatic. Along the port at a chosen

instant, total- as well as static-pressure drops and so the equilibrium com-

position can change. But, by the frozen flow assumption we have to adopt

a fixed composition for the port flow at that instant. Generally, for the

chosen instant, the head-end total pressure or the average of the total pres-

sures at head- and nozzle-end can be selected to calculate the equilibrium


composition that is to be kept constant. But these pressures are not known

a priori at the specified instant. Therefore for the first time-instant, we can

assume a trial total-pressure and calculate the corresponding trial compo-

sition of combustion products. Keeping this composition frozen, through

a spatial iteration we can evaluate the port flow, which gives a new trial

total-pressure. For this new trial total-pressure calculate the composition

of the combustion products to continue the spatial iteration. Within a few

cycles of this “trial total pressure, its composition, and spatial iteration,”

we should be able to arrive at the convergence. As the trial values for the

subsequent incremented time, we start with the total pressure (head-end

value or the average of head-end and nozzle-end values) and its composition

of the preceding instant.

Regarding the above third and fourth simplifying assumptions, the

points that we discussed under the equivalent second and third assumptions

under the equilibrium-pressure analysis hold good, Section 2.1.

Note that the fifth simplifying assumption is different from the equiva-

lent fourth simplifying assumption under the equilibrium-pressure analysis.

Here we additionally mention the spatial pressure variation and neglect the

structural effects of spatial and temporal pressure-variations on the pro-

pellant grain as well as the casing, saying that these are infinitely rigid. In

reality however, these structural members are flexible. Among the casing

materials, metallic ones are less flexible than composites. Metal casings

are generally adopted for boost- and lower-stages. But, in order to make

them lighter, upper stages are invariably with composite material casings

and are of higher performance. Therefore, in high performance motors,

structural effects may be considerable, more so for propellant grains since

these are viscoelastic materials and less rigid than casing materials. Note

that in a hardware carrying high speed flows the local structural deflection

is due to the static pressure there. In a high performance motor, since the

port flow velocity can increase from near zero at the head end to a very

high value at the nozzle end, the static-pressure drop from head end to

nozzle end can be considerable, leading to a constriction of the port end.

A situation that can arise in a motor having a propellant grain of very low

structural strength and low port-to-throat area ratio is shown exaggerated

in Fig. 3.2 — the possibility of the port-end exit-area becoming close to

the nozzle throat area or less! In industries, incremental analysis programs

3.4. EROSIVE BURNING 73

considering deflections of propellant grain and casing are in use. However,

for simplicity of the present analysis we assume that the motor casing and

the propellant grain are infinitely rigid so that there is no deflection of

these structural members during the motor operation.

Figure 3.2: Deflection (exaggerated) due to static pressure fallalong the port of a pure cylindrical grain of low structuralstrength.

3.4 Erosive Burning

Erosive burning is the dependence of the burning rate of solid propellants

on the crossflow properties of the burned products over the burning sur-

face. This is in addition to the burning rate that is dependent on the static

pressure experienced by the surface. Previously we wrote Eq. 2.7 with the

implicit notion that there was no erosive burning effect and the burning

rate was purely dependent on the static pressure experienced by the burn-

ing surface. As the erosive burning effect is considered now, let us denote

the static pressure dependent burning rate as the one at zero crossflow or

the “normal” burning rate r0. Therefore r0 as per Saint-Roberts equation

is,

r0 = apn (3.1)

where a is the pre-exponent factor, n is the combustion index, and p is the

static pressure experienced by the burning surface.

The most famous and widely recognized erosive burning model was

developed by Lenoir and Robillard based on heat transfer theory [Ref.

[4]]. In this model they proposed the following mechanism. To maintain

combustion, the solid propellant receives heat from two sources to bring

each succeeding layer of propellant to the burning surface temperature Ts

from the base temperature Ti.


The first source of heat is from the primary burning zone. The mech-

anism of heat transfer from this primary zone to the propellant is by a

complex combination of conduction, heterogeneous turbulent convection,

and radiation. The narrower the primary burning zone, the less resistance

exists to heat transfer by this complex mechanism. Increased static pres-

sure is considered to narrow the primary burning zone through an increase

in the gas phase reaction rate. This mechanism of heat transfer rate is thus

static-pressure dependent but it is independent of the crossflow velocity.

The second source of heat is from the crossflow of combustion prod-

ucts through the convective heat transfer and is therefore dependent upon

crossflow rates. Thus the burning rate is proposed to be the sum of the

two effects, a rate dependent on static pressure r0 and an erosive rate

dependent upon the combustion-products crossflow rate re. Thus,

r = r0 + re (3.2)

where re is the erosive burning rate component. The erosive burning rate

component is postulated to be proportional to the convective heat transfer

coefficient h under the condition of blowing and can be written with respect

to convective heat transfer coefficient with zero blowing h0 as

h = h0e

−βrρpG (3.3)

where β, ρp, and G are respectively dimensionless exponential constant,

propellant density, and mass flux of crossflow ρu. Note that rρp is the

blowing mass-flux against the crossflow mass-flux G = ρu. Substituting

Eq. 3.3 into Eq. 3.2, we get,

r = apn + kh = apn + kh0e

−βrρpG (3.4)

where k is the proportionality constant to be determined. The convective

heat transfer coefficient under zero blowing h0 is correlated by Chilton-

Colborn equation for flow over a flat plate,

h0 = 0.0288GcpRe−0.2Pr−0.667 (3.5)


where Re and Pr are respectively Reynolds number and Prandtl number.

Although this equation is originally proposed for flow over flat plate, it can

be applied to flow through grain ports by incorporating the characteristic

dimension as the hydraulic diameter D. Combining Eqs. 3.5and 3.4,

r = apn + 0.0288GcpRe−0.2Pr−0.667ke

−βrρpG (3.6)

Noting G = ρu and Re = uDρ/µ, Eq. 3.6 is simplified to,

r = apn +αG0.8

D0.2e

βrρp

G

(3.7)

where

α = 0.0288cpµ0.2Pr−0.667k (3.8)

Recall Pr ≡ µcp/λ. An expression for the proportionality constant k is de-

rived by considering the energy balance between the heat transfer from the

flame to the propellant surface and the heat required to raise the propellant

temperature from its initial temperature Ti to the surface temperature Ts.

The heat balance per unit area is given by,

h (T0 − Ts) = reρpcs (Ts − Ti) (3.9)

This equation assumes that there is no significant exothermic or endother-

mic process occurring in the solid phase during the heating from Ti to the

burning-surface temperature Ts. Solving Eq. 3.9 and comparing with the

earlier expression for re in Eq. 3.4,

re =h

ρpcs

(T0 − TsTs − Ti

)= kh (3.10)

k =1

ρpcs

(T0 − TsTs − Ti

)(3.11)


Therefore, the erosive burning rate equation due to Lenoir and Robillard

can be written as

r = apn +αG0.8

D0.2e

βrρp

G

(3.12)

where α the erosive constant of dimension (m2.8/kg0.8 − s0.2) is,

α =0.0288cpµ

0.2Pr−0.667

ρpcs

(T0 − TsTs − Ti

)(3.13)

Although the value of β was proposed to be 53 by Lenoir and Robillard

based on their experiments, the value of β can be chosen based on the

experimental results of the motor and propellant under investigation.

To characterize the erosive burning effect, often the ratio of total burn-

ing rate to normal burning rate, termed as erosive burning ratio ε, is

adopted.

ε ≡r

r0= 1 +

re

r0(3.14)

Example 3. 2

An aluminized composite propellant has the following properties. Specific

heat of solid propellant, cs = 1400J/kg-K. Density ρp = 1750 kg/m3. Pre-

exponent factor a in the burning rate equation r0 = apn is 3×10−5m/s. Burning

rate index n = 0.4. Adiabatic flame temperature (stagnation temperature) T0 =

3610 K. Stagnation pressure p0 = 7 MPa. Molar mass of combustion products

m = 29.7 kg/kg-mole. Specific heat at constant pressure of combustion products

cp = 1975 J/kg-K. Viscosity of combustion products µ = 1.0049 × 10−3Poise.

Prandtl number Pr = 0.4922. Average surface temperature of burning propellant

Ts = 1000 K. Propellant base temperature Ti = 300K

The hydraulic diameter of the grain port is 0.1m. If the propellant is assumed

to follow the Lenoir-Robillard erosive burning rate model, calculate the total

burning rate of the propellant for two crossflow Mach numbers of 0.5 and 0.7 at

the given stagnation temperature. Distinguish the normal and erosive component

of the burning rates. Assume that the dimensionless exponential constant in the

Lenoir-Robillard equation to be 60. The Lenoir-Robillard equation is given by


r = apn +αG0.8

D0.2e

βrρp

G

where

α =0.0288cpµ

0.2Pr−0.667

ρpcs

(T0 − TsTs − Ti

)

Solution The total burning rate (normal burning component plus the erosive

burning component) has be calculated for a fixed stagnation pressure under two

crossflow Mach numbers. From the given values, the ratio of specific heats γ, and

static pressures and mass fluxes for the two crossflow Mach numbers have to be

calculated. Since the total burning rate r is implicit, the total burning rate has

to be calculated through a suitable iteration.

Assumptions Although it is known that propellant surface temperature in-

creases as the static pressure increases, its variation is small in the rocket operating

pressure variation during equilibrium operation. Therefore, here the propellant

surface temperature is assumed constant.

Analysis

The mass flux G is given by,

G = ρu =p

RTu

By routine gas-dynamic manipulations we get

G = Mp0

√γ

RT0

(1 +

γ − 1

2M2

)− (γ + 1)

2 (γ − 1)

γ =cp

cp −R=

1975

1975− 8314.51/29.7= 1.1652


GM=0.5 = 0.5× 7× 106

√√√√√√ 1.1652

8314.51

29.7× 3610

(1 +

0.1652

2× 0.25

) −2.1652

2× 0.1652

= 3287.06kg

m2s

GM=0.7 = 0.7× 7× 106

√√√√√√ 1.1652

8314.51

29.7× 3610

(1 +

0.1652

2× 0.49

) −2.1652

2× 0.1652

= 4056.834kg

m2s

µ = 1.0049× 10−3Poise = 1.0049× 10−3g

cm− s= 1.0049× 10−4

kg

m− s

α =0.0288× 1975×

(1.0049× 10−4

)0.20.4922−0.667

1750× 1400×

(3610− 1000

1000− 300

)

= 2.20344× 10−5

p = p0

(1 +

γ − 1

2M2

) −γγ − 1

pM=0.5 = 7× 106 ×

(1 +

0.1652

2× 0.25

)−1.1652

0.1652= 6060205Pa


pM=0.7 = 7× 106 ×

(1 +

0.1652

2× 0.49

)−1.1652

0.1652= 5291290Pa

The normal burning rates are given by,

r0M=0.5= 3× 10−5 × 60602050.4 = 0.01549

m

s

r0M=0.7= 3× 10−5 × 52912900.4 = 0.01467

m

s

For the crossflow Mach number of 0.5, the total burning rate is given by,

rM=0.5 = 0.01549 +2.20344× 10−5 × 3287.060.8

0.10.2e

60× 1750rM=0.5

3287.06

= 0.01549 + 0.022727× e−31.9434rM=0.5

Solving iteratively we get the total burning rates at the crossflow Mach number

= 0.5 as,

rM=0.5 = 0.025540m

s

The erosive burning rate component at the crossflow Mach number of 0.5 is,

reM=0.5 = rM=0.5 − r0M=0.5= 0.025540− 0.01549 = 0.010051

m

s

The erosive burning ratio ε, defined as the ratio of the total burning rate and

normal burning rate, for the crossflow Mach number of 0.5 is given by

εM=0.5 =0.025540

0.01549= 1.6488


For the crossflow Mach number of 0.7, the total burning rate is given by,

rM=0.7 = 0.01467 +2.20344× 10−5 × 4056.8340.8

0.10.2e

60× 1750rM=0.7

4056..834

= 0.01467 + 0.026894× e−25.8823rM=0.7

Solving iteratively we get the total burning rate at the crossflow Mach number =

0.7 as,

rM=0.7 = 0.027775m

s

The erosive burning rate component at the crossflow Mach number of 0.7 is

reM=0.7 = rM=0.7 − r0M=0.7= 0.027775− 0.01467 = 0.013105

m

s

The erosive burning ratio ε for the crossflow Mach number of 0.7 is given by,

εM=0.7 =0.027775

0.01467= 1.8933

Discussion As often said, a successful model need not be of non-tractable

mathematics; nor should it be fully correct. After Lenoir-Robillard model, al-

though quite a few modeling efforts and improvements have been done for erosive

burning effect, the model of Lenoir-Robillard captures the most observed behav-

iors of erosive burning phenomenon. Here in this example we find that the erosive

burning effect is more for higher mass flux, which is a known fact. On further

analysis you will find that the model predicts the erosive burning effect to be more

for smaller motors (for smaller characteristic dimension D ) and slower burning

propellant, which are the observed behavior in rocket motor operations.

The properties given in the example such as adiabatic flame temperature

T0, specific heat of combustion products at constant pressure cp, molar mass,

viscosity, and Prandtl number can be determined for the chosen propellant by

adopting standard codes such as CEC71 [Ref. [1]].

3.5. INCREMENTAL-ANALYSIS EQUATIONS 81

3.5 Incremental-Analysis Equations

3.5.1 Unsteady Flow Equations

Let us consider the unsteady flow conditions through the jth incremental

segment (the control volume) between the jth and (j+1)st stations, shown

in Figures 3.1 and 3.3. Into this segment propellant combustion products

of mass flow rate mj is entering the incremental station j. From the

burning grain-surface, additional mass flow rate of dmj is entering with

zero axial-momentum. Mass flow rate of combustion products exiting the

segment is m(j+1). The mass accumulation rate, the unsteady term, is

dm/dt = ∂∂t

(j+1)∫j

ρAdx. From fundamentals, the mass conservation for the

Figure 3.3: Typical incremental segment with flow conditions.

control volume can be written as,

∂

∂x(ρuA) +

∂

∂t(ρA) = 0 (3.15)

Integrating with respect to x,

m(j+1) − mj − dmj +∂

∂t

(j+1)∫j

ρAdx = 0 (3.16)


That is,

ρ(j+1)u(j+1)A(j+1) − ρjujAj − dmj +∂

∂t

(j+1)∫j

ρAdx = 0 (3.17)

Assuming an average density ρm, the last term on the left side of the

equation, representing the rate of change of mass within the control volume,

can be written as,

∂

∂t

(j+1)∫j

ρAdx =∂

∂t(ρmV ) = ρm

dV

dt+ V

dρm

dt(3.18)

Using finite differences and adopting ideal gas equation of state, the con-

tinuity equation, Eq. 3.17, on simplification becomes,

ρ(j+1)u(j+1)A(j+1) =

ρjujAj + dmj −

(pj + p(j+1)

) (2Vj − V

′j

)− Vj

(pj

′+ p

′

(j+1)

)R(Tj + T(j+1)

)∆t

(3.19)

where V′j , p

′j , and p

′

(j+1) correspond to the preceding time instant and are

respectively the jth segment′s control volume, and inlet and exit pressures.

The mass addition rate with zero axial momentum dmj can be written as,

dmj = ρp ¯rj

(sj + s(j+1)

)2

dx (3.20)

Here ¯rj is the average burning rate of propellant for the jth segment. In

high performance rocket motors erosive burning effect is frequently ob-

served. Therefore, the equation for dmj , Eq. 3.20 can be written as,

dmj = ρp

(εj + ε(j+1)

2

)a

(pj + p(j+1)

2

)n (sj + s(j+1)

)2

dx (3.21)

As the fluid dynamic properties change along the jth segment from the

station j to (j + 1), we have to know the fluid dynamic properties at the


station (j + 1) to evaluate dmj . But the properties at the station (j + 1)

are the very unknowns that we have to evaluate. Since the incremental

stations as well as the temporal increments are to be sufficiently closely-

spaced to give the global mass conservation within an acceptable error,

[see item 10 under Section 3.2], we approximate that the average burning

rate for the jth segment is based on the fluid dynamic properties at the

station j. Thus, the approximated equation for dmj is,

dmj∼= ρpεjap

nj

(sj + s(j+1)

)2

dx (3.21a)

With this approximation and on adopting perfect gas equation of state,

the continuity equation becomes,

p(j+1)u(j+1)A(j+1)

RT(j+1)=pjujAj

RTj+ ρpεjap

nj

(sj + s(j+1)

)2

∆x

−

(pj + p(j+1)

) (2V − V ′

)− V

(pj

′+ p

′

(j+1)

)R(Tj + T(j+1)

)∆t

(3.22)

Note that the last term on the right hand side of the above equation

represents the unsteady condition.

From fundamentals, the momentum equation for the control volume

can be written as,

∂ (ρuA)

∂t+∂(ρu2A

)∂x

= −∂ (pA)

∂x+ p

∂A

∂x(3.23)

Integrating the above equation with respect to x between the stations j

and (j + 1),

∂

∂t

(j+1)∫j

(ρuA) dx+(ρ(j+1)u

2(j+1)A(j+1) − ρju2jAj

)

=(pjAj − p(j+1)A(j+1)

)+

(j+1)∫j

pdA

(3.24)

The last integral can be written as,


(j+1)∫j

pdA =pj + p(j+1)

2

(A(j+1) −Aj

)(3.25)

Considering mean density ρm and mean velocity um as,

ρm =

(pj + p(j+1)

)R(Tj + T(j+1)

) (3.26)

um =(uj + uj+1)

2(3.27)

The unsteady momentum term can be written as,

∂

∂t

j+1∫j

(ρuA) dx =∂ (ρmumV )

∂t=

ρmumdV

dt+ ρmV

dum

dt+ umV

dρm

dt

(3.28)

Since mean acceleration dum/dt is found be comparatively negligible,

∂

∂t

j+1∫j

(ρuA) dx =∂ (ρmumV )

∂t∼= ρmum

dV

dt+ umV

dρm

dt(3.29)

Substituting Eqs. 3.25 and 3.29 into Eq. 3.24, and adopting ideal gas

equation of state and finite differences, the momentum equation becomes,


p(j+1)A(j+1) =

pjAju2j

RTj−p(j+1)A(j+1)u

2(j+1)

RT(j+1)+ pjAj

+

(pj + p(j+1)

)2

(A(j+1) −Aj

)

−(pj + p(j+1)

) (uj + u(j+1)

)2R(Tj + T(j+1)

)(V − V ′

)∆t

−V(uj + u(j+1)

)2R(Tj + T(j+1)

)(pj + p(j+1) − pj

′ − p′(j+1)

)∆t

(3.30)

Note that the last two terms on the right hand side of the above equation

represent the unsteady condition.

The general energy equation can be written as,

T0 = T j +u2j2cp

= T(j+1) +u2(j+1)

2cp(3.31)

3.5.2 Steady Flow Equations

For steady flow conditions the governing equations are readily obtained

from the above by dropping the unsteady terms from Eqs. 3.22 and 3.30.

And, the mass- and momentum-conservation equations on simplifications

become,

p(j+1) =RT(j+1)

A(j+1)u(j+1)

[pjujAj

RTj+ ρpεjap

nj

(sj + s(j+1)

)2

dx

](3.32)

p(j+1) =

pjAju

2j

RTj− m(j+1)u(j+1)

+pj

(A(j+1) +Aj

)2

2(A(j+1) +Aj

) (3.33)


The energy conservation Eq. 3.31 remains the same

T0 = T j +u2j2cp

= T(j+1) +u2(j+1)

2cp(3.31)

3.5.3 Solution of Steady Port-Flow

Let us first consider the solution of port flow for steady flow situations. The

more involved solution-procedure for unsteady port-flow will be considered

next.

The three steady conservation-equations have the three unknowns p(j+1),

T(j+1), and u(j+1). By elimination, we deduce one equation with one un-

known. With the approximations that we adopted to calculate dmj , Eq.

3.21, the exiting mass flow rate can be readily calculated. By the en-

ergy conservation Eq. 3.31 and ideal gas equation of state, p(j+1) can be

expressed as,

p(j+1) =RT(j+1)m(j+1)

A(j+1)u(j+1)=

R

T0 − u2(j+1)

2cp

m(j+1)

A(j+1)u(j+1)(3.34)

Equating Eq. 3.34 with Eq. 3.33 to eliminate p(j+1), we get,

R

T0 − u2(j+1)

2cp

m(j+1)

A(j+1)u(j+1)=

pjAju

2j

RTj− m(j+1)u(j+1)

+pj

(A(j+1) +Aj

)2

2(

A(j+1) +Aj)

(3.35)

On simplification, we get a quadratic equation in u(j+1).


[2(

A(j+1) +Aj)− γ − 1

2γA(j+1)

]u2(j+1)−

[pj

m(j+1)+

2mjuj

m(j+1)

(A(j+1) +Aj

)]u(j+1) +RT0

A(j+1)= 0

(3.36)

We are all aware of the solution to a quadratic equation:

x =−b±

√b2 − 4ac

2a(3.37)

Here,

x = u(j+1) (3.38)

a =

[2(

A(j+1) +Aj)− γ − 1

2γA(j+1)

](3.39)

b = −

[pj

m(j+1)+

2mjuj

m(j+1)

(A(j+1) +Aj

)] (3.40)

c =RT0

A(j+1)(3.41)

On solving for u(j+1), the value of T(j+1) can be determined from the en-

ergy conservation equation, Eq. 3.31. Note that the quadratic equation

Eq. 3.36 will give two solutions for u(j+1). While analyzing compressible

flows, we get usually more than one solutions and we have to adopt the

one that is practical. In most situations, the practical solution can be de-

termined by applying the increase of entropy principle of the second law of

thermodynamics. However, in the case of isentropic flow through nozzles,

note that a given area of the nozzle with respect to the choked throat-area

can have a subsonic flow in the nozzle convergence and a supersonic flow

in the nozzle divergence. Both are practical! And, we have to choose the

one that is appropriate for the nozzle section.

With the known values of u(j+1) and T(j+1), p(j+1) can be calculated

from,


p(j+1) =RT(j+1)m(j+1)

A(j+1)u(j+1)(3.42)

When we determine the exit values for segment by hand calculations, it

is a good practice to check the calculated values by substituting them on

the right hand side of the following momentum-equation, where the term

m(j+1) in Eq. 3.33has been expanded.

p(j+1) =

pjAju

2j

RTj−p(j+1)A(j+1)u

2(j+1)

RT(j+1)

+pj

(A(j+1) +Aj

)2

2(A(j+1) +Aj

) (3.43)

With the known values of T(j+1) and p(j+1), the density ρ(j+1) can be

determined using perfect gas equation. With the known values of T(j+1)

and p(j+1), and the constant total temperature T0, p0(j+1) is determined

from,

p0(j+1) = p(j+1)

(T0

T(j+1)

)γ/(γ−1)(3.44)

Note that the first incremental station is fixed a little away from the

port head, Fig. 3.1. Propellant combustion products of the burning surface

from the port head region enter the first incremental station. To initiate the

incremental analysis, we require the fluid dynamic properties at the first

incremental station. In the port head region ahead of the first incremental

station, the velocity of combustion products is minimal. Therefore, the

region can be assumed to have a uniform total pressure and the propellant

burning rate is governed by this assumed total pressure. Let us denote

this port head pressure as p0h and the burning area of the region as Abh.

Therefore the mass flow rate from this region entering the first incremental

station is,

ρpAbhapn0h = m1 =

p1

RT1A1u1 (3.45)

From the above mass conservation, applying the routine gas-dynamic ma-

nipulations, we get,


m1 =√γ

(1 +

γ − 1

2M2

1

)−(γ+1)/(2(γ−1))

M1

p0hA1√RT0

(3.46)

Determining M1 iteratively, other fluid dynamic properties can be easily

calculated as,

T1 =T0(

1 + γ−12 M2

1

) (3.47)

p1 =p0h(

1 +γ − 1

2M2

1

)γ/(γ−1) (3.48)

u1 =√

2cp (T0 − T1) (3.49)

Example 3. 3

Propellant combustion products enter an incremental station of a conical cir-

cular cylindrical port with the following properties: total temperature = 3643K,

total pressure = 6.925MPa, static pressure = 6.1787 MPa, specific heat at con-

stant pressure = 1973.7 J/kg-K and molar mass = 29.926 kg/kg-mole. Inlet

station diameter = 100mm. Exit station diameter = 102mm. Incremental dis-

tance = 5mm. Propellant properties are: density = 1750 kg/m3, burning rate

r = 2.814 × 10−5p0.35m/s with p expressed in Pa, erosive burning ratio ε = 1.1.

Assuming the port flow to be steady and frozen, determine (a) velocity, static

temperature, density, and mass flow rate at the entry, (b) mass flow added into

the incremental segment, and (c) mass flow rate, velocity, static temperature,

static pressure, total pressure, and density at the exit of the segment.

Solution Total and static pressures, total temperature, specific heat at con-

stant pressure, and molar mass of combustion products are all given at the entry

station of a port flow segment. Port segment dimensions are given. Propellant

burning rate equation and erosive burning ratio are given. Steady frozen-flow is

to be assumed. (a) Velocity, static pressure, density, and mass flow rate at the

entry to the port segment have to be calculated. (b) Mass flow added into the

incremental segment is to be determined. (c) Mass flow rate, velocity, static tem-

perature, static pressure, total pressure, and density at the exit of the segment

are to be finally calculated.


Assumptions Port flow is steady and frozen. Mass flow added into the seg-

mental volume from the burning surface is to be calculated based on the properties

at the entry station.

Analysis

(a)

The ratio of specific heats is given by,

γ =cp

cp −R=

1973.7

1973.7− (8314.51/29.926)= 1.1638

The static temperature at the entry,

Tj = T0j

(pj

p0j

)γ − 1

γ= 3643

(6.1787

6.925

)0.1638

1.1638= 3585K

The density of the combustion products at the entry,

ρj =pj

RTj=

6.1787× 106

(8314.51/29.926)× 3585= 6.2034

kg

m3

The entry velocity is given by,

uj =√

2cp (T0 − Tj) =√

2× 1973.7× (3643− 3585) = 478.49m

s

Thus the mass flow rate at the entry to the incremental segment,

mj = ρjAjuj = 6.2034×π

4× 0.12 × 478.49 = 23.313

kg

s

The entry Mach number is given by,

Mj =uj√γRTj

=478.49√

1.1638× (8314.51/29.926)× 3585

=478.49

1076.64= 0.444


(b)

The mass of combustion products entering into the segment with zero axial

momentum is given by,

dmj = ρpεjapnj

(sj + s(j+1)

)2

dx

= 1750× 1.1× 2.814× 10−5 ×(6.1787× 106

)0.35 × π × 0.101× 0.005

= 0.02046kg

s

(c)

The mass flow rate exiting the segment,

m(j+1) = 23.313 + 0.02046 = 23.3335kg

s

Now we have to calculate the exit velocity using Eq. 3.36.

a =

[2(

A(j+1) +Aj)− γ − 1

2γA(j+1)

]

=

2

π

4×(0.1022 + 0.12

)− 0.1638

π

2× 1.1638× 0.1022

= 116.191

b = −

[pj

m(j+1)+

2mjuj

m(j+1)

(A(j+1) +Aj

)] =

−

6.1787× 106

23.3335+

2× 23.313× 478.49

23.3335×π

4×(0.1022 + 0.12

) = −324464.03


c =RT0

A(j+1)=

8314.51× 3643

29.926×π

4× 0.1022

= 123864246.7

u(j+1) =−b±

√b2 − 4ac

2a

=324464.03±

√324464.032 − 4× 116.191× 123864246.7

2× 116.191

= 456.32m

sOR 2336.19

m

s

A quadratic equation has two solutions. Here we have two values of the exit

velocity as solutions. Since the entry velocity to the segment is only 478.49 m/s

and also there is a divergence of the port from 100mm to 102mm, the exit velocity

of 456.32 m/s looks appropriate. However, we continue to calculate all the values

corresponding to the two values of exit velocity and finally by entropy principle

show that the velocity of 2336.19 m/s is not possible.

T(j+1)1= 3643−

456.322

2× 1973.7= 3590.25K

T(j+1)2= 3643−

2336.192

2× 1973.7= 2260.37K

Exit Mach number can be calculated.

M(j+1)1=

u(j+1)1√γRT(j+1)1

=456.32√

1.1638× (8314.51/29.926)× 3590.25

= 0.424


M(j+1)2=

u(j+1)2√γRT(j+1)2

=2336.19√

1.1638× (8314.51/29.926)× 2260.37

= 2.733

Exit static pressure,

p(j+1)1=

8314.51× 3590.25× 23.3335× 4

29.926× π × 0.1022 × 456.32

= 6241972.76Pa

p(j+1)2=

8314.51× 2260.37× 23.3335× 4

29.926× π × 0.1022 × 2336.19

= 767605.5Pa

Exit density,

ρ(j+1)1=

p(j+1)1

RT(j+1)1

=6242196.5× 29.926

8314.51× 3590.3= 6.2579

kg

m3

ρ(j+1)2=

p(j+1)2

RT(j+1)2

=767568.5× 29.926

8314.51× 2260.3= 1.2223

kg

m3

Exit total pressure,

p0(j+1)1= p(j+1)1

(T0

T(j+1)1

)γ/(γ−1)

= 6241972.76×

(3643

3590.25

)1.1638

0.1638= 6923543Pa


p0(j+1)2= p(j+1)2

(T0

T(j+1)2

)γ/(γ−1)

= 767605.5×

(3643

2260.37

)1.1638

0.1638= 22796132Pa

We find that the total pressure of 6923543 Pa at the segment exit corresponding to

the first solution of subsonic exit-velocity 456.32m/s is less than the total pressure

of 6925000 Pa at the segment entry. Whereas for the second solution of supersonic

exit velocity of 2260.37 m/s the corresponding total pressure of 22796132 Pa is

greater than the total pressure of 6925000 Pa at the segment entry. The latter is

not possible because there should be total pressure loss with mass addition.

The entropy change is given by,

∆s = cp ln

(T(j+1)

Tj

)−R ln

(p(j+1)

pj

)

Therefore entropy change,

∆s1 = 1973.7× ln3590.25

3585−

8314.51

29.926ln

(6241972.76

6.1787× 106

)

= 0.0576J

kg −K

∆s2 = 1973.7× ln2260.37

3585−

8314.51

29.926ln

(767605.5

6.1787× 106

)

= −330.89J

kg −K

Here again we see that the second solution gives a decrease in entropy, which is

not possible.

Discussion The present example is the one considered under steady flow

operation. Steady flow consideration is generally applicable during equilibrium

operation while the unsteady flow assumption is more appropriate during ignition

and tail-off transients.


While analyzing compressible flows, more than one solution is possible. A typ-

ically well-known situation is the isentropic flow in a choked convergent-divergent-

nozzle. Here, for a chosen area ratio with respect to the throat, we find the

possibility of a subsonic flow in the convergent portion and a supersonic flow in

the divergent portion. Therefore, while analyzing a compressible flow we should

choose an appropriate solution. In the present Example, the flow is not isentropic

but an adiabatic one with mass addition. On solving the quadratic equation, we

obtained two solutions, one corresponding to a subsonic flow and the other cor-

responding to a supersonic flow. The latter is inappropriate because of decrease

in entropy and increase in total pressure.

In the present example, it is very obvious that the subsonic velocity at the exit

of the segment is the correct solution. Nevertheless, we extended the analysis to

show by entropy principle that the supersonic velocity at the exit of the segment

is the incorrect solution. While developing a computer code to calculate the

internal ballistics of rocket, this approach of selecting the correct solution by

entropy principle is found to be safe and hence recommended.

In the centrally perforated cylindrical grains, as the mass flow rate of combus-

tion products being minimal at the head end and maximum at the nozzle end, in

order to enhance the propellant loading fraction a tapered port is adopted with

a minimal port area at the head end. Furthermore, such an arrangement can

keep the erosive burning effect within an acceptable level. In the present exam-

ple, the entry diameter of the segment is found to be less than the exit diameter,

representing a segment of a tapered grain port.

3.5.4 Solution of Unsteady Port-Flow

For the easy following of the ensuing discussion we list below the three

unsteady conservation-equations of mass Eq. 3.22, momentum Eq. 3.30,

and energy Eq. 3.31 that we derived previously.

p(j+1)u(j+1)A(j+1)

RT(j+1)=pjujAj

RTj+ ρpεjap

nj

(sj + s(j+1)

)2

∆x

−

(pj + p(j+1)

) (2V − V ′

)− V

(pj

′+ p

′

(j+1)

)R(Tj + T(j+1)

)∆t

(3.22)


p(j+1)A(j+1) =

pjAju2j

RTj−p(j+1)A(j+1)u

2(j+1)

RT(j+1)+ pjAj

+

(pj + p(j+1)

)2

(A(j+1) −Aj

)

−(pj + p(j+1)

) (uj + u(j+1)

)2R(Tj + T(j+1)

)(V − V ′

)∆t

−V(uj + u(j+1)

)2R(Tj + T(j+1)

)(pj + p(j+1) − pj

′ − p′(j+1)

)∆t

(3.30)

T0 = T j +u2j2cp

= T(j+1) +u2(j+1)

2cp(3.31)

The solution objective remains the same for the steady port flow as

well as the unsteady port flow. It is to find the values of three unknowns:

p(j+1), T(j+1) and u(j+1) . For this we have the above three equations. But

the solution procedure is more involved than what we just learned in the

case of steady port flow. The procedure involves two-step iterations. For

easy understanding of this two-step iteration-procedure we rewrite below

the above conservation equations in slightly different forms.

Propellant properties ρp, a, n, T0, m, γ, and ε are specified. All the

gas dynamic values corresponding to the previous time instant are known,

namely, p0j′, pj

′, p0(j+1)

′, and p(j+1)

′. The envelope dimensions of the port

for the previous as well as the current instant are also known: Aj′, A(j+1)

′, V

′, Aj , A(j+1),

and V . For the known inlet total and static pressures p0j and pj and the

specified total temperature T0 (adiabatic flow) we have to calculate exit

conditions, namely, p(j+1), T(j+1) and u(j+1). Trial values for p(j+1) and

T(j+1) are p(j+1)g and T(j+1)g. To start with, generally it is sufficient to

take p(j+1)g = pj and T(j+1)g = Tj . The suggested iteration procedure is

summarized below.

1. Start with the trial value T(j+1)g.


2. Start with the trial value p(j+1)g.

3. Substituting T(j+1)g and p(j+1)g in the following mass conservation

equation calculate the trial value m(j+1)g.

m(j+1)g =pjujAj

RTj+ ρpεjap

nj

(sj + s(j+1)

)2

∆x

−

(pj + p(j+1)g

) (2V − V ′

)− V

(pj

′+ p

′

(j+1)

)R(Tj + T(j+1)g

)∆t

(3.22a)

4. Calculate u(j+1)g using the following form of mass conservation equa-

tion.

u(j+1)g =m(j+1)gRT(j+1)g

p(j+1)gA(j+1)(3.22b)

5. Substitute the trial values of p(j+1)g and the T(j+1)g,and the calcu-

lated u(j+1)g into the right hand of the following momentum equation

to find p(j+1) in the left hand side of the equation.

p(j+1)A(j+1) =

mjuj − m(j+1)gu(j+1)g + pjAj

+

(pj + p(j+1)g

)2

(A(j+1) −Aj

)

−(pj + p(j+1)g

) (uj + u(j+1)g

)2R(Tj + T(j+1)g

)(V − V ′

)∆t

−V(uj + u(j+1)g

)2R(Tj + T(j+1)g

)(pj + p(j+1)g − pj

′ − p′(j+1)

)∆t

(3.30a)

6. If the modulus

∣∣∣∣∣(p(j+1) − p(j+1)g

)p(j+1)g

∣∣∣∣∣ ≤ δp where δp is the allowable

error fraction go to step 7. If NO, p(j+1)g = p(j+1) and go to step 2.


7. Calculate T(j+1) from the following form of energy conservation equa-

tion.

T(j+1) = T0 −u2(j+1)g

2cp(3.31a)

8. If the modulus

∣∣∣∣∣T(j+1) − T(j+1)g

T(j+1)g

∣∣∣∣∣ ≤ δT , where δT is the allowable

error fraction, u(j+1) = u(j+1)g go to step 9 to END iteration. If NO,

T(j+1)g = T(j+1) and go to step 1.

9. Solution has been reached. p(j+1), T(j+1) and u(j+1) are the solutions.

Substitute the solved values of p(j+1), T(j+1) and u(j+1) into Eqs.

3.22, 3.30 and 3.31to check for the adequacy of convergence error δ.

Example 3. 4

Assuming unsteady port-flow, exit quantities are to be calculated. Conver-

gence error fraction to be adopted is 1.0x10−6.

Propellant properties are the following. Propellant density, ρp = 1700 kg/m3.

Pre-exponent factor a in the normal burning rate equation = 3.23x10−7m/s. Com-

bustion index n in the normal burning rate equation = 0.7. Adiabatic flame tem-

perature, T0 = 3700K. Molar mass of combustion products, m = 25 kg/kg-mole.

Specific heat ratio, γ = 1.25. Erosive burning ratio ε = 1.1.

Input data for the previous time instant are the following. Time step, ∆t =

0.1s. Spatial step, ∆x = 0.020m. Inlet diameter, dj′

= 0.5m. Exit diameter,

d(j+1)′

= 0.505m. Inlet static pressure, pj′

= 6.0x106Pa. Exit static pressure,

p(j+1)′

= 5.95x106Pa.

Input data for the current instant are the following. Inlet diameter, dj =

0.505m. Exit diameter, d(j+1) = 0.510m Inlet static pressure, pj = 6.3x106Pa.

Inlet total pressure, p0j = 7.07x106Pa

Solution You may write a suitable computer-program to solve the unsteady

port-flow. Here, however, we adopt the procedure of spreadsheet (Excel) to get

the feel for the convergence. The procedure following the steps 1 to 9 given above

is straight forward. Although the procedure as per steps 1 to 9 is “safe”, the

convergence is generally seen to be slow. In a departure from the steps 1 to 9

given previously, we may also follow the shortened steps as given below.

1. Start with the trial value T(j+1)g.

2. Start with the trial value p(j+1)g.


3. Substituting T(j+1)g and p(j+1)g in the following mass conservation equation

calculate the trial value m(j+1)g.

m(j+1)g =pjujAj

RTj+ ρpεjap

nj

(sj + s(j+1)

)2

∆x

−

(pj + p(j+1)g

) (2V − V ′

)− V

(pj′+ p

′

(j+1)

)R(Tj + T(j+1)g

)∆t

(3.22a)

4. Calculate u(j+1)g using the following form of mass conservation equation.

u(j+1)g =m(j+1)gRT(j+1)g

p(j+1)gA(j+1)(3.22b)

5. Substitute the trial values of p(j+1)g and the T(j+1)g,and the calculated

u(j+1)g into the right hand of the following momentum equation to find

p(j+1) in the left hand side of the equation.

p(j+1)A(j+1) =

mjuj − m(j+1)gu(j+1)g + pjAj

+

(pj + p(j+1)g

)2

(A(j+1) −Aj

)

−(pj + p(j+1)g

) (uj + u(j+1)g

)2R(Tj + T(j+1)g

)(V − V ′

)∆t

−V(uj + u(j+1)g

)2R(Tj + T(j+1)g

)(pj + p(j+1)g − pj

′ − p′(j+1)

)∆t

(3.30a)

6. Calculate T(j+1) from the following form of energy conservation equation.

T(j+1) = T0 −u2(j+1)g

2cp(3.31a)

7. If

∣∣∣∣∣(p(j+1) − p(j+1)g

)p(j+1)g

∣∣∣∣∣ ≤ δp and

∣∣∣∣∣T(j+1) − T(j+1)g

T(j+1)g

∣∣∣∣∣ ≤ δT , u(j+1) = u(j+1)g


go to step 8 to END iteration. If NO p(j+1)g = p(j+1) and T(j+1)g = T(j+1),

and go to step 1.

8. Solution has been reached. p(j+1), T(j+1), and u(j+1) are the solutions.

Substitute the solved values of p(j+1), T(j+1), and u(j+1) into Eqs. 3.22,

3.30 and 3.31 to check for the adequacy of convergence error δ.

The iteration results adopting the shortened steps are given in Table 3.2.

Assumptions Adopted assumptions have already been discussed. Briefly,

however, we will recall the assumptions. (1) Frozen one dimensional adiabatic

flow with specific heat and molar mass held constant, (2) Frictionless port wall,

(3) Mass addition with zero axial momentum, (4) Ideal gas equation of state, and

(5) Infinitely rigid grain.

Analysis

Specific gas constant:

R =Rum

=8314.51

25= 332.5804

J

kg −K

Specific heat at constant pressure:

cp =Rγ

γ − 1=

332.5804× 1.25

0.25= 1662.902

J

kg −K

Burning area:

(sj + s(j+1)

)2

dx = π ×

(0.505 + 0.510

2

)× 0.02 = 0.031887m2

Burning rate:

r = εjapnj = 1.1× 3.23× 10−7 ×

(6.3× 106

)0.7= 0.02042m/s


dm = ρpεjapnj

(sj + s(j+1)

)2

dx

= 1700× 1.1× 3.23× 10−7 ×(6.3× 106

)0.7 × 0.031887

= 1.107129kg/s

Inlet static temperature:

Tj = T0

(pj

p0j

)γ − 1

γ= 3700×

(6.3

7.07

)0.25

1.25= 3615.6464K

Inlet velocity:

uj =√

2cp (T0 − Tj) =√

2× 1662.902× (3700− 3615.646)

= 529.665m /s

Current inlet area:

Aj =π

4d2j =

π

4× 0.5052 = 0.200296m2

Current exit area:

Aj+1 =π

4d2j+1 =

π

4× 0.512 = 0.204282m2

Entry mass flow rate:

mj =pjujAj

RTj=

6.3× 106 × 529.665× 0.200296

332.5804× 3615.6464= 555.817kg/s

Previous control volume:


V ′ =π

4×

(0.52 + 0.5052

2

)× 0.02 = 3.966457× 10−3m3

Current control volume:

V =π

4×

(0.5052 + 0.5102

2

)× 0.02 = 4.0457823× 10−3m3

Table 3.2: Typical iteration results for the unsteady flow throughgrain port segmental volume, Example 3. 4

No. p(j+1)g T(j+1)g p(j+1) T(j+1) u(j+1) δp δT(Pa) (K) (Pa) (K) (m/s)

1 6300000 3615.646 6322525 3618.587 520.348 3.6E-03 8.2E-042 6322525 3618.587 6326647 3619.035 518.917 6.5E-04 1.2E-043 6326647 3619.035 6327434 3619.120 518.643 1.2E-04 2.4E-054 6327434 3619.120 6327585 3619.137 518.591 2.4E-05 4.6E-065 6327585 3619.137 6327613 3619.140 518.581 4.5E-06 7.2E-076 6327613 3619.140 6327619 3619.140 518.579 8.6E-07 5.2E-08

The convergence has been attained at the sixth iteration (Table 3.2).

The related results are the following.

Mass accumulation rate,

dm

dt=

(pj + p(j+1)

) (2V − V ′

)− V

(p′+p

′(j+1)

j

)R(Tj + T(j+1)

)∆t

= 0.015557kg

s

Exit mass flow rate,

m(j+1) = mj + dm−dm

dt

= 555.817 + 1.107129− 0.015557 = 556.9086kg/s

Check on exit mass flow rate,

m(j+1) =p(j+1)u(j+1)A(j+1)

RT(j+1)

=6327619× 518.579× 0.204282

332.5804× 3619.14= 556.9078kg/s


Exit static pressure (see Table 3.2),

p(j+1) = 6327619Pa

Exit total pressure,

p0(j+1) = p(j+1)

(T0

T(j+1)

) γ

γ − 1

= 6327619×

(3700

3619.14

)5

= 7066787Pa

Entry Mach number,

Mj =

√√√√(T0Tj− 1

)2

γ − 1=

√√√√( 3700

3615.64− 1

)×

2

0.25= 0.432

Exit Mach number,

M(j+1) =

√√√√( T0

T(j+1)− 1

)2

γ − 1

=

√√√√( 3700

3619.14− 1

)×

2

0.25= 0.4228

FOR THE LAST TIME INSTANT CONTROL VOLUME GEOMEN-

TRY AND GAS DYNAMICS

Time step (s) = 1.000000E-01

Entry diameter (m) = 5.000000E-01

Exit diameter (m) = 5.050000E-01

Incremental step length (m) = 2.000000E-02

Entry static pressure (Pa) = 6.000000E+06

Exit static pressure (Pa) = 5.950000E+06


FOR THE CURRENT TIME INSTANT CONTROL VOLUME GE-

OMENTRY AND GAS DYNAMICS

Entry diameter (m) = 5.050000E-01

Exit diameter (m) = 5.100000E-01

Entry total pressure (Pa) = 7.070000E+06


FOR THE CURRENT TIME INSTANT CONTROL VOLUME GE-

OMETRY AND GAS DYNAMICS

Adiabatic flame temperature (checked against the input) after convergence

(K) = 3.700000E+03

Entry total pressure (Pa) = 7.070000E+06

Exit total pressure (Pa) = 7.066787E+06

Entry velocity (m/s) = 5.29665E+02

Exit velocity (m/s) = 5.18579E+02


Exit static pressure (Pa) = 6.327619E+06

Entry static temperature (K) = 3.615646E+03

Exit static temperature (K) = 3.619140E+03

Entry Mach number = 4.320E-01

Exit Mach number = 4.228E-01

Entry mass flow rate (kg/s) = 5.55817E+02

Exit mass flow rate (kg/s) = 5.56908E+02

Mass addition rate into

the control volume (kg/s) = 1.107129E+00

Mass accumulation rate in

the control volume (kg/s) = 1.5557E-02

Discussion

Convergence error fraction is sufficient as the stagnation temperature

and exit static pressure have been checked after the solution: check=

3700.000K = 3700.000K; check= 6327619Pa. = 6327619Pa. Note that

the flow is actually decelerating, despite the mass addition. This is be-

cause the port area is enlarging and acting as a diffuser. Nevertheless,

there is a total pressure loss due to mass addition. Mass conservation is

also satisfied: Exit mass flow rate = Entry mass flow rate + Mass addi-


tion rate into the control volume - Mass accumulation rate in the control

volume.

Solution of the unsteady flow in a segmental volume of a propellant-

grain port is explained. Such a solution is one of the many of such segmen-

tal volumes in a single iteration march to be solved in the propellant-grain

port flow. A few such marches have to be completed to reach convergence

(port exit flow = nozzle flow) and determine the flow field in the grain port

for a given instant. Many such instants interposed with incremental time

steps have to be analysed to predict the rocket motor performance from

ignition to complete burnout.

The iteration procedure we followed for the unsteady port flow in this

Example 3. 4 [shortened steps 1 8] can also be followed for the steady

port flow, after dropping the unsteady terms in Eqs. (3.22a) and (3.30a).

Equations (3.22a) and (3.30a) without the unsteady terms are nothing but

Eqs. 3.32 and 3.33 respectively. The iteration results and the solution for

the steady flow are given in Table 3. 3.

Table 3.3: Typical iteration results for the steady flow throughgrain port segmental volume, Example 3. 4

No. p(j+1)g T(j+1)g p(j+1) T(j+1) u(j+1) δp δT(Pa) (K) (Pa) (K) (m/s)

1 6300000 3615.646 6322561 3618.583 520.363 3.6E-03 8.1E-042 6322561 3618.583 6326697 3619.031 518.928 6.5E-04 1.2E-043 6326697 3619.031 6327487 3619.117 518.653 1.2E-04 2.4E-054 6327487 3619.117 6327638 3619.134 518.6 2.4E-05 4.5E-065 6327638 3619.134 6327667 3619.137 518.59 4.6E-06 8.6E-076 6327667 3619.137 6327672 3619.137 518.588 8.7E-07 1.7E-07


Chapter 4

Computer Program

Adopting 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, ig-

nition 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 here. The

source code, typical examples along with their outputs, and an exe file of

the code are stored in the attached USB-device.

For quick understanding of the program logic, you may have a cursory

glance of the source code and of any pair of output files given in the

attached USB device.

4.1 Computer Program

4.1.1 Main Program Steadyfull

There are two input files: xxxxxxinput1.dat and xxxxxxinput2.dat. The 6

alphanumeric-space xxxxxx is for problem identification. Examples of file

names are: case10input1.dat and case10input2.dat. xxxxxxinput1.dat file

contains propellant data under the namelist-name “prplnt” and xxxxxxin-

put2.dat file contains motor data under the namelist-name “motor”. Typ-

ical examples are:

case10input1.dat file containing

&prplnt a = 8.544e-6, n = 0.45, rhop = 1760., mbar = 25, cs = 1350.,

cp = 1779., t0 = 3110., ts = 950., ti = 300., beta = 55., mu = 0.904e-04,

107

108 CHAPTER 4. COMPUTER PROGRAM

pr = 0.51, eros = f/

case10input2.dat file containing

&motor od = 0.16, grainl = 0.19, dt = 0.025, aebyat = 8., tapangle =

.2, tailoffend = 0.05, deltatime = 0.02, error = 1.e-04, increments = 100,

idhinitial = 0.1/

On clicking the exe file, you will be asked to enter names of the input

files that you have created for the problem. In case your problem input

files are of the names case10input1.dat and case10input2.dat, you enter

sequentially only the names: case10input1 and case10input2. On execution

of the problem two output files: xxxxxxout1.out and xxxxxxout2.out are

printed. Related file names are case10out1.out and case10out2.out. The

former contains the detailed output along with iteration details for mass

convergence at the nozzle end and the latter contains the summary output.

The following are the steps by the main program (Refer to the Source

Code).

1. Set burnout = .false.; solution = .false. Read the two input files, and

call the subprogram propellant to calculate the propellant prop-

erties. This subprogram, on this calculation, prints the calculated

propellant data in both the output files.

2. Call the subprogram prsrratio to calculate nozzle pressure ratio

for the given nozzle area ratio aebyat and the ratio of specific heats,

gamma. The subprogram prsrratio calls the subprogram falci,

which uses the method of regula falci to converge on nozzle pressure

ratio.

3. Initiate motor operation from time 0 to burnout and then to complete

burnout: (Do ii=1,iiburnout); ii=1 represents 0th time.

4. Call the subprogram geometry (which is written for the ends-

inhibited tapered-circular-port cylindrical grain). If ii=1 (that is for

the 0th time instant) the subprogram calculates the motor configu-

ration data (burning area, chamber free-volume, loaded propellant

mass, and propellant volumetric loading fraction). The volume of

plenum is taken as 10 per cent of the initial port volume. Nozzle

convergence angle is taken to be 45o. End if. For all time instants,

including the 0th time, the subprogram calculates port envelop prop-

4.1. COMPUTER PROGRAM 109

erties (port diameter, burning perimeter, port area, and taper angle

at all incremental stations). For time instants >0, it returns the

port-envelop properties on regressing the envelop for the incremental

time step under quasi steady state assumption.

5. If ii=1 (0th time), on return from the subprogram geometry, print

the motor configuration data in both the output files under the head-

ing MOTOR CONFIGURATION. Else continue.

6. For all times, including the 0th time, set two trial total-pressures at

the head end, p0ht1 p0ht2.

7. If burnout has just happened (port diameter at the last incremental

station ≥ grain outer diameter), record nozzle-end total pressure at

burnout (“p0burnout”), pinpoint the burnout location and set at all

incremental stations downstream of it: grain taper angle = 0, port

diameter = grain outer diameter, burning perimeter = 0, and port

area = chamber area. Write in both the output files xxxxxxout1.out

and xxxxxxout2.out burnout information: time of burnout, port di-

ameter just upstream of burnout location, burnout location, sliver

mass, sliver fraction, and nozzle-end total pressure at burnout. Set

the logical variable burnout = .true.. Initiate printing tail-off tran-

sient performance in xxxxxxout2.out. Else continue.

8. Print the port envelop properties in the xxxxxxout1.out file for the

time instant.

9. Set j = 1.

10. If j = 1, p0ht=p0ht1. Else p0ht=p0ht2

11. Assuming that the port volume at the head end upstream of the

first incremental station is having p0ht, calculate the gas dynamic

properties at the first incremental station using the subprogram

falci or the subprogram bisection for the required iteration.

12. With these gas dynamic properties at the first incremental station,

call the subprogram segsteady to calculate the gas dynamic prop-

erties for all the remaining incremental stations along the port length.

If the logical variable eros = .true., the subprogram segsteady uses


subprogram erosive which in turn calls subprogram falci for the

calculation of total burning rates (including the erosive component)

for each control volume.

13. If burnout = .true. continue. Else go to step 15.

14. For all incremental stations downstream of the station where the

grain burning has crossed the casing wall, set (i) erosive component

of burning rate and total burning rate = 0, and (ii) gas dynamic

properties equal to the properties at that station.

15. If solution is .false. continue. Else go to step 22.

16. If j=1 continue. Else go to step 19.

17. Taking the port end total pressure as the nozzle entry total pressure,

calculate nozzle flow rate.

18. Store p0ht1 = p0ht, mdotp1 = port end mass flow rate, mdotn1

= nozzle flow rate, and the error fraction delmdot1 [=(mdotp1 -

mdotn1)/mdotp1]. Set j=2. Go to step 10.

19. Taking the port end total pressure as the nozzle entry total pressure,

calculate nozzle flow rate.

20. Store p0ht2 = p0ht, mdotp2 = port end mass flow rate, mdotn2

= nozzle flow rate, and the error fraction delmdot2 [=(mdotp2 -

mdotn2)/mdotp2].

21. If delmdot1 ≤ allowable error, solution = .true., p0ht=p0ht1, and

go to step 11. Else if delmdot2 ≤ allowable error, solution = .true.

p0ht = p0ht2., go to step 11. Else call subprogram falci to get

improved values of p0ht1 and p0ht2. Set j = 1. Go to step 10.

22. Add the mass exited from the port end for the time step to the

cumulative mass already exited from the port end. Similarly, add

the mass exited through the nozzle throat for the time step to the

cumulative mass that already exited though the nozzle throat. Store

nozzle mass flow rate and head end total pressure p0ht for the instant.

Calculate the erosive properties for the instant at the port exit by


calling subprogram erosive and store the same for the instant. Set

solution = .false..

23. Store the nozzle exit static pressure and the thrust for the instant.

24. If ii=1, call subprogram starttransienteql (which prints ignition

transient data in both the output files). On return prepare to print

motor performance for equilibrium operation. Else continue.

25. Print in the output file xxxxxxout1.out the details of converged so-

lution for the instant. Also print in that file the port dynamics for

the instant along the port: total pressure (p0), Mach number (M),

mass flow rate (mdot), erosive burning ratio (epsilon).

26. Print the related motor performance (equilibrium operation or tail-

off transient) in summary output file xxxxxxout2.out [head end total

pressure (p0h), nozzle end total pressure (p0n), motor mass flow

rate (mdot), erosive burning ratio (epsilon), head end port diameter

(dhprt), nozzle end port diameter (dnprt), nozzle exit static pressure

(pe), vacuum thrust (fvac), port end location (Lnprt), mass flow error

fraction at convergence (delmdot)].

27. If port end total pressure is not less than the specified fraction

of burnout total pressure (tailoffend*p0burnout), go to next time in-

stant, that is, go to step 3 to stay in the do-loop Do ii=1,iiburnout.

Else continue.

28. Print in both the output files that the total burnout has been reached

and print the following: (i) final sliver mass, and (ii) final sliver frac-

tion. To validate the adequacy of the time step (delta time) and

incremental-step length (delx), the following mass balance checks

are also printed in both the output files: (i) initial propellant mass,

(ii) cumulative port flow mass, (iii) cumulative nozzle flow mass, (iv)

Mass balance error2 = [(cumulative port-flow)-(cumulative nozzle-

flow)]/cumulative port-flow, and (v) Mass balance error3 = [(cumu-

lative nozzle-flow)+(final sliver)] - (initial propellant-mass)/(initial

propellant-mass). Also total impulse, calculated vacuum specific im-

pulse through incremental analysis, and the theoretical vacuum spe-

cific impulse (= c∗ × CFvac) are printed.


4.1.2 Subprogram Propellant

This subprogram calculates the propellant properties: ratio of specific

heats, experimental characteristic velocity c*, beta and alpha in the Lenoir-

Robillard erosive burning equation. Calculated and other-specified propellant-

properties are printed. Experimental characteristic velocity is calculated

by assuming a c*-efficiency of 0.98.

4.1.3 Subprogram Prsrratio

Subprogram prsrratio calculates the nozzle pressure ratio p0bype of the

nozzle of given area ratio aebyat. The value of the ratio of specific heats

gamma comes from the main. For convergence the subprogram uses the

subprogram falci.

4.1.4 Subprogram Geometry

This subprogram is for an ends-inhibited tapered circular port grain. For

0th time (ii=1) burning area, free volume, propellant mass, and propellant

volumetric loading fraction are calculated. For all time instants, including

0th time, this subprogram calculates the port envelop. Included under port

envelop are: port diameter, burning perimeter, port area, and taper angle

at all incremental stations. For time instants > 0, the subprogram returns

the port-envelop properties on regressing the envelop for the incremental

time step under quasi steady state assumption.

4.1.5 Subprogram Segsteady

The Subprogram Segsteady calculates under steady flow assumption the

exit properties of the incremental control volume by applying the equations

of continuity, momentum, energy, and ideal-gas. Calculated exit properties

transferred to the calling program are: velocity, static-temperature and -

pressure, Mach number, and total pressure; also the increase in entropy

across the incremental control volume is calculated. Burning rate for the

incremental surface is approximated to the one at its entry station. Ero-

sive burning at the entry station is accounted through the Lenoir-Robillard

model. Erosive burning characteristics are calculated by calling Subpro-

gram Erosive


4.1.6 Subprogram Erosive

This program calculates the erosive burning rate using Lenoir-Robillard

model. The implicit equation in erosive burning rate r is solved by Regula-

Falci by calling Subroutine FALCI. Subprogram Erosive is called by the

Subprogram Segsteady and also directly by the main program.

4.1.7 Subprogram Starttransienteql

The start transient is calculated assuming the “equilibrium pressure anal-

ysis”, that is assuming that there is one uniform pressure for the entire

chamber-cavity. Furthermore it is assumed that (i) for the duration of

start transient the burned distance is negligible, that is, the burning area

is constant and (ii) entire grain surface is instantaneously ignited with neg-

ligible igniter mass. This subprogram prints the calculated start transient.

4.1.8 Subprogram Falci

Given the trial pair of iterative points (x1, y1) and (x2, y2), the program

uses the method of regula falci for giving improved x2 after retaining one of

the given points in the place of (x1,y1). Adequacy of the method of regula

falci for various sample problems has been observed. However, in rare

cases of non-convergence, the pair of points with negative and positive y-

values of x1 and x2 in the solution vicinity are referred to the Subprogram

Bisection.

4.1.9 Error Messages

As per the given input, in case the initial port-diameter at the nozzle end

is greater than the grain outer diameter or in case the port-diameter at

the nozzle end diameter is less than the throat diameter, error messages

are printed and the execution is aborted. The error messages are:

**** INPUT ERROR **** INPUT ERROR ****

INITIAL PORT EXIT DIAMETER = xxxxxx(m) IS GREATER THAN

GRAIN OD = xxxxxx(m)

PROGRAM EXECUTION ABORTED

**** INPUT ERROR **** INPUT ERROR ****


INITIAL PORT EXIT DIAMETER = xxxxxx(m) IS LESS THAN

NOZZLE THROAT DIAMETER = xxxxxx(m)

PROGRAM EXECUTION ABORTED

The maximum number of time steps given in the source code = 500

(iiburnout = 500). If this is not sufficient for the given time interval (delta-

time) the complete burnout will not be reached by the program and the

following error message is printed. You may have to increase the time step,

however, without sacrificing the overall mass convergence accuracy (refer

Mass balance error2 and Mass balance error3 in the program-output).

**** UNEXPECTED COMPLETE-BURNOUT ****

CHOOSE FINER TIMESTEP “deltatime” under NAMELIST “motor”

PROGRAM-EXECUTION ABORTED ** PROGRAM-EXECUTION

ABORTED

In the use of subprogram falci and/or subprogram bisection there is

a limit on number of iterations. If this number is exceeded the following

error message will be printed. On solving varied problems, such a situa-

tion has rarely been met. If such a situation arises, the detailed output

xxxxxxout1.out can be studied to solve the issue.

***FALCI BISECTION combinedly have been called more than 60

times by the main program for the head end Mach number convergence.

Calculations abandoned.***

4.1.10 Outputs

On execution of the problem two output files:

xxxxxxout1.out

and

xxxxxxout2.out

are printed. The former contains the detailed output along with iteration

details for mass convergence at the nozzle end and the latter contains

the summary output. The program has been run successfully on many

configurations. However, in case of non-convergence for any problem the

iteration details found in xxxxxxout1.out may be studied to solve the issue.

xxxxxxout1.out file, being detailed, is rather large. The summary of

the calculation is given in xxxxxxout2.out and hence it is relatively small.


4.1.11 Sample Problems

Problem01

prob01input1.dat

&prplnt a = 3.51e-05, n = 0.36, rhop = 1765., mbar = 26.1, cs = 1400.,

cp = 2880., t0 = 3390., ts = 1000., ti = 300., beta = 60., mu = 1.0e-04,

pr = 0.49, eros = .f./

prob01input2.dat


.5, tailoffend = 0.05, deltatime = 0.050, idhinitial = 0.05 /

Problem02

prob02input1.dat


cp = 2880., t0 = 3390., ts = 1000., ti = 300., beta = 60., mu = 1.0e-04,

pr = 0.49, eros = .t./

prob02input2.dat



Problem03

prob03input1.dat


cp = 2880., t0 = 3390., ts = 1000., ti = 300., beta = 60., mu = 1.0e-04,

pr = 0.49, eros = .t./

prob03input2.dat



Problem01 and Problem02 are of same data except the choice of ero-

sive burning consideration. The outputs bring out the difference in motor

performance for specifying eros = .false. (erosive burning not to be con-

sidered) and eros = .true. (erosive burning to be considered) for a very

low port-to-throat-ratio motor.

Problem02 and Problem03 bring out the difference in the outputs by

having a higher burning rate propellant in Problem03. Problem02 and


Problem03 have same motor dimensions. Erosive burning ratio for faster

burning propellant (in Problem03) is less.

Case10

case10input1.dat


cp = 1779., t0 = 3110., ts = 950., ti = 300., beta = 55., mu = 0.904e-04,

pr = 0.51, eros = f/

case10input2.dat



idhinitial = 0.1/

Case11

case11input1.dat


cp = 1779., t0 = 3110., ts = 950., ti = 300., beta = 55., mu = 0.904e-04,

pr = 0.51, eros = t/

case11input2.dat



idhinitial = 0.1/

Case10 and Case 11 depict the difference in the inputs by specifying

eros=.true. and eros=.false. for a very high port-to-throat-ratio motor.

The outputs are not different as there is no erosive burning for very high

port to throat ratio motors.

Case13

case13input1.dat


cp = 2289., t0 = 3146., ts = 1000., ti = 300., beta = 60., mu = 0.904e-04,

pr = 0.51, eros = t/

case13input2.dat


1.5, tailoffend = 0.05, deltatime = 0.065, error = 1.e-04, increments = 100,


idhinitial = 0.05/

Case13 depict the performance of comparatively a large motor with

a slower burning propellant. In spite of large port-to-throat area ratio,

certain amount erosive burning is noticed. The data of the Case13 are

same as Example 2.5 in Chapter 2. Example 2.5 has been calculated under

equilibrium pressure analysis assuming no erosive effect. The outputs of

Example 2.5 and of Case13 are different because in Case13 (i) erosive effect

is taken into account and (ii) incremental analysis is adopted.

As mentioned previously the outputs of these sample problems are large

and hence these outputs are given in the USB device attached. However,

for ready readability and quick understanding of the outputs, the summary

output of Case 13 is given here.

4.1.12 Sample Output: Case13

case13input1.dat

&prplnt a=2.814e-5, n=0.35, rhop=1780., mbar=25, cs=1350., cp=2289.,

t0=3146., ts=1000., ti=300., beta=60., mu=0.904e-04, pr=0.51, eros=t/

case13input2.dat

&motor od=0.4, grainl=2.4, dt=0.077, aebyat=8., tapangle=1.5, tai-

loffend=0.05, deltatime=0.065, error=1.e-04, increments=100, idhinitial=0.05/

Case13 depicts the performance of comparatively a large motor with

a slower burning propellant. In spite of large port-to-throat area ratio,

certain amount erosive burning is noticed because of slow burning propel-

lant. The data of the Case13 are basically same as Example 2.5 in Chapter

2. Example 2.5 has been calculated under equilibrium pressure analysis

assuming no erosive effect. The outputs of Example 2.5 and of Case13 are

different because in Case13 (i) the erosive effect is taken into account and

(ii) incremental analysis is adopted. Besides, the program assumes (i) the

nozzle convergence angle = 45o, and (ii) plenum volume is 10 percent of

initial port volume. Accordingly initial free volume and final empty vol-

umes are different between the solution of example 2.5 and the program

output: 0.1064 m3 against 0.03745m3 and 0.3815m3 against 0.34007m3

respectively. Due to the above, ignition transient is longer in Example 2.5.

Case14 is run with the data of Case13, but assuming no erosive effect.

The input and output files of Case14 are also given in the attached USB

device.


Computer Output for Case13

&PRPLNT

A = 2.814000000000000E-005,

N = 0.350000000000000 ,

RHOP = 1780.00000000000 ,

MBAR = 25.0000000000000 ,

CS = 1350.00000000000 ,

CP = 2289.00000000000 ,

T0 = 3146.00000000000 ,

TS = 1000.00000000000 ,

TI = 300.000000000000 ,

BETA = 60.0000000000000 ,

MU = 9.040000000000000E-005,

PR = 0.510000000000000 ,

EROS = T /

&MOTOR

OD = 0.400000000000000 ,

GRAINL = 2.40000000000000 ,

DT = 7.700000000000000E-002,

AEBYAT = 8.00000000000000 ,

TAPANGLE = 1.50000000000000,

DELTATIME = 6.500000000000000E-002,

INCREMENTS = 100,

IDHINITIAL = 5.000000000000000E-002,

ERROR = 1.000000000000000E-004,

TAILOFFEND = 5.000000000000000E-002

/

SOLID PROPELLANT ROCKET MOTOR

INTERNAL BALLISTICS PREDICTION

(Start transients are calculated through Lumped Chamber Pressure

Model or p(t) Model)

(Equilibrium operations are calculated throgh quasi steady flow model or

p(x) Model)

NOTE 1. Rocket performance parameters are calculated for operations in

vacuum.


NOTE 2. For non-vacuum operation, calculated vacuum-thrust has to be

corrected.

NOTE 3. Under non-vacuum operations nozzle exit plane pressure below

certain level will lead to nozzle separation.

PROPELLANT DATA

Pre-exponent factor a in the burning rate

equation r0 = apn(m/s) = 2.814000E-05

Burning rate index n in the burning rate

equation r0 = apn = 3.500000E-01

Propellant density (kg/m3) = 1.780000E+03

Molar mass of combustion

products (kg/kg −mole) = 2.500000E+01

Ratio of specific heats = 1.169994E+00

Experimental Characteristic velocity (m/s) = 1.559864E+03

Adiabatic flame temperature (K) = 3.146000E+03

Beta in the Lenoir-Robillard equation = 6.000000E+01

Alpha in the Lenoir Robillard equation = 2.046918E-05

Erosive Burning is not considered if eros=.FALSE., ie, F

Eros = T

Vacuum specific impulse theoretical

[= c ∗ ×CFvac](N − s/kg) = 2.697695E+03

MOTOR CONFIGARATION INCREMENTS

Grain outer diameter (m) = 4.000000E-01

Head end port diameter (m) = 5.000000E-02

Grain length (m) = 2.400000E+00

Grain taper angle (degree) = 1.500000E+00

Initial propellant mass(kg) = 4.896918E+02

Nozzle throat diameter(m) = 7.700000E-02

Nozzle area ratio = 8.000000E+00

Nozzle pressure ratio = 5.487120E+01


Incremental distance (m) = 2.400000E-02

Incremental time (s) = 0.065

Port end diameter (m) = 1.756924E-01

Initial-port to throat area ratio = 5.206245E+00

Initial burning area (m2) = 8.511321E-01

Port volume, initial (m3) = 2.648517E-02

Chamber free volume, initial (m3) = 3.745150E-02

Chamber empty-volume (m3) = 3.400700E-01

Propellant volumetric loadinng fraction = 8.089738E-01

4.1.

COMPUTER

PROGRAM

121

START TRANSIENT CALCULATED ASSUMING “EQUILIBRIUM PRESSURE ANALYSIS”

Time(s) p0h(Pa) mdot(kg/s) pe(Pa) Vac-Thrust(N)

0.000000E+00 1.099373E+05 2.478231E+00 2.003553E+03 8.853658E+02

4.035485E-04 1.366081E+05 2.673978E+00 2.489614E+03 1.100155E+03

7.880553E-04 1.632789E+05 2.846207E+00 2.975675E+03 1.314945E+03

1.158290E-03 1.899496E+05 3.000990E+00 3.461736E+03 1.529734E+03

1.517458E-03 2.166204E+05 3.142214E+00 3.947798E+03 1.744523E+03

1.867846E-03 2.432911E+05 3.272542E+00 4.433859E+03 1.959313E+03

2.211158E-03 2.699619E+05 3.393883E+00 4.919920E+03 2.174102E+03

2.548711E-03 2.966326E+05 3.507661E+00 5.405981E+03 2.388892E+03

2.881548E-03 3.233034E+05 3.614969E+00 5.892042E+03 2.603681E+03

3.210519E-03 3.499742E+05 3.716667E+00 6.378103E+03 2.818470E+03

3.536329E-03 3.766449E+05 3.813443E+00 6.864165E+03 3.033260E+03

3.859570E-03 4.033157E+05 3.905861E+00 7.350226E+03 3.248049E+03

4.180753E-03 4.299864E+05 3.994388E+00 7.836287E+03 3.462839E+03

4.500320E-03 4.566572E+05 4.079413E+00 8.322348E+03 3.677628E+03

4.818659E-03 4.833279E+05 4.161268E+00 8.808409E+03 3.892418E+03

5.136116E-03 5.099987E+05 4.240238E+00 9.294470E+03 4.107207E+03

5.453002E-03 5.366695E+05 4.316567E+00 9.780532E+03 4.321996E+03

5.769599E-03 5.633402E+05 4.390468E+00 1.026659E+04 4.536786E+03

6.086166E-03 5.900110E+05 4.462129E+00 1.075265E+04 4.751575E+03

122

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4.COMPUTER

PROGRAM

6.402942E-03 6.166817E+05 4.531713E+00 1.123872E+04 4.966365E+03

6.720150E-03 6.433525E+05 4.599368E+00 1.172478E+04 5.181154E+03

7.037998E-03 6.700233E+05 4.665224E+00 1.221084E+04 5.395943E+03

7.356684E-03 6.966940E+05 4.729397E+00 1.269690E+04 5.610733E+03

7.676397E-03 7.233648E+05 4.791992E+00 1.318296E+04 5.825522E+03

7.997317E-03 7.500355E+05 4.853105E+00 1.366902E+04 6.040312E+03

8.319618E-03 7.767063E+05 4.912821E+00 1.415508E+04 6.255101E+03

8.643470E-03 8.033770E+05 4.971218E+00 1.464114E+04 6.469890E+03

8.969038E-03 8.300478E+05 5.028369E+00 1.512720E+04 6.684680E+03

9.296483E-03 8.567186E+05 5.084338E+00 1.561327E+04 6.899469E+03

9.625966E-03 8.833893E+05 5.139185E+00 1.609933E+04 7.114259E+03

9.957644E-03 9.100601E+05 5.192967E+00 1.658539E+04 7.329048E+03

1.029168E-02 9.367308E+05 5.245734E+00 1.707145E+04 7.543838E+03

1.062822E-02 9.634016E+05 5.297532E+00 1.755751E+04 7.758627E+03

1.096743E-02 9.900723E+05 5.348407E+00 1.804357E+04 7.973416E+03

1.130947E-02 1.016743E+06 5.398399E+00 1.852963E+04 8.188206E+03

1.165450E-02 1.043414E+06 5.447545E+00 1.901569E+04 8.402995E+03

1.200269E-02 1.070085E+06 5.495882E+00 1.950176E+04 8.617785E+03

1.235419E-02 1.096755E+06 5.543441E+00 1.998782E+04 8.832574E+03

1.270919E-02 1.123426E+06 5.590255E+00 2.047388E+04 9.047363E+03

1.306784E-02 1.150097E+06 5.636352E+00 2.095994E+04 9.262153E+03

4.1.

COMPUTER

PROGRAM

123

1.343034E-02 1.176768E+06 5.681759E+00 2.144600E+04 9.476942E+03

1.379687E-02 1.203438E+06 5.726502E+00 2.193206E+04 9.691732E+03

1.416761E-02 1.230109E+06 5.770605E+00 2.241812E+04 9.906521E+03

1.454276E-02 1.256780E+06 5.814090E+00 2.290418E+04 1.012131E+04

1.492252E-02 1.283451E+06 5.856980E+00 2.339024E+04 1.033610E+04

1.530711E-02 1.310121E+06 5.899295E+00 2.387631E+04 1.055089E+04

1.569675E-02 1.336792E+06 5.941053E+00 2.436237E+04 1.076568E+04

1.609165E-02 1.363463E+06 5.982273E+00 2.484843E+04 1.098047E+04

1.649207E-02 1.390134E+06 6.022972E+00 2.533449E+04 1.119526E+04

1.689824E-02 1.416804E+06 6.063167E+00 2.582055E+04 1.141005E+04

1.731043E-02 1.443475E+06 6.102873E+00 2.630661E+04 1.162484E+04

1.772892E-02 1.470146E+06 6.142105E+00 2.679267E+04 1.183963E+04

1.815400E-02 1.496817E+06 6.180876E+00 2.727873E+04 1.205442E+04

1.858596E-02 1.523487E+06 6.219202E+00 2.776480E+04 1.226920E+04

1.902512E-02 1.550158E+06 6.257094E+00 2.825086E+04 1.248399E+04

1.947184E-02 1.576829E+06 6.294564E+00 2.873692E+04 1.269878E+04

1.992646E-02 1.603500E+06 6.331625E+00 2.922298E+04 1.291357E+04

2.038936E-02 1.630171E+06 6.368287E+00 2.970904E+04 1.312836E+04

2.086096E-02 1.656841E+06 6.404561E+00 3.019510E+04 1.334315E+04

2.134168E-02 1.683512E+06 6.440458E+00 3.068116E+04 1.355794E+04

2.183198E-02 1.710183E+06 6.475986E+00 3.116722E+04 1.377273E+04

124

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4.COMPUTER

PROGRAM

2.233234E-02 1.736854E+06 6.511157E+00 3.165328E+04 1.398752E+04

2.284330E-02 1.763524E+06 6.545978E+00 3.213935E+04 1.420231E+04

2.336542E-02 1.790195E+06 6.580459E+00 3.262541E+04 1.441710E+04

2.389929E-02 1.816866E+06 6.614607E+00 3.311147E+04 1.463189E+04

2.444558E-02 1.843537E+06 6.648431E+00 3.359753E+04 1.484668E+04

2.500497E-02 1.870207E+06 6.681938E+00 3.408359E+04 1.506147E+04

2.557824E-02 1.896878E+06 6.715136E+00 3.456965E+04 1.527626E+04

2.616620E-02 1.923549E+06 6.748033E+00 3.505571E+04 1.549105E+04

2.676974E-02 1.950220E+06 6.780634E+00 3.554177E+04 1.570584E+04

2.738985E-02 1.976890E+06 6.812946E+00 3.602784E+04 1.592062E+04

2.802757E-02 2.003561E+06 6.844976E+00 3.651390E+04 1.613541E+04

2.868407E-02 2.030232E+06 6.876731E+00 3.699996E+04 1.635020E+04

2.936064E-02 2.056903E+06 6.908215E+00 3.748602E+04 1.656499E+04

3.005868E-02 2.083573E+06 6.939435E+00 3.797208E+04 1.677978E+04

3.077975E-02 2.110244E+06 6.970397E+00 3.845814E+04 1.699457E+04

3.152557E-02 2.136915E+06 7.001105E+00 3.894420E+04 1.720936E+04

3.229808E-02 2.163586E+06 7.031565E+00 3.943026E+04 1.742415E+04

3.309943E-02 2.190256E+06 7.061782E+00 3.991632E+04 1.763894E+04

3.393204E-02 2.216927E+06 7.091760E+00 4.040239E+04 1.785373E+04

3.479864E-02 2.243598E+06 7.121505E+00 4.088845E+04 1.806852E+04

3.570233E-02 2.270269E+06 7.151021E+00 4.137451E+04 1.828331E+04

4.1.

COMPUTER

PROGRAM

125

3.664667E-02 2.296939E+06 7.180313E+00 4.186057E+04 1.849810E+04

3.763571E-02 2.323610E+06 7.209384E+00 4.234663E+04 1.871289E+04

3.867417E-02 2.350281E+06 7.238240E+00 4.283269E+04 1.892768E+04

3.976751E-02 2.376952E+06 7.266883E+00 4.331875E+04 1.914247E+04

4.092219E-02 2.403622E+06 7.295318E+00 4.380481E+04 1.935726E+04

4.214583E-02 2.430293E+06 7.323549E+00 4.429087E+04 1.957204E+04

4.344758E-02 2.456964E+06 7.351579E+00 4.477694E+04 1.978683E+04

4.483851E-02 2.483635E+06 7.379412E+00 4.526300E+04 2.000162E+04

4.633224E-02 2.510306E+06 7.407051E+00 4.574906E+04 2.021641E+04

4.794574E-02 2.536976E+06 7.434500E+00 4.623512E+04 2.043120E+04

4.970054E-02 2.563647E+06 7.461763E+00 4.672118E+04 2.064599E+04

5.162455E-02 2.590318E+06 7.488841E+00 4.720724E+04 2.086078E+04

5.375482E-02 2.616989E+06 7.515739E+00 4.769330E+04 2.107557E+04

5.614210E-02 2.643659E+06 7.542459E+00 4.817936E+04 2.129036E+04

5.885845E-02 2.670330E+06 7.569005E+00 4.866543E+04 2.150515E+04

6.201126E-02 2.697001E+06 7.595378E+00 4.915149E+04 2.171994E+04

6.577080E-02 2.723672E+06 7.621583E+00 4.963755E+04 2.193473E+04

7.043173E-02 2.750342E+06 7.647622E+00 5.012361E+04 2.214952E+04

7.657437E-02 2.777013E+06 7.673497E+00 5.060967E+04 2.236431E+04

126

CHAPTER

4.COMPUTER

PROGRAM

Time integral of propellant mass consumed during start transient (kg) = 5.126751E-01

Impulse = time integral of thrust during start transient (N-s) = 1.250759E+03

EQUILIBRIUM PERFORMANCE

p0h=head-end total pressure; p0n=nozzle end total pressure; dhprt=head-end port dia; dnprt=nozzle-end port dia Lnprt=exit-

port location; delmdot=convergence error

Time (s) p0h(Pa) p0n(Pa) mdot(kg/s) epsilon dhprt(m) dnprt(m) pe(Pa) Fvac(N) Lnprt (m) delmdot

0.089 2.8482E+06 2.7688E+06 8.266E+00 1.106E+00 5.000E-02 1.757E-01 5.0460E+04 2.2298E+04 2.400E+00 1.351E-05

0.154 2.8697E+06 2.7915E+06 8.333E+00 1.105E+00 5.066E-02 1.764E-01 5.0874E+04 2.2481E+04 2.400E+00 1.219E-06

0.219 2.8911E+06 2.8141E+06 8.401E+00 1.105E+00 5.133E-02 1.771E-01 5.1286E+04 2.2663E+04 2.400E+00 2.513E-05

0.284 2.9125E+06 2.8367E+06 8.469E+00 1.104E+00 5.200E-02 1.779E-01 5.1698E+04 2.2845E+04 2.400E+00 8.420E-05

0.349 2.9348E+06 2.8601E+06 8.538E+00 1.103E+00 5.267E-02 1.786E-01 5.2124E+04 2.3033E+04 2.400E+00 2.507E-07

0.414 2.9570E+06 2.8834E+06 8.607E+00 1.102E+00 5.334E-02 1.793E-01 5.2549E+04 2.3221E+04 2.400E+00 4.718E-05

0.479 2.9792E+06 2.9067E+06 8.677E+00 1.101E+00 5.401E-02 1.801E-01 5.2973E+04 2.3409E+04 2.400E+00 5.887E-05

0.544 3.0014E+06 2.9300E+06 8.747E+00 1.101E+00 5.469E-02 1.808E-01 5.3398E+04 2.3596E+04 2.400E+00 3.592E-05

0.609 3.0236E+06 2.9532E+06 8.816E+00 1.100E+00 5.537E-02 1.815E-01 5.3821E+04 2.3784E+04 2.400E+00 2.063E-05

0.674 3.0463E+06 2.9770E+06 8.887E+00 1.099E+00 5.604E-02 1.823E-01 5.4254E+04 2.3975E+04 2.400E+00 1.516E-07

0.739 3.0690E+06 3.0007E+06 8.958E+00 1.098E+00 5.672E-02 1.830E-01 5.4686E+04 2.4166E+04 2.400E+00 1.331E-05

0.804 3.0917E+06 3.0244E+06 9.029E+00 1.097E+00 5.741E-02 1.838E-01 5.5118E+04 2.4356E+04 2.400E+00 5.935E-05

0.869 3.1150E+06 3.0487E+06 9.101E+00 1.097E+00 5.809E-02 1.845E-01 5.5561E+04 2.4552E+04 2.400E+00 1.899E-07

0.934 3.1383E+06 3.0730E+06 9.173E+00 1.096E+00 5.878E-02 1.853E-01 5.6003E+04 2.4748E+04 2.400E+00 2.546E-05

0.999 3.1617E+06 3.0972E+06 9.246E+00 1.095E+00 5.946E-02 1.860E-01 5.6445E+04 2.4943E+04 2.400E+00 1.832E-05

1.064 3.1850E+06 3.1215E+06 9.319E+00 1.094E+00 6.015E-02 1.867E-01 5.6887E+04 2.5138E+04 2.400E+00 2.064E-05

4.1.

COMPUTER

PROGRAM

127

1.129 3.2083E+06 3.1457E+06 9.392E+00 1.094E+00 6.084E-02 1.875E-01 5.7329E+04 2.5333E+04 2.400E+00 9.048E-05

1.194 3.2326E+06 3.1708E+06 9.466E+00 1.093E+00 6.154E-02 1.882E-01 5.7786E+04 2.5536E+04 2.400E+00 2.624E-07

1.259 3.2568E+06 3.1959E+06 9.540E+00 1.092E+00 6.223E-02 1.890E-01 5.8244E+04 2.5738E+04 2.400E+00 5.707E-05

1.324 3.2811E+06 3.2210E+06 9.615E+00 1.092E+00 6.293E-02 1.898E-01 5.8701E+04 2.5940E+04 2.400E+00 8.246E-05

1.389 3.3053E+06 3.2461E+06 9.690E+00 1.091E+00 6.363E-02 1.905E-01 5.9158E+04 2.6142E+04 2.400E+00 7.654E-05

1.454 3.3296E+06 3.2711E+06 9.765E+00 1.090E+00 6.433E-02 1.913E-01 5.9615E+04 2.6344E+04 2.400E+00 4.031E-05

1.519 3.3538E+06 3.2962E+06 9.840E+00 1.089E+00 6.503E-02 1.920E-01 6.0071E+04 2.6545E+04 2.400E+00 2.564E-05

1.584 3.3787E+06 3.3218E+06 9.917E+00 1.089E+00 6.573E-02 1.928E-01 6.0539E+04 2.6752E+04 2.400E+00 1.617E-07

1.649 3.4035E+06 3.3474E+06 9.993E+00 1.088E+00 6.644E-02 1.935E-01 6.1005E+04 2.6958E+04 2.400E+00 4.920E-06

1.714 3.4284E+06 3.3730E+06 1.007E+01 1.087E+00 6.714E-02 1.943E-01 6.1472E+04 2.7164E+04 2.400E+00 3.934E-05

1.779 3.4537E+06 3.3992E+06 1.015E+01 1.087E+00 6.785E-02 1.951E-01 6.1948E+04 2.7375E+04 2.400E+00 1.372E-07

1.844 3.4791E+06 3.4253E+06 1.023E+01 1.086E+00 6.856E-02 1.958E-01 6.2424E+04 2.7585E+04 2.400E+00 9.191E-06

1.909 3.5045E+06 3.4514E+06 1.030E+01 1.085E+00 6.928E-02 1.966E-01 6.2900E+04 2.7795E+04 2.400E+00 1.080E-05

1.974 3.5299E+06 3.4775E+06 1.038E+01 1.085E+00 6.999E-02 1.974E-01 6.3375E+04 2.8005E+04 2.400E+00 5.928E-05

2.039 3.5560E+06 3.5042E+06 1.046E+01 1.084E+00 7.071E-02 1.982E-01 6.3863E+04 2.8221E+04 2.400E+00 1.803E-07

2.104 3.5821E+06 3.5310E+06 1.054E+01 1.083E+00 7.143E-02 1.989E-01 6.4351E+04 2.8437E+04 2.400E+00 2.928E-05

2.169 3.6082E+06 3.5578E+06 1.062E+01 1.083E+00 7.215E-02 1.997E-01 6.4839E+04 2.8652E+04 2.400E+00 2.999E-05

2.234 3.6343E+06 3.5845E+06 1.070E+01 1.082E+00 7.287E-02 2.005E-01 6.5327E+04 2.8868E+04 2.400E+00 2.463E-06

2.299 3.6604E+06 3.6113E+06 1.078E+01 1.081E+00 7.359E-02 2.013E-01 6.5814E+04 2.9083E+04 2.400E+00 5.252E-05

2.364 3.6872E+06 3.6388E+06 1.086E+01 1.081E+00 7.432E-02 2.020E-01 6.6315E+04 2.9304E+04 2.400E+00 1.766E-07

2.429 3.7141E+06 3.6662E+06 1.094E+01 1.080E+00 7.504E-02 2.028E-01 6.6815E+04 2.9525E+04 2.400E+00 2.353E-05

2.494 3.7409E+06 3.6937E+06 1.103E+01 1.079E+00 7.577E-02 2.036E-01 6.7315E+04 2.9747E+04 2.400E+00 1.947E-05

2.559 3.7677E+06 3.7211E+06 1.111E+01 1.079E+00 7.650E-02 2.044E-01 6.7816E+04 2.9968E+04 2.400E+00 1.190E-05

2.624 3.7946E+06 3.7486E+06 1.119E+01 1.078E+00 7.724E-02 2.052E-01 6.8316E+04 3.0188E+04 2.400E+00 6.985E-05

2.689 3.8223E+06 3.7769E+06 1.127E+01 1.078E+00 7.797E-02 2.060E-01 6.8831E+04 3.0416E+04 2.400E+00 2.007E-07

2.754 3.8500E+06 3.8052E+06 1.136E+01 1.077E+00 7.871E-02 2.067E-01 6.9347E+04 3.0644E+04 2.400E+00 4.132E-05

128

CHAPTER

4.COMPUTER

PROGRAM

2.819 3.8777E+06 3.8334E+06 1.144E+01 1.076E+00 7.945E-02 2.075E-01 6.9863E+04 3.0872E+04 2.400E+00 5.554E-05

2.884 3.9055E+06 3.8617E+06 1.153E+01 1.076E+00 8.019E-02 2.083E-01 7.0378E+04 3.1100E+04 2.400E+00 4.291E-05

2.949 3.9332E+06 3.8900E+06 1.161E+01 1.075E+00 8.093E-02 2.091E-01 7.0893E+04 3.1327E+04 2.400E+00 4.143E-06

3.014 3.9609E+06 3.9182E+06 1.170E+01 1.075E+00 8.167E-02 2.099E-01 7.1408E+04 3.1555E+04 2.400E+00 6.007E-05

3.079 3.9895E+06 3.9474E+06 1.178E+01 1.074E+00 8.242E-02 2.107E-01 7.1939E+04 3.1790E+04 2.400E+00 1.923E-07

3.144 4.0182E+06 3.9765E+06 1.187E+01 1.073E+00 8.317E-02 2.115E-01 7.2470E+04 3.2024E+04 2.400E+00 3.283E-05

3.209 4.0468E+06 4.0057E+06 1.196E+01 1.073E+00 8.392E-02 2.123E-01 7.3001E+04 3.2259E+04 2.400E+00 3.939E-05

3.274 4.0754E+06 4.0348E+06 1.204E+01 1.072E+00 8.467E-02 2.131E-01 7.3532E+04 3.2494E+04 2.400E+00 2.019E-05

3.339 4.1040E+06 4.0639E+06 1.213E+01 1.072E+00 8.542E-02 2.139E-01 7.4062E+04 3.2728E+04 2.400E+00 2.410E-05

3.404 4.1326E+06 4.0930E+06 1.222E+01 1.071E+00 8.618E-02 2.147E-01 7.4593E+04 3.2962E+04 2.400E+00 9.311E-05

3.469 4.1624E+06 4.1233E+06 1.231E+01 1.070E+00 8.693E-02 2.155E-01 7.5144E+04 3.3206E+04 2.400E+00 2.383E-07

3.534 4.1922E+06 4.1535E+06 1.240E+01 1.070E+00 8.769E-02 2.163E-01 7.5696E+04 3.3450E+04 2.400E+00 6.583E-05

3.599 4.2213E+06 4.1831E+06 1.249E+01 1.069E+00 8.845E-02 2.172E-01 7.6234E+04 3.3688E+04 2.400E+00 2.751E-06

3.664 4.2504E+06 4.2126E+06 1.258E+01 1.069E+00 8.922E-02 2.180E-01 7.6773E+04 3.3926E+04 2.400E+00 9.494E-05

3.729 4.2809E+06 4.2435E+06 1.267E+01 1.068E+00 8.998E-02 2.188E-01 7.7336E+04 3.4175E+04 2.400E+00 2.674E-07

3.794 4.3113E+06 4.2744E+06 1.276E+01 1.068E+00 9.075E-02 2.196E-01 7.7899E+04 3.4424E+04 2.400E+00 6.829E-05

3.859 4.3410E+06 4.3046E+06 1.285E+01 1.067E+00 9.152E-02 2.204E-01 7.8449E+04 3.4666E+04 2.400E+00 2.700E-06

3.924 4.3707E+06 4.3347E+06 1.294E+01 1.067E+00 9.229E-02 2.212E-01 7.8998E+04 3.4909E+04 2.400E+00 9.676E-05

3.989 4.4019E+06 4.3663E+06 1.303E+01 1.066E+00 9.306E-02 2.220E-01 7.9573E+04 3.5163E+04 2.400E+00 2.695E-07

4.054 4.4330E+06 4.3978E+06 1.313E+01 1.066E+00 9.383E-02 2.229E-01 8.0148E+04 3.5417E+04 2.400E+00 7.049E-05

4.119 4.4633E+06 4.4285E+06 1.322E+01 1.065E+00 9.461E-02 2.237E-01 8.0708E+04 3.5665E+04 2.400E+00 2.651E-06

4.184 4.4936E+06 4.4593E+06 1.331E+01 1.064E+00 9.539E-02 2.245E-01 8.1268E+04 3.5912E+04 2.400E+00 9.856E-05

4.249 4.5254E+06 4.4915E+06 1.341E+01 1.064E+00 9.616E-02 2.253E-01 8.1855E+04 3.6172E+04 2.400E+00 2.712E-07

4.314 4.5573E+06 4.5237E+06 1.350E+01 1.063E+00 9.695E-02 2.262E-01 8.2442E+04 3.6431E+04 2.400E+00 7.262E-05

4.379 4.5882E+06 4.5550E+06 1.360E+01 1.063E+00 9.773E-02 2.270E-01 8.3013E+04 3.6683E+04 2.400E+00 2.603E-06

4.444 4.6199E+06 4.5871E+06 1.369E+01 1.062E+00 9.851E-02 2.278E-01 8.3597E+04 3.6941E+04 2.400E+00 1.221E-07

4.1.

COMPUTER

PROGRAM

129

4.509 4.6515E+06 4.6191E+06 1.379E+01 1.062E+00 9.930E-02 2.287E-01 8.4180E+04 3.7199E+04 2.400E+00 2.091E-05

4.574 4.6832E+06 4.6511E+06 1.389E+01 1.061E+00 1.001E-01 2.295E-01 8.4764E+04 3.7457E+04 2.400E+00 6.467E-05

4.639 4.7157E+06 4.6841E+06 1.398E+01 1.061E+00 1.009E-01 2.303E-01 8.5364E+04 3.7722E+04 2.400E+00 1.606E-07

4.704 4.7483E+06 4.7170E+06 1.408E+01 1.060E+00 1.017E-01 2.312E-01 8.5965E+04 3.7988E+04 2.400E+00 4.018E-05

4.769 4.7809E+06 4.7499E+06 1.418E+01 1.060E+00 1.025E-01 2.320E-01 8.6565E+04 3.8253E+04 2.400E+00 5.697E-05

4.834 4.8135E+06 4.7829E+06 1.428E+01 1.059E+00 1.033E-01 2.329E-01 8.7165E+04 3.8518E+04 2.400E+00 5.079E-05

4.899 4.8461E+06 4.8158E+06 1.438E+01 1.059E+00 1.041E-01 2.337E-01 8.7766E+04 3.8783E+04 2.400E+00 2.191E-05

4.964 4.8787E+06 4.8487E+06 1.448E+01 1.058E+00 1.049E-01 2.345E-01 8.8366E+04 3.9049E+04 2.400E+00 2.899E-05

5.029 4.9120E+06 4.8824E+06 1.458E+01 1.058E+00 1.057E-01 2.354E-01 8.8979E+04 3.9320E+04 2.400E+00 1.225E-07

5.094 4.9454E+06 4.9161E+06 1.468E+01 1.058E+00 1.065E-01 2.362E-01 8.9593E+04 3.9591E+04 2.400E+00 5.991E-06

5.159 4.9787E+06 4.9497E+06 1.478E+01 1.057E+00 1.073E-01 2.371E-01 9.0207E+04 3.9862E+04 2.400E+00 1.038E-05

5.224 5.0121E+06 4.9834E+06 1.488E+01 1.057E+00 1.081E-01 2.379E-01 9.0820E+04 4.0133E+04 2.400E+00 4.858E-05

5.289 5.0462E+06 5.0179E+06 1.498E+01 1.056E+00 1.089E-01 2.388E-01 9.1449E+04 4.0411E+04 2.400E+00 1.301E-07

5.354 5.0804E+06 5.0524E+06 1.508E+01 1.056E+00 1.097E-01 2.397E-01 9.2077E+04 4.0689E+04 2.400E+00 2.565E-05

5.419 5.1146E+06 5.0869E+06 1.519E+01 1.055E+00 1.105E-01 2.405E-01 9.2705E+04 4.0966E+04 2.400E+00 2.902E-05

5.484 5.1488E+06 5.1213E+06 1.529E+01 1.055E+00 1.113E-01 2.414E-01 9.3334E+04 4.1244E+04 2.400E+00 1.063E-05

5.549 5.1829E+06 5.1558E+06 1.539E+01 1.054E+00 1.122E-01 2.422E-01 9.3962E+04 4.1522E+04 2.400E+00 2.928E-05

5.614 5.2171E+06 5.1903E+06 1.550E+01 1.054E+00 1.130E-01 2.431E-01 9.4590E+04 4.1799E+04 2.400E+00 9.006E-05

5.679 5.2526E+06 5.2261E+06 1.560E+01 1.053E+00 1.138E-01 2.440E-01 9.5243E+04 4.2088E+04 2.400E+00 2.055E-07

5.744 5.2882E+06 5.2619E+06 1.571E+01 1.053E+00 1.146E-01 2.448E-01 9.5896E+04 4.2376E+04 2.400E+00 6.690E-05

5.809 5.3228E+06 5.2968E+06 1.581E+01 1.053E+00 1.154E-01 2.457E-01 9.6532E+04 4.2657E+04 2.400E+00 2.399E-06

5.874 5.3574E+06 5.3317E+06 1.592E+01 1.052E+00 1.163E-01 2.466E-01 9.7168E+04 4.2938E+04 2.400E+00 9.175E-05

5.939 5.3937E+06 5.3683E+06 1.603E+01 1.052E+00 1.171E-01 2.474E-01 9.7834E+04 4.3233E+04 2.400E+00 2.391E-07

6.004 5.4300E+06 5.4048E+06 1.613E+01 1.051E+00 1.179E-01 2.483E-01 9.8500E+04 4.3527E+04 2.400E+00 6.887E-05

6.069 5.4653E+06 5.4404E+06 1.624E+01 1.051E+00 1.188E-01 2.492E-01 9.9148E+04 4.3813E+04 2.400E+00 2.357E-06

6.134 5.5005E+06 5.4759E+06 1.635E+01 1.050E+00 1.196E-01 2.500E-01 9.9796E+04 4.4100E+04 2.400E+00 9.326E-05

130

CHAPTER

4.COMPUTER

PROGRAM

6.199 5.5376E+06 5.5132E+06 1.646E+01 1.050E+00 1.204E-01 2.509E-01 1.0048E+05 4.4400E+04 2.400E+00 2.406E-07

6.264 5.5746E+06 5.5505E+06 1.657E+01 1.050E+00 1.213E-01 2.518E-01 1.0115E+05 4.4700E+04 2.400E+00 7.071E-05

6.329 5.6105E+06 5.5867E+06 1.668E+01 1.049E+00 1.221E-01 2.527E-01 1.0181E+05 4.4991E+04 2.400E+00 2.317E-06

6.394 5.6465E+06 5.6229E+06 1.679E+01 1.049E+00 1.230E-01 2.536E-01 1.0247E+05 4.5283E+04 2.400E+00 9.466E-05

6.459 5.6842E+06 5.6609E+06 1.690E+01 1.048E+00 1.238E-01 2.544E-01 1.0317E+05 4.5589E+04 2.400E+00 2.419E-07

6.524 5.7220E+06 5.6989E+06 1.701E+01 1.048E+00 1.246E-01 2.553E-01 1.0386E+05 4.5895E+04 2.400E+00 7.243E-05

6.589 5.7586E+06 5.7357E+06 1.712E+01 1.048E+00 1.255E-01 2.562E-01 1.0453E+05 4.6192E+04 2.400E+00 2.278E-06

6.654 5.7952E+06 5.7726E+06 1.723E+01 1.047E+00 1.263E-01 2.571E-01 1.0520E+05 4.6489E+04 2.400E+00 9.596E-05

6.719 5.8337E+06 5.8113E+06 1.735E+01 1.047E+00 1.272E-01 2.580E-01 1.0591E+05 4.6801E+04 2.400E+00 2.429E-07

6.784 5.8722E+06 5.8501E+06 1.746E+01 1.046E+00 1.281E-01 2.589E-01 1.0661E+05 4.7113E+04 2.400E+00 7.405E-05

6.849 5.9095E+06 5.8876E+06 1.758E+01 1.046E+00 1.289E-01 2.598E-01 1.0730E+05 4.7415E+04 2.400E+00 2.240E-06

6.914 5.9469E+06 5.9252E+06 1.769E+01 1.046E+00 1.298E-01 2.607E-01 1.0798E+05 4.7717E+04 2.400E+00 9.716E-05

6.979 5.9861E+06 5.9646E+06 1.781E+01 1.045E+00 1.306E-01 2.616E-01 1.0870E+05 4.8035E+04 2.400E+00 2.436E-07

7.044 6.0254E+06 6.0041E+06 1.792E+01 1.045E+00 1.315E-01 2.625E-01 1.0942E+05 4.8353E+04 2.400E+00 7.556E-05

7.109 6.0634E+06 6.0423E+06 1.804E+01 1.044E+00 1.324E-01 2.634E-01 1.1012E+05 4.8661E+04 2.400E+00 2.203E-06

7.174 6.1014E+06 6.0806E+06 1.815E+01 1.044E+00 1.332E-01 2.643E-01 1.1082E+05 4.8969E+04 2.400E+00 9.827E-05

7.239 6.1414E+06 6.1208E+06 1.827E+01 1.044E+00 1.341E-01 2.652E-01 1.1155E+05 4.9293E+04 2.400E+00 2.442E-07

7.304 6.1814E+06 6.1610E+06 1.839E+01 1.043E+00 1.350E-01 2.661E-01 1.1228E+05 4.9617E+04 2.400E+00 7.697E-05

7.369 6.2201E+06 6.1999E+06 1.851E+01 1.043E+00 1.358E-01 2.670E-01 1.1299E+05 4.9930E+04 2.400E+00 2.167E-06

7.434 6.2589E+06 6.2388E+06 1.863E+01 1.043E+00 1.367E-01 2.679E-01 1.1370E+05 5.0244E+04 2.400E+00 9.929E-05

7.499 6.2996E+06 6.2798E+06 1.875E+01 1.042E+00 1.376E-01 2.688E-01 1.1445E+05 5.0574E+04 2.400E+00 2.502E-07

7.564 6.3404E+06 6.3208E+06 1.887E+01 1.042E+00 1.385E-01 2.697E-01 1.1519E+05 5.0904E+04 2.400E+00 7.806E-05

7.629 6.3798E+06 6.3604E+06 1.899E+01 1.042E+00 1.393E-01 2.706E-01 1.1592E+05 5.1223E+04 2.400E+00 2.132E-06

7.694 6.4202E+06 6.4010E+06 1.911E+01 1.041E+00 1.402E-01 2.716E-01 1.1666E+05 5.1550E+04 2.400E+00 1.113E-07

7.759 6.4607E+06 6.4416E+06 1.923E+01 1.041E+00 1.411E-01 2.725E-01 1.1740E+05 5.1877E+04 2.400E+00 1.676E-05

7.824 6.5011E+06 6.4822E+06 1.935E+01 1.041E+00 1.420E-01 2.734E-01 1.1814E+05 5.2204E+04 2.400E+00 5.218E-05

4.1.

COMPUTER

PROGRAM

131

7.889 6.5425E+06 6.5239E+06 1.948E+01 1.040E+00 1.429E-01 2.743E-01 1.1889E+05 5.2539E+04 2.400E+00 1.181E-07

7.954 6.5840E+06 6.5655E+06 1.960E+01 1.040E+00 1.438E-01 2.752E-01 1.1965E+05 5.2875E+04 2.400E+00 3.237E-05

8.019 6.6255E+06 6.6072E+06 1.972E+01 1.040E+00 1.446E-01 2.762E-01 1.2041E+05 5.3210E+04 2.400E+00 4.586E-05

8.084 6.6669E+06 6.6488E+06 1.985E+01 1.039E+00 1.455E-01 2.771E-01 1.2117E+05 5.3545E+04 2.400E+00 4.061E-05

8.149 6.7084E+06 6.6905E+06 1.997E+01 1.039E+00 1.464E-01 2.780E-01 1.2193E+05 5.3881E+04 2.400E+00 1.684E-05

8.214 6.7499E+06 6.7321E+06 2.010E+01 1.039E+00 1.473E-01 2.790E-01 1.2269E+05 5.4216E+04 2.400E+00 2.489E-05

8.279 6.7913E+06 6.7738E+06 2.022E+01 1.038E+00 1.482E-01 2.799E-01 1.2345E+05 5.4552E+04 2.400E+00 8.445E-05

8.344 6.8345E+06 6.8171E+06 2.035E+01 1.038E+00 1.491E-01 2.808E-01 1.2424E+05 5.4900E+04 2.400E+00 1.817E-07

8.409 6.8777E+06 6.8604E+06 2.048E+01 1.038E+00 1.500E-01 2.818E-01 1.2503E+05 5.5249E+04 2.400E+00 6.453E-05

8.474 6.9196E+06 6.9025E+06 2.061E+01 1.037E+00 1.509E-01 2.827E-01 1.2580E+05 5.5589E+04 2.400E+00 2.060E-06

8.539 6.9616E+06 6.9447E+06 2.073E+01 1.037E+00 1.518E-01 2.836E-01 1.2656E+05 5.5928E+04 2.400E+00 8.586E-05

8.604 7.0055E+06 6.9888E+06 2.086E+01 1.037E+00 1.528E-01 2.846E-01 1.2737E+05 5.6283E+04 2.400E+00 2.058E-07

8.669 7.0495E+06 7.0329E+06 2.099E+01 1.036E+00 1.537E-01 2.855E-01 1.2817E+05 5.6639E+04 2.400E+00 6.627E-05

8.734 7.0922E+06 7.0758E+06 2.112E+01 1.036E+00 1.546E-01 2.865E-01 1.2895E+05 5.6984E+04 2.400E+00 2.027E-06

8.799 7.1349E+06 7.1186E+06 2.125E+01 1.036E+00 1.555E-01 2.874E-01 1.2973E+05 5.7329E+04 2.400E+00 8.703E-05

8.864 7.1797E+06 7.1636E+06 2.139E+01 1.035E+00 1.564E-01 2.884E-01 1.3055E+05 5.7691E+04 2.400E+00 2.070E-07

8.929 7.2244E+06 7.2085E+06 2.152E+01 1.035E+00 1.573E-01 2.893E-01 1.3137E+05 5.8053E+04 2.400E+00 6.780E-05

8.994 7.2679E+06 7.2521E+06 2.165E+01 1.035E+00 1.582E-01 2.903E-01 1.3217E+05 5.8404E+04 2.400E+00 1.995E-06

9.059 7.3113E+06 7.2957E+06 2.178E+01 1.034E+00 1.592E-01 2.912E-01 1.3296E+05 5.8755E+04 2.400E+00 8.845E-05

9.124 7.3569E+06 7.3414E+06 2.192E+01 1.034E+00 1.601E-01 2.922E-01 1.3379E+05 5.9123E+04 2.400E+00 2.083E-07

9.189 7.4025E+06 7.3872E+06 2.205E+01 1.034E+00 1.610E-01 2.931E-01 1.3463E+05 5.9491E+04 2.400E+00 6.924E-05

9.254 7.4467E+06 7.4315E+06 2.219E+01 1.034E+00 1.619E-01 2.941E-01 1.3544E+05 5.9849E+04 2.400E+00 1.964E-06

9.319 7.4909E+06 7.4758E+06 2.232E+01 1.033E+00 1.629E-01 2.950E-01 1.3624E+05 6.0206E+04 2.400E+00 8.944E-05

9.384 7.5374E+06 7.5224E+06 2.246E+01 1.033E+00 1.638E-01 2.960E-01 1.3709E+05 6.0581E+04 2.400E+00 2.091E-07

9.449 7.5838E+06 7.5690E+06 2.259E+01 1.033E+00 1.647E-01 2.970E-01 1.3794E+05 6.0956E+04 2.400E+00 7.072E-05

9.514 7.6287E+06 7.6141E+06 2.273E+01 1.032E+00 1.657E-01 2.979E-01 1.3876E+05 6.1319E+04 2.400E+00 1.933E-06

132

CHAPTER

4.COMPUTER

PROGRAM

9.579 7.6737E+06 7.6592E+06 2.287E+01 1.032E+00 1.666E-01 2.989E-01 1.3958E+05 6.1682E+04 2.400E+00 9.049E-05

9.644 7.7210E+06 7.7066E+06 2.301E+01 1.032E+00 1.676E-01 2.999E-01 1.4045E+05 6.2064E+04 2.400E+00 2.096E-07

9.709 7.7682E+06 7.7540E+06 2.315E+01 1.032E+00 1.685E-01 3.008E-01 1.4131E+05 6.2445E+04 2.400E+00 7.176E-05

9.774 7.8140E+06 7.7998E+06 2.328E+01 1.031E+00 1.694E-01 3.018E-01 1.4215E+05 6.2815E+04 2.400E+00 1.903E-06

9.839 7.8597E+06 7.8457E+06 2.342E+01 1.031E+00 1.704E-01 3.028E-01 1.4298E+05 6.3184E+04 2.400E+00 9.144E-05

9.904 7.9078E+06 7.8939E+06 2.357E+01 1.031E+00 1.713E-01 3.038E-01 1.4386E+05 6.3573E+04 2.400E+00 2.100E-07

9.969 7.9559E+06 7.9422E+06 2.371E+01 1.030E+00 1.723E-01 3.047E-01 1.4474E+05 6.3961E+04 2.400E+00 7.307E-05

10.034 8.0024E+06 7.9888E+06 2.385E+01 1.030E+00 1.732E-01 3.057E-01 1.4559E+05 6.4337E+04 2.400E+00 1.874E-06

10.099 8.0489E+06 8.0354E+06 2.399E+01 1.030E+00 1.742E-01 3.067E-01 1.4644E+05 6.4712E+04 2.400E+00 9.222E-05

10.164 8.0979E+06 8.0845E+06 2.413E+01 1.030E+00 1.752E-01 3.077E-01 1.4734E+05 6.5107E+04 2.400E+00 2.102E-07

10.229 8.1468E+06 8.1336E+06 2.428E+01 1.029E+00 1.761E-01 3.087E-01 1.4823E+05 6.5503E+04 2.400E+00 7.395E-05

10.294 8.1941E+06 8.1810E+06 2.442E+01 1.029E+00 1.771E-01 3.097E-01 1.4909E+05 6.5885E+04 2.400E+00 1.846E-06

10.359 8.2414E+06 8.2284E+06 2.457E+01 1.029E+00 1.780E-01 3.107E-01 1.4996E+05 6.6266E+04 2.400E+00 9.292E-05

10.424 8.2912E+06 8.2783E+06 2.471E+01 1.029E+00 1.790E-01 3.116E-01 1.5087E+05 6.6669E+04 2.400E+00 2.103E-07

10.489 8.3411E+06 8.3283E+06 2.486E+01 1.028E+00 1.800E-01 3.126E-01 1.5178E+05 6.7071E+04 2.400E+00 7.512E-05

10.554 8.3891E+06 8.3765E+06 2.501E+01 1.028E+00 1.809E-01 3.136E-01 1.5266E+05 6.7459E+04 2.400E+00 1.819E-06

10.619 8.4372E+06 8.4247E+06 2.515E+01 1.028E+00 1.819E-01 3.146E-01 1.5354E+05 6.7847E+04 2.400E+00 9.369E-05

10.684 8.4879E+06 8.4755E+06 2.530E+01 1.028E+00 1.829E-01 3.156E-01 1.5446E+05 6.8256E+04 2.400E+00 2.102E-07

10.749 8.5386E+06 8.5263E+06 2.545E+01 1.027E+00 1.838E-01 3.166E-01 1.5539E+05 6.8665E+04 2.400E+00 7.588E-05

10.814 8.5875E+06 8.5753E+06 2.560E+01 1.027E+00 1.848E-01 3.176E-01 1.5628E+05 6.9060E+04 2.400E+00 1.792E-06

10.879 8.6364E+06 8.6243E+06 2.575E+01 1.027E+00 1.858E-01 3.186E-01 1.5717E+05 6.9455E+04 2.400E+00 9.439E-05

10.944 8.6880E+06 8.6760E+06 2.590E+01 1.027E+00 1.868E-01 3.196E-01 1.5812E+05 6.9871E+04 2.400E+00 2.208E-07

11.009 8.7395E+06 8.7276E+06 2.605E+01 1.026E+00 1.878E-01 3.206E-01 1.5906E+05 7.0287E+04 2.400E+00 7.670E-05

11.074 8.7892E+06 8.7774E+06 2.620E+01 1.026E+00 1.887E-01 3.217E-01 1.5996E+05 7.0688E+04 2.400E+00 1.766E-06

11.139 8.8389E+06 8.8272E+06 2.635E+01 1.026E+00 1.897E-01 3.227E-01 1.6087E+05 7.1089E+04 2.400E+00 9.467E-05

11.204 8.8914E+06 8.8798E+06 2.651E+01 1.026E+00 1.907E-01 3.237E-01 1.6183E+05 7.1512E+04 2.400E+00 2.093E-07

4.1.

COMPUTER

PROGRAM

133

11.269 8.9438E+06 8.9324E+06 2.666E+01 1.026E+00 1.917E-01 3.247E-01 1.6279E+05 7.1936E+04 2.400E+00 7.747E-05

11.334 8.9943E+06 8.9830E+06 2.682E+01 1.025E+00 1.927E-01 3.257E-01 1.6371E+05 7.2343E+04 2.400E+00 1.740E-06

11.399 9.0449E+06 9.0336E+06 2.697E+01 1.025E+00 1.937E-01 3.267E-01 1.6463E+05 7.2751E+04 2.400E+00 9.527E-05

11.464 9.0982E+06 9.0870E+06 2.713E+01 1.025E+00 1.947E-01 3.277E-01 1.6561E+05 7.3181E+04 2.400E+00 2.090E-07

11.529 9.1516E+06 9.1405E+06 2.728E+01 1.025E+00 1.957E-01 3.288E-01 1.6658E+05 7.3612E+04 2.400E+00 7.823E-05

11.594 9.2029E+06 9.1919E+06 2.744E+01 1.024E+00 1.967E-01 3.298E-01 1.6752E+05 7.4026E+04 2.400E+00 1.715E-06

11.659 9.2542E+06 9.2434E+06 2.760E+01 1.024E+00 1.977E-01 3.308E-01 1.6846E+05 7.4440E+04 2.400E+00 9.583E-05

11.724 9.3085E+06 9.2977E+06 2.776E+01 1.024E+00 1.987E-01 3.318E-01 1.6945E+05 7.4878E+04 2.400E+00 2.087E-07

11.789 9.3627E+06 9.3520E+06 2.792E+01 1.024E+00 1.997E-01 3.329E-01 1.7044E+05 7.5315E+04 2.400E+00 7.900E-05

11.854 9.4149E+06 9.4043E+06 2.807E+01 1.024E+00 2.007E-01 3.339E-01 1.7139E+05 7.5736E+04 2.400E+00 1.690E-06

11.919 9.4671E+06 9.4566E+06 2.823E+01 1.023E+00 2.017E-01 3.349E-01 1.7234E+05 7.6157E+04 2.400E+00 9.634E-05

11.984 9.5222E+06 9.5118E+06 2.840E+01 1.023E+00 2.027E-01 3.360E-01 1.7335E+05 7.6602E+04 2.400E+00 2.082E-07

12.049 9.5774E+06 9.5671E+06 2.856E+01 1.023E+00 2.037E-01 3.370E-01 1.7436E+05 7.7047E+04 2.400E+00 7.970E-05

12.114 9.6304E+06 9.6202E+06 2.872E+01 1.023E+00 2.048E-01 3.381E-01 1.7532E+05 7.7475E+04 2.400E+00 1.666E-06

12.179 9.6834E+06 9.6733E+06 2.888E+01 1.023E+00 2.058E-01 3.391E-01 1.7629E+05 7.7902E+04 2.400E+00 9.680E-05

12.244 9.7395E+06 9.7294E+06 2.905E+01 1.022E+00 2.068E-01 3.401E-01 1.7731E+05 7.8355E+04 2.400E+00 2.077E-07

12.309 9.7956E+06 9.7856E+06 2.921E+01 1.022E+00 2.078E-01 3.412E-01 1.7834E+05 7.8807E+04 2.400E+00 8.038E-05

12.374 9.8494E+06 9.8396E+06 2.937E+01 1.022E+00 2.088E-01 3.422E-01 1.7932E+05 7.9242E+04 2.400E+00 1.643E-06

12.439 9.9033E+06 9.8935E+06 2.954E+01 1.022E+00 2.099E-01 3.433E-01 1.8030E+05 7.9676E+04 2.400E+00 9.722E-05

12.504 9.9603E+06 9.9506E+06 2.971E+01 1.022E+00 2.109E-01 3.443E-01 1.8134E+05 8.0136E+04 2.400E+00 2.071E-07

12.569 1.0017E+07 1.0008E+07 2.987E+01 1.021E+00 2.119E-01 3.454E-01 1.8238E+05 8.0596E+04 2.400E+00 8.099E-05

12.634 1.0072E+07 1.0062E+07 3.004E+01 1.021E+00 2.130E-01 3.464E-01 1.8338E+05 8.1037E+04 2.400E+00 1.620E-06

12.699 1.0127E+07 1.0117E+07 3.021E+01 1.021E+00 2.140E-01 3.475E-01 1.8438E+05 8.1478E+04 2.400E+00 9.763E-05

12.764 1.0185E+07 1.0175E+07 3.038E+01 1.021E+00 2.150E-01 3.485E-01 1.8544E+05 8.1945E+04 2.400E+00 2.065E-07

12.829 1.0243E+07 1.0233E+07 3.055E+01 1.021E+00 2.161E-01 3.496E-01 1.8650E+05 8.2413E+04 2.400E+00 8.157E-05

12.894 1.0298E+07 1.0289E+07 3.072E+01 1.020E+00 2.171E-01 3.507E-01 1.8751E+05 8.2861E+04 2.400E+00 1.597E-06

134

CHAPTER

4.COMPUTER

PROGRAM

12.959 1.0354E+07 1.0345E+07 3.088E+01 1.020E+00 2.181E-01 3.517E-01 1.8853E+05 8.3309E+04 2.400E+00 9.794E-05

13.024 1.0413E+07 1.0404E+07 3.106E+01 1.020E+00 2.192E-01 3.528E-01 1.8960E+05 8.3784E+04 2.400E+00 2.057E-07

13.089 1.0472E+07 1.0463E+07 3.123E+01 1.020E+00 2.202E-01 3.539E-01 1.9068E+05 8.4259E+04 2.400E+00 8.210E-05

13.154 1.0528E+07 1.0519E+07 3.140E+01 1.020E+00 2.213E-01 3.549E-01 1.9171E+05 8.4714E+04 2.400E+00 1.575E-06

13.219 1.0584E+07 1.0576E+07 3.157E+01 1.019E+00 2.223E-01 3.560E-01 1.9274E+05 8.5169E+04 2.400E+00 9.825E-05

13.284 1.0644E+07 1.0636E+07 3.175E+01 1.019E+00 2.234E-01 3.571E-01 1.9383E+05 8.5652E+04 2.400E+00 2.049E-07

13.349 1.0704E+07 1.0695E+07 3.193E+01 1.019E+00 2.244E-01 3.581E-01 1.9492E+05 8.6134E+04 2.400E+00 8.259E-05

13.414 1.0762E+07 1.0753E+07 3.210E+01 1.019E+00 2.255E-01 3.592E-01 1.9597E+05 8.6597E+04 2.400E+00 1.554E-06

13.479 1.0819E+07 1.0810E+07 3.227E+01 1.019E+00 2.266E-01 3.603E-01 1.9701E+05 8.7059E+04 2.400E+00 9.851E-05

13.544 1.0880E+07 1.0871E+07 3.245E+01 1.019E+00 2.276E-01 3.614E-01 1.9812E+05 8.7549E+04 2.400E+00 2.127E-07

13.609 1.0940E+07 1.0932E+07 3.263E+01 1.018E+00 2.287E-01 3.625E-01 1.9923E+05 8.8040E+04 2.400E+00 8.283E-05

13.674 1.0999E+07 1.0990E+07 3.281E+01 1.018E+00 2.297E-01 3.635E-01 2.0029E+05 8.8509E+04 2.400E+00 1.533E-06

13.739 1.1057E+07 1.1049E+07 3.299E+01 1.018E+00 2.308E-01 3.646E-01 2.0136E+05 8.8978E+04 2.400E+00 9.854E-05

13.804 1.1119E+07 1.1110E+07 3.317E+01 1.018E+00 2.319E-01 3.657E-01 2.0248E+05 8.9476E+04 2.400E+00 2.027E-07

13.869 1.1180E+07 1.1172E+07 3.335E+01 1.018E+00 2.329E-01 3.668E-01 2.0361E+05 8.9974E+04 2.400E+00 8.326E-05

13.934 1.1240E+07 1.1231E+07 3.353E+01 1.018E+00 2.340E-01 3.679E-01 2.0469E+05 9.0451E+04 2.400E+00 1.512E-06

13.999 1.1299E+07 1.1291E+07 3.371E+01 1.017E+00 2.351E-01 3.690E-01 2.0577E+05 9.0928E+04 2.400E+00 9.875E-05

14.064 1.1361E+07 1.1353E+07 3.389E+01 1.017E+00 2.362E-01 3.701E-01 2.0691E+05 9.1434E+04 2.400E+00 2.018E-07

14.129 1.1424E+07 1.1416E+07 3.408E+01 1.017E+00 2.373E-01 3.712E-01 2.0806E+05 9.1940E+04 2.400E+00 8.365E-05

14.194 1.1484E+07 1.1476E+07 3.426E+01 1.017E+00 2.383E-01 3.723E-01 2.0915E+05 9.2423E+04 2.400E+00 1.492E-06

14.259 1.1544E+07 1.1536E+07 3.444E+01 1.017E+00 2.394E-01 3.734E-01 2.1025E+05 9.2907E+04 2.400E+00 9.893E-05

14.324 1.1608E+07 1.1600E+07 3.463E+01 1.017E+00 2.405E-01 3.745E-01 2.1141E+05 9.3421E+04 2.400E+00 2.008E-07

14.389 1.1672E+07 1.1664E+07 3.482E+01 1.016E+00 2.416E-01 3.756E-01 2.1257E+05 9.3935E+04 2.400E+00 8.401E-05

14.454 1.1733E+07 1.1725E+07 3.500E+01 1.016E+00 2.427E-01 3.767E-01 2.1368E+05 9.4426E+04 2.400E+00 1.472E-06

14.519 1.1794E+07 1.1786E+07 3.519E+01 1.016E+00 2.438E-01 3.778E-01 2.1480E+05 9.4917E+04 2.400E+00 9.908E-05

14.584 1.1858E+07 1.1851E+07 3.538E+01 1.016E+00 2.449E-01 3.789E-01 2.1598E+05 9.5439E+04 2.400E+00 1.998E-07

4.1.

COMPUTER

PROGRAM

135

14.649 1.1923E+07 1.1916E+07 3.557E+01 1.016E+00 2.460E-01 3.800E-01 2.1716E+05 9.5961E+04 2.400E+00 8.433E-05

14.714 1.1985E+07 1.1978E+07 3.576E+01 1.016E+00 2.470E-01 3.811E-01 2.1829E+05 9.6460E+04 2.400E+00 1.453E-06

14.779 1.2047E+07 1.2039E+07 3.594E+01 1.016E+00 2.481E-01 3.822E-01 2.1941E+05 9.6958E+04 2.400E+00 9.921E-05

14.844 1.2112E+07 1.2105E+07 3.614E+01 1.015E+00 2.492E-01 3.834E-01 2.2061E+05 9.7488E+04 2.400E+00 1.988E-07

14.909 1.2178E+07 1.2171E+07 3.633E+01 1.015E+00 2.504E-01 3.845E-01 2.2181E+05 9.8018E+04 2.400E+00 8.463E-05

14.974 1.2241E+07 1.2234E+07 3.652E+01 1.015E+00 2.515E-01 3.856E-01 2.2296E+05 9.8524E+04 2.400E+00 1.434E-06

15.039 1.2304E+07 1.2297E+07 3.671E+01 1.015E+00 2.526E-01 3.867E-01 2.2410E+05 9.9030E+04 2.400E+00 9.931E-05

15.104 1.2371E+07 1.2364E+07 3.691E+01 1.015E+00 2.537E-01 3.879E-01 2.2532E+05 9.9568E+04 2.400E+00 1.976E-07

15.169 1.2437E+07 1.2430E+07 3.710E+01 1.015E+00 2.548E-01 3.890E-01 2.2654E+05 1.0011E+05 2.400E+00 8.496E-05

15.234 1.2501E+07 1.2494E+07 3.730E+01 1.015E+00 2.559E-01 3.901E-01 2.2770E+05 1.0062E+05 2.400E+00 1.415E-06

15.299 1.2565E+07 1.2558E+07 3.749E+01 1.014E+00 2.570E-01 3.912E-01 2.2886E+05 1.0113E+05 2.400E+00 9.938E-05

15.364 1.2633E+07 1.2626E+07 3.769E+01 1.014E+00 2.581E-01 3.924E-01 2.3010E+05 1.0168E+05 2.400E+00 1.965E-07

15.429 1.2700E+07 1.2694E+07 3.789E+01 1.014E+00 2.592E-01 3.935E-01 2.3133E+05 1.0223E+05 2.400E+00 8.515E-05

15.494 1.2765E+07 1.2758E+07 3.809E+01 1.014E+00 2.604E-01 3.946E-01 2.3251E+05 1.0275E+05 2.400E+00 1.397E-06

15.559 1.2830E+07 1.2823E+07 3.828E+01 1.014E+00 2.615E-01 3.958E-01 2.3369E+05 1.0327E+05 2.400E+00 9.944E-05

15.624 1.2898E+07 1.2892E+07 3.849E+01 1.014E+00 2.626E-01 3.969E-01 2.3495E+05 1.0382E+05 2.400E+00 1.953E-07

15.689 1.2967E+07 1.2961E+07 3.869E+01 1.014E+00 2.637E-01 3.981E-01 2.3620E+05 1.0438E+05 2.400E+00 8.537E-05

15.754 1.3033E+07 1.3026E+07 3.889E+01 1.013E+00 2.649E-01 3.992E-01 2.3740E+05 1.0491E+05 2.400E+00 1.379E-06

BURNOUT HAS OCCURED AT 1.581886E+01seconds

PORT DIAMETER = 3.990262E-01 AT PORT LOCATION 2.376000E+00(m)

Sliver mass at burnout (kg) = 1.693337E+02

Sliver fraction at burnout (sliver mass/initial propellant mass) = 3.457966E-01

Burnout nozzle-end total-pressure (Pa) = 1.302640E+07

136

CHAPTER

4.COMPUTER

PROGRAM

TAIL-OFF PERFORMANCE

p0h=head-end total pressure; p0n = nozzle end total pressure;

Lnprt = exit-port location; delmdot = convergence error

Time (s) p0h(Pa) p0n(Pa) mdot(kg/s) epsilon dhprt(m) dnprt(m) pe(Pa) Fvac(N) Lnprt (m) delmdot

15.819 1.2857E+07 1.2850E+07 3.836E+01 1.013E+00 2.660E-01 3.990E-01 2.3419E+05 1.0349E+05 2.376E+00 2.895E-05

15.884 1.2680E+07 1.2674E+07 3.784E+01 1.013E+00 2.671E-01 3.988E-01 2.3098E+05 1.0207E+05 2.352E+00 6.050E-05

15.949 1.2746E+07 1.2740E+07 3.803E+01 1.013E+00 2.683E-01 4.000E-01 2.3218E+05 1.0260E+05 2.352E+00 1.133E-05

16.014 1.2570E+07 1.2564E+07 3.751E+01 1.012E+00 2.694E-01 3.998E-01 2.2897E+05 1.0118E+05 2.328E+00 3.047E-05

16.079 1.2393E+07 1.2387E+07 3.698E+01 1.012E+00 2.705E-01 3.996E-01 2.2575E+05 9.9757E+04 2.304E+00 5.568E-05

16.144 1.2216E+07 1.2210E+07 3.645E+01 1.012E+00 2.716E-01 3.994E-01 2.2253E+05 9.8335E+04 2.280E+00 8.831E-05

16.209 1.2042E+07 1.2036E+07 3.593E+01 1.011E+00 2.727E-01 3.991E-01 2.1935E+05 9.6930E+04 2.256E+00 1.171E-05

16.274 1.1867E+07 1.1862E+07 3.541E+01 1.011E+00 2.738E-01 3.989E-01 2.1617E+05 9.5525E+04 2.232E+00 5.770E-05

16.339 1.1691E+07 1.1685E+07 3.488E+01 1.010E+00 2.749E-01 3.987E-01 2.1295E+05 9.4103E+04 2.208E+00 3.119E-07

16.404 1.1748E+07 1.1743E+07 3.506E+01 1.010E+00 2.760E-01 3.998E-01 2.1400E+05 9.4567E+04 2.208E+00 1.125E-05

16.469 1.1571E+07 1.1566E+07 3.453E+01 1.010E+00 2.771E-01 3.995E-01 2.1078E+05 9.3145E+04 2.184E+00 3.464E-05

16.534 1.1395E+07 1.1389E+07 3.400E+01 1.010E+00 2.782E-01 3.993E-01 2.0757E+05 9.1724E+04 2.160E+00 7.225E-05

16.599 1.1220E+07 1.1215E+07 3.348E+01 1.009E+00 2.793E-01 3.990E-01 2.0439E+05 9.0317E+04 2.136E+00 1.352E-05

16.664 1.1045E+07 1.1040E+07 3.296E+01 1.009E+00 2.803E-01 3.988E-01 2.0120E+05 8.8911E+04 2.112E+00 3.037E-05

16.729 1.1098E+07 1.1093E+07 3.312E+01 1.009E+00 2.814E-01 3.999E-01 2.0216E+05 8.9334E+04 2.112E+00 1.113E-05

16.794 1.0921E+07 1.0916E+07 3.259E+01 1.009E+00 2.825E-01 3.996E-01 1.9895E+05 8.7914E+04 2.088E+00 3.791E-05

16.859 1.0745E+07 1.0740E+07 3.206E+01 1.008E+00 2.835E-01 3.993E-01 1.9573E+05 8.6494E+04 2.064E+00 8.537E-05

16.924 1.0571E+07 1.0566E+07 3.154E+01 1.008E+00 2.846E-01 3.990E-01 1.9256E+05 8.5093E+04 2.040E+00 1.508E-05

16.989 1.0397E+07 1.0392E+07 3.102E+01 1.008E+00 2.856E-01 3.987E-01 1.8939E+05 8.3692E+04 2.016E+00 3.405E-05

17.054 1.0445E+07 1.0440E+07 3.117E+01 1.008E+00 2.867E-01 3.998E-01 1.9026E+05 8.4077E+04 2.016E+00 1.123E-05

17.119 1.0269E+07 1.0264E+07 3.064E+01 1.007E+00 2.877E-01 3.995E-01 1.8706E+05 8.2663E+04 1.992E+00 4.173E-05

4.1.

COMPUTER

PROGRAM

137

17.184 1.0095E+07 1.0090E+07 3.012E+01 1.007E+00 2.888E-01 3.992E-01 1.8389E+05 8.1259E+04 1.968E+00 1.669E-05

17.249 9.9201E+06 9.9157E+06 2.960E+01 1.007E+00 2.898E-01 3.989E-01 1.8071E+05 7.9855E+04 1.944E+00 2.001E-05

17.314 9.9656E+06 9.9612E+06 2.974E+01 1.007E+00 2.908E-01 3.999E-01 1.8154E+05 8.0221E+04 1.944E+00 1.162E-05

17.379 9.7905E+06 9.7861E+06 2.922E+01 1.006E+00 2.919E-01 3.996E-01 1.7835E+05 7.8811E+04 1.920E+00 4.514E-05

17.444 9.6167E+06 9.6124E+06 2.870E+01 1.006E+00 2.929E-01 3.993E-01 1.7518E+05 7.7412E+04 1.896E+00 1.823E-05

17.509 9.4429E+06 9.4387E+06 2.818E+01 1.006E+00 2.939E-01 3.989E-01 1.7202E+05 7.6014E+04 1.872E+00 2.609E-05

17.574 9.4853E+06 9.4812E+06 2.830E+01 1.006E+00 2.949E-01 4.000E-01 1.7279E+05 7.6356E+04 1.872E+00 1.177E-05

17.639 9.3109E+06 9.3069E+06 2.778E+01 1.006E+00 2.959E-01 3.996E-01 1.6961E+05 7.4952E+04 1.848E+00 4.878E-05

17.704 9.1381E+06 9.1341E+06 2.727E+01 1.005E+00 2.969E-01 3.993E-01 1.6646E+05 7.3560E+04 1.824E+00 1.996E-05

17.769 8.9652E+06 8.9613E+06 2.675E+01 1.005E+00 2.979E-01 3.989E-01 1.6331E+05 7.2169E+04 1.800E+00 3.313E-05

17.834 9.0046E+06 9.0008E+06 2.687E+01 1.005E+00 2.989E-01 3.999E-01 1.6403E+05 7.2487E+04 1.800E+00 1.193E-05

17.899 8.8313E+06 8.8275E+06 2.635E+01 1.005E+00 2.999E-01 3.996E-01 1.6088E+05 7.1091E+04 1.776E+00 5.286E-05

17.964 8.6596E+06 8.6558E+06 2.584E+01 1.005E+00 3.009E-01 3.992E-01 1.5775E+05 6.9709E+04 1.752E+00 2.190E-05

18.029 8.4878E+06 8.4842E+06 2.533E+01 1.004E+00 3.019E-01 3.988E-01 1.5462E+05 6.8326E+04 1.728E+00 4.112E-05

18.094 8.5244E+06 8.5208E+06 2.544E+01 1.004E+00 3.029E-01 3.998E-01 1.5529E+05 6.8621E+04 1.728E+00 1.211E-05

18.159 8.3523E+06 8.3488E+06 2.492E+01 1.004E+00 3.038E-01 3.994E-01 1.5215E+05 6.7236E+04 1.704E+00 5.747E-05

18.224 8.1820E+06 8.1785E+06 2.442E+01 1.004E+00 3.048E-01 3.990E-01 1.4905E+05 6.5865E+04 1.680E+00 2.410E-05

18.289 8.2165E+06 8.2130E+06 2.452E+01 1.004E+00 3.058E-01 4.000E-01 1.4968E+05 6.6143E+04 1.680E+00 1.208E-05

18.354 8.0452E+06 8.0418E+06 2.401E+01 1.004E+00 3.067E-01 3.996E-01 1.4656E+05 6.4764E+04 1.656E+00 6.077E-05

18.419 7.8757E+06 7.8724E+06 2.350E+01 1.003E+00 3.077E-01 3.992E-01 1.4347E+05 6.3399E+04 1.632E+00 2.573E-05

18.484 7.7062E+06 7.7030E+06 2.300E+01 1.003E+00 3.086E-01 3.988E-01 1.4038E+05 6.2035E+04 1.608E+00 5.709E-05

18.549 7.7382E+06 7.7350E+06 2.309E+01 1.003E+00 3.096E-01 3.997E-01 1.4097E+05 6.2293E+04 1.608E+00 1.246E-05

18.614 7.5687E+06 7.5655E+06 2.259E+01 1.003E+00 3.105E-01 3.993E-01 1.3788E+05 6.0928E+04 1.584E+00 6.654E-05

18.679 7.4010E+06 7.3980E+06 2.209E+01 1.003E+00 3.115E-01 3.988E-01 1.3482E+05 5.9578E+04 1.560E+00 2.846E-05

18.744 7.4309E+06 7.4279E+06 2.217E+01 1.003E+00 3.124E-01 3.998E-01 1.3537E+05 5.9820E+04 1.560E+00 1.236E-05

18.809 7.2624E+06 7.2595E+06 2.167E+01 1.003E+00 3.133E-01 3.993E-01 1.3230E+05 5.8463E+04 1.536E+00 7.061E-05

138

CHAPTER

4.COMPUTER

PROGRAM

18.874 7.0959E+06 7.0930E+06 2.118E+01 1.002E+00 3.142E-01 3.989E-01 1.2927E+05 5.7123E+04 1.512E+00 3.050E-05

18.939 7.1241E+06 7.1212E+06 2.126E+01 1.002E+00 3.151E-01 3.998E-01 1.2978E+05 5.7350E+04 1.512E+00 1.249E-05

19.004 6.9568E+06 6.9540E+06 2.076E+01 1.002E+00 3.161E-01 3.994E-01 1.2673E+05 5.6003E+04 1.488E+00 7.521E-05

19.069 6.7916E+06 6.7889E+06 2.027E+01 1.002E+00 3.170E-01 3.989E-01 1.2372E+05 5.4674E+04 1.464E+00 3.274E-05

19.134 6.8181E+06 6.8154E+06 2.035E+01 1.002E+00 3.179E-01 3.998E-01 1.2421E+05 5.4887E+04 1.464E+00 1.263E-05

19.199 6.6522E+06 6.6495E+06 1.985E+01 1.002E+00 3.188E-01 3.993E-01 1.2118E+05 5.3551E+04 1.440E+00 8.027E-05

19.264 6.4884E+06 6.4859E+06 1.936E+01 1.002E+00 3.197E-01 3.989E-01 1.1820E+05 5.2233E+04 1.416E+00 3.520E-05

19.329 6.5133E+06 6.5107E+06 1.944E+01 1.002E+00 3.206E-01 3.997E-01 1.1865E+05 5.2433E+04 1.416E+00 1.277E-05

19.394 6.3488E+06 6.3463E+06 1.895E+01 1.002E+00 3.214E-01 3.993E-01 1.1566E+05 5.1109E+04 1.392E+00 8.585E-05

19.459 6.1866E+06 6.1842E+06 1.846E+01 1.002E+00 3.223E-01 3.988E-01 1.1270E+05 4.9804E+04 1.368E+00 3.793E-05

19.524 6.2099E+06 6.2075E+06 1.853E+01 1.002E+00 3.232E-01 3.996E-01 1.1313E+05 4.9991E+04 1.368E+00 1.293E-05

19.589 6.0471E+06 6.0447E+06 1.805E+01 1.001E+00 3.241E-01 3.991E-01 1.1016E+05 4.8680E+04 1.344E+00 9.203E-05

19.654 6.0700E+06 6.0677E+06 1.811E+01 1.001E+00 3.249E-01 4.000E-01 1.1058E+05 4.8865E+04 1.344E+00 1.342E-05

19.719 5.9078E+06 5.9055E+06 1.763E+01 1.001E+00 3.258E-01 3.995E-01 1.0762E+05 4.7559E+04 1.320E+00 9.559E-05

19.784 5.7479E+06 5.7457E+06 1.715E+01 1.001E+00 3.267E-01 3.990E-01 1.0471E+05 4.6272E+04 1.296E+00 4.259E-05

19.849 5.7689E+06 5.7667E+06 1.722E+01 1.001E+00 3.275E-01 3.998E-01 1.0509E+05 4.6441E+04 1.296E+00 1.319E-05

19.914 5.6097E+06 5.6076E+06 1.674E+01 1.001E+00 3.284E-01 3.993E-01 1.0219E+05 4.5160E+04 1.272E+00 4.273E-05

19.979 5.4505E+06 5.4484E+06 1.627E+01 1.001E+00 3.292E-01 3.988E-01 9.9295E+04 4.3878E+04 1.248E+00 4.823E-05

20.044 5.4701E+06 5.4680E+06 1.632E+01 1.001E+00 3.300E-01 3.996E-01 9.9651E+04 4.4036E+04 1.248E+00 1.350E-05

20.109 5.3128E+06 5.3108E+06 1.585E+01 1.001E+00 3.309E-01 3.991E-01 9.6787E+04 4.2770E+04 1.224E+00 4.085E-05

20.174 5.3309E+06 5.3289E+06 1.591E+01 1.001E+00 3.317E-01 3.999E-01 9.7117E+04 4.2916E+04 1.224E+00 1.291E-05

20.239 5.1745E+06 5.1725E+06 1.544E+01 1.001E+00 3.325E-01 3.993E-01 9.4267E+04 4.1656E+04 1.200E+00 4.021E-05

20.304 5.0180E+06 5.0161E+06 1.498E+01 1.001E+00 3.333E-01 3.988E-01 9.1417E+04 4.0397E+04 1.176E+00 7.974E-05

20.369 5.0357E+06 5.0338E+06 1.503E+01 1.001E+00 3.342E-01 3.996E-01 9.1738E+04 4.0539E+04 1.176E+00 1.401E-05

20.434 4.8814E+06 4.8796E+06 1.457E+01 1.001E+00 3.350E-01 3.990E-01 8.8928E+04 3.9297E+04 1.152E+00 3.694E-05

20.499 4.8975E+06 4.8956E+06 1.462E+01 1.001E+00 3.358E-01 3.998E-01 8.9221E+04 3.9426E+04 1.152E+00 1.321E-05

4.1.

COMPUTER

PROGRAM

139

20.564 4.7441E+06 4.7424E+06 1.416E+01 1.001E+00 3.366E-01 3.992E-01 8.6427E+04 3.8192E+04 1.128E+00 3.603E-05

20.629 4.5912E+06 4.5895E+06 1.370E+01 1.000E+00 3.374E-01 3.987E-01 8.3642E+04 3.6961E+04 1.104E+00 6.164E-05

20.694 4.6066E+06 4.6049E+06 1.375E+01 1.000E+00 3.382E-01 3.994E-01 8.3923E+04 3.7085E+04 1.104E+00 1.419E-05

20.759 4.4558E+06 4.4541E+06 1.330E+01 1.000E+00 3.389E-01 3.989E-01 8.1174E+04 3.5871E+04 1.080E+00 3.178E-05

20.824 4.4699E+06 4.4683E+06 1.334E+01 1.000E+00 3.397E-01 3.996E-01 8.1432E+04 3.5985E+04 1.080E+00 1.355E-05

20.889 4.3201E+06 4.3185E+06 1.289E+01 1.000E+00 3.405E-01 3.990E-01 7.8702E+04 3.4778E+04 1.056E+00 3.017E-05

20.954 4.3336E+06 4.3320E+06 1.293E+01 1.000E+00 3.413E-01 3.998E-01 7.8949E+04 3.4887E+04 1.056E+00 1.367E-05

21.019 4.1849E+06 4.1833E+06 1.249E+01 1.000E+00 3.420E-01 3.992E-01 7.6239E+04 3.3690E+04 1.032E+00 2.776E-05

21.084 4.1978E+06 4.1963E+06 1.253E+01 1.000E+00 3.428E-01 3.999E-01 7.6475E+04 3.3794E+04 1.032E+00 1.379E-05

21.149 4.0502E+06 4.0487E+06 1.209E+01 1.000E+00 3.436E-01 3.993E-01 7.3786E+04 3.2606E+04 1.008E+00 2.507E-05

21.214 3.9034E+06 3.9020E+06 1.165E+01 1.000E+00 3.443E-01 3.987E-01 7.1111E+04 3.1424E+04 9.840E-01 7.816E-05

21.279 3.9158E+06 3.9144E+06 1.169E+01 1.000E+00 3.450E-01 3.994E-01 7.1337E+04 3.1524E+04 9.840E-01 1.494E-05

21.344 3.7710E+06 3.7697E+06 1.125E+01 1.000E+00 3.458E-01 3.988E-01 6.8700E+04 3.0359E+04 9.600E-01 1.803E-05

21.409 3.7823E+06 3.7809E+06 1.129E+01 1.000E+00 3.465E-01 3.995E-01 6.8905E+04 3.0449E+04 9.600E-01 1.425E-05

21.474 3.6388E+06 3.6375E+06 1.086E+01 1.000E+00 3.473E-01 3.989E-01 6.6292E+04 2.9294E+04 9.360E-01 1.494E-05

21.539 3.6495E+06 3.6482E+06 1.089E+01 1.000E+00 3.480E-01 3.996E-01 6.6487E+04 2.9381E+04 9.360E-01 1.441E-05

21.604 3.5075E+06 3.5062E+06 1.047E+01 1.000E+00 3.487E-01 3.990E-01 6.3899E+04 2.8237E+04 9.120E-01 1.076E-05

21.669 3.5176E+06 3.5164E+06 1.050E+01 1.000E+00 3.494E-01 3.997E-01 6.4084E+04 2.8319E+04 9.120E-01 1.458E-05

21.734 3.3769E+06 3.3757E+06 1.008E+01 1.000E+00 3.501E-01 3.990E-01 6.1521E+04 2.7186E+04 8.880E-01 6.082E-06

21.799 3.3866E+06 3.3854E+06 1.011E+01 1.000E+00 3.508E-01 3.997E-01 6.1698E+04 2.7264E+04 8.880E-01 1.477E-05

21.864 3.2474E+06 3.2462E+06 9.691E+00 1.000E+00 3.515E-01 3.991E-01 5.9161E+04 2.6143E+04 8.640E-01 8.470E-07

21.929 3.2566E+06 3.2554E+06 9.718E+00 1.000E+00 3.522E-01 3.998E-01 5.9328E+04 2.6217E+04 8.640E-01 1.497E-05

21.994 3.1188E+06 3.1177E+06 9.307E+00 1.000E+00 3.529E-01 3.991E-01 5.6819E+04 2.5108E+04 8.400E-01 5.018E-06

22.059 3.1275E+06 3.1264E+06 9.333E+00 1.000E+00 3.536E-01 3.998E-01 5.6978E+04 2.5178E+04 8.400E-01 1.519E-05

22.124 2.9913E+06 2.9903E+06 8.927E+00 1.000E+00 3.543E-01 3.991E-01 5.4497E+04 2.4082E+04 8.160E-01 1.160E-05

22.189 2.9996E+06 2.9985E+06 8.952E+00 1.000E+00 3.550E-01 3.998E-01 5.4647E+04 2.4148E+04 8.160E-01 1.543E-05

140

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4.COMPUTER

PROGRAM

22.254 2.8650E+06 2.8640E+06 8.550E+00 1.000E+00 3.557E-01 3.991E-01 5.2195E+04 2.3065E+04 7.920E-01 1.899E-05

22.319 2.8728E+06 2.8718E+06 8.573E+00 1.000E+00 3.563E-01 3.997E-01 5.2337E+04 2.3128E+04 7.920E-01 1.569E-05

22.384 2.7399E+06 2.7390E+06 8.177E+00 1.000E+00 3.570E-01 3.990E-01 4.9916E+04 2.2058E+04 7.680E-01 2.731E-05

22.449 2.7473E+06 2.7463E+06 8.199E+00 1.000E+00 3.577E-01 3.997E-01 5.0051E+04 2.2117E+04 7.680E-01 1.597E-05

22.514 2.6161E+06 2.6152E+06 7.807E+00 1.000E+00 3.583E-01 3.990E-01 4.7661E+04 2.1061E+04 7.440E-01 3.670E-05

22.579 2.6231E+06 2.6222E+06 7.828E+00 1.000E+00 3.590E-01 3.996E-01 4.7788E+04 2.1117E+04 7.440E-01 1.627E-05

22.644 2.4937E+06 2.4929E+06 7.442E+00 1.000E+00 3.596E-01 3.989E-01 4.5431E+04 2.0076E+04 7.200E-01 4.732E-05

22.709 2.5003E+06 2.4994E+06 7.462E+00 1.000E+00 3.602E-01 3.995E-01 4.5551E+04 2.0129E+04 7.200E-01 1.661E-05

22.774 2.3728E+06 2.3720E+06 7.081E+00 1.000E+00 3.609E-01 3.988E-01 4.3228E+04 1.9102E+04 6.960E-01 5.934E-05

22.839 2.3789E+06 2.3781E+06 7.099E+00 1.000E+00 3.615E-01 3.994E-01 4.3340E+04 1.9152E+04 6.960E-01 1.698E-05

22.904 2.2534E+06 2.2526E+06 6.725E+00 1.000E+00 3.621E-01 3.987E-01 4.1053E+04 1.8141E+04 6.720E-01 7.301E-05

22.969 2.2591E+06 2.2584E+06 6.742E+00 1.000E+00 3.627E-01 3.993E-01 4.1158E+04 1.8188E+04 6.720E-01 1.740E-05

23.034 2.2649E+06 2.2642E+06 6.759E+00 1.000E+00 3.633E-01 3.999E-01 4.1263E+04 1.8234E+04 6.720E-01 3.509E-05

23.099 2.1409E+06 2.1401E+06 6.389E+00 1.000E+00 3.640E-01 3.991E-01 3.9003E+04 1.7235E+04 6.480E-01 8.918E-05

23.164 2.1463E+06 2.1456E+06 6.405E+00 1.000E+00 3.646E-01 3.997E-01 3.9102E+04 1.7279E+04 6.480E-01 1.788E-05

23.229 2.0248E+06 2.0241E+06 6.043E+00 1.000E+00 3.652E-01 3.990E-01 3.6889E+04 1.6301E+04 6.240E-01 6.128E-09

23.294 2.0296E+06 2.0289E+06 6.057E+00 1.000E+00 3.657E-01 3.996E-01 3.6976E+04 1.6339E+04 6.240E-01 1.710E-05

23.359 1.9103E+06 1.9096E+06 5.701E+00 1.000E+00 3.663E-01 3.988E-01 3.4802E+04 1.5379E+04 6.000E-01 8.069E-09

23.424 1.9147E+06 1.9140E+06 5.714E+00 1.000E+00 3.669E-01 3.994E-01 3.4882E+04 1.5414E+04 6.000E-01 1.739E-05

23.489 1.9190E+06 1.9184E+06 5.727E+00 1.000E+00 3.675E-01 4.000E-01 3.4962E+04 1.5450E+04 6.000E-01 3.712E-05

23.554 1.8018E+06 1.8012E+06 5.377E+00 1.000E+00 3.681E-01 3.992E-01 3.2826E+04 1.4506E+04 5.760E-01 1.107E-08

23.619 1.8058E+06 1.8052E+06 5.389E+00 1.000E+00 3.686E-01 3.998E-01 3.2899E+04 1.4538E+04 5.760E-01 1.768E-05

23.684 1.6908E+06 1.6903E+06 5.046E+00 1.000E+00 3.692E-01 3.990E-01 3.0805E+04 1.3612E+04 5.520E-01 1.507E-08

23.749 1.6945E+06 1.6940E+06 5.057E+00 1.000E+00 3.698E-01 3.995E-01 3.0872E+04 1.3642E+04 5.520E-01 1.802E-05

23.814 1.5820E+06 1.5814E+06 4.721E+00 1.000E+00 3.703E-01 3.987E-01 2.8821E+04 1.2736E+04 5.280E-01 2.024E-08

23.879 1.5853E+06 1.5848E+06 4.731E+00 1.000E+00 3.709E-01 3.993E-01 2.8883E+04 1.2763E+04 5.280E-01 1.838E-05

4.1.

COMPUTER

PROGRAM

141

23.944 1.5887E+06 1.5882E+06 4.741E+00 1.000E+00 3.714E-01 3.998E-01 2.8944E+04 1.2790E+04 5.280E-01 3.877E-05

24.009 1.4785E+06 1.4780E+06 4.412E+00 1.000E+00 3.719E-01 3.990E-01 2.6935E+04 1.1903E+04 5.040E-01 2.698E-08

24.074 1.4815E+06 1.4810E+06 4.421E+00 1.000E+00 3.725E-01 3.995E-01 2.6991E+04 1.1927E+04 5.040E-01 1.875E-05

24.139 1.3738E+06 1.3733E+06 4.100E+00 1.000E+00 3.730E-01 3.987E-01 2.5028E+04 1.1060E+04 4.800E-01 3.611E-08

24.204 1.3766E+06 1.3761E+06 4.108E+00 1.000E+00 3.735E-01 3.992E-01 2.5079E+04 1.1082E+04 4.800E-01 1.917E-05

24.269 1.3793E+06 1.3789E+06 4.117E+00 1.000E+00 3.740E-01 3.998E-01 2.5130E+04 1.1105E+04 4.800E-01 4.013E-05

24.334 1.2743E+06 1.2739E+06 3.803E+00 1.000E+00 3.745E-01 3.989E-01 2.3216E+04 1.0259E+04 4.560E-01 8.841E-05

24.399 1.2766E+06 1.2762E+06 3.810E+00 1.000E+00 3.750E-01 3.994E-01 2.3258E+04 1.0278E+04 4.560E-01 1.821E-05

24.464 1.2792E+06 1.2788E+06 3.818E+00 1.000E+00 3.755E-01 3.999E-01 2.3305E+04 1.0299E+04 4.560E-01 4.457E-08

24.529 1.1765E+06 1.1762E+06 3.511E+00 1.000E+00 3.760E-01 3.991E-01 2.1435E+04 9.4721E+03 4.320E-01 5.102E-05

24.594 1.1787E+06 1.1783E+06 3.518E+00 1.000E+00 3.765E-01 3.996E-01 2.1474E+04 9.4894E+03 4.320E-01 1.926E-05

24.659 1.0789E+06 1.0785E+06 3.220E+00 1.000E+00 3.770E-01 3.988E-01 1.9656E+04 8.6857E+03 4.080E-01 7.033E-06

24.724 1.0808E+06 1.0805E+06 3.226E+00 1.000E+00 3.775E-01 3.992E-01 1.9692E+04 8.7017E+03 4.080E-01 2.052E-05

24.789 1.0828E+06 1.0825E+06 3.232E+00 1.000E+00 3.780E-01 3.997E-01 1.9728E+04 8.7176E+03 4.080E-01 4.955E-05

24.854 9.8577E+05 9.8545E+05 2.942E+00 1.000E+00 3.784E-01 3.988E-01 1.7959E+04 7.9362E+03 3.840E-01 5.079E-05

24.919 9.8760E+05 9.8728E+05 2.947E+00 1.000E+00 3.789E-01 3.993E-01 1.7993E+04 7.9509E+03 3.840E-01 2.217E-05

24.984 9.8943E+05 9.8911E+05 2.953E+00 1.000E+00 3.794E-01 3.998E-01 1.8026E+04 7.9657E+03 3.840E-01 4.828E-06

25.049 8.9529E+05 8.9501E+05 2.672E+00 1.000E+00 3.798E-01 3.989E-01 1.6311E+04 7.2078E+03 3.600E-01 4.133E-08

25.114 8.9683E+05 8.9654E+05 2.676E+00 1.000E+00 3.803E-01 3.994E-01 1.6339E+04 7.2202E+03 3.600E-01 2.190E-05

25.179 8.9836E+05 8.9808E+05 2.681E+00 1.000E+00 3.807E-01 3.998E-01 1.6367E+04 7.2326E+03 3.600E-01 4.506E-05

25.244 8.0736E+05 8.0711E+05 2.409E+00 1.000E+00 3.811E-01 3.989E-01 1.4709E+04 6.4999E+03 3.360E-01 8.177E-08

25.309 8.0869E+05 8.0844E+05 2.413E+00 1.000E+00 3.816E-01 3.994E-01 1.4733E+04 6.5107E+03 3.360E-01 2.262E-05

25.374 8.1003E+05 8.0977E+05 2.418E+00 1.000E+00 3.820E-01 3.998E-01 1.4758E+04 6.5214E+03 3.360E-01 4.635E-05

25.439 7.2224E+05 7.2201E+05 2.155E+00 1.000E+00 3.824E-01 3.989E-01 1.3158E+04 5.8146E+03 3.120E-01 1.476E-07

25.504 7.2338E+05 7.2315E+05 2.159E+00 1.000E+00 3.828E-01 3.993E-01 1.3179E+04 5.8238E+03 3.120E-01 2.345E-05

25.569 7.2452E+05 7.2430E+05 2.162E+00 1.000E+00 3.833E-01 3.997E-01 1.3200E+04 5.8330E+03 3.120E-01 4.784E-05

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PROGRAM

25.634 6.4010E+05 6.3990E+05 1.910E+00 1.000E+00 3.837E-01 3.988E-01 1.1662E+04 5.1533E+03 2.880E-01 2.544E-07

Nozzle entry total-pressure is less than 5.000000E-02 of the burnout nozzle-end total-pressure. 1.302640E+07 (PA)

Tailoff calculation is terminated.

Final sliver mass = Mass of propellant left-out at the termination of tailoff (kg) = 2.794027E+00

Sliver fraction2 = (Final sliver mass)/(initial propellant mass) = 5.705685E-03

CHECK ON MASS BALANCE

Initial propellant mass (kg) = 4.896918E+02

Cumulative port flows (kg) = 4.853493E+02

Cumulative nozzle flows (kg) = 4.853444E+02

Final sliver mass (kg) = 2.794027E+00

Mass balance error2=[(cumulative port-flow)-(cumulative nozzle-flow)]/cumulative port-flow = 9.940034E-06

Mass balance error3 =

[(cumulative nozzle-flow)+(final sliver)]-(initial propellant-mass) /(initial propellant-mass) = -3.171972E-03

4.1.

COMPUTER

PROGRAM

143

**** MOTOR PERFORMANCE **** MOTOR PERFORMANCE ****

TOTAL IMPULSE IN VACUUM (N-s) = 1.309311E+06

VACUUM SPECIFIC-IMPULSE CALCULATED

[Total impulse in vacuum divided by propellant mass] (N-s/kg) = 2.673746E+03

CHECK: Vacuum specific impulse theoretical [= c ∗ ×CFvac](N − s/kg) (N-s/kg) = 2.697695E+03

144

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PROGRAM

4.2 Source Code

Program steadyfull

IMPLICIT NONE

! This program calculates the theoretical rocket performance

! of solid rocket motors under the incremental analysis.

! GRAIN GEOMETRY

! Although various grain geometries can be considered, this

! program restricts the application to any

! tapered-circular-port grain with its ends inhibited. Grain

! geometrical properties are calculated by the subprogram

! GEOMETRY. For other grain geometries, suitable

! subprogram can be written to replace the existing subprogram.

! A positive taper (tapangle > 0) is to be provided from head

! end to nozzle end.

! EQUILIBRIUM OPERATION & TAIL-OFF TRANSIENT

! The program considers mass, momentum, energy, and ideal gas

! equation of state. Only steady flow conditions are considered.

! Equilibrium operation and tail-off transient are calculated

! assuming the steady flow in the incremental control volumes.

! START TRANSIENT

! But the start transient is calculated assuming the

4.2.

SOURCE

CODE

145

! "equilibrium pressure analysis", that is assuming that there

! is one uniform pressure for the entire chamber-cavity.

! Furthermore it is assumed that (i) for the duration of start

! transient the burned distance is negligible, that is, the

! burning area is constant and (ii) entire grain surface is

! instantaneously ignited with negligible igniter mass.

! INPUTS

! Inputs are under unit 40 & 41 of names xxxxxxinput1.dat and

! xxxxxxinput2.dat under namelist inputs. xxxxxxinput1.dat file

! contains propellant data under the namelist-name "prplnt" and

! xxxxxxinput2.dat file contains motor data under the

! namelist-name "motor". You have to create these two files to

! run this program. For xxxxxx you have to choose a name of

! 6 characters(alphanumeric) to identify your problem. Examples

! are case01input1 & case01input2; prob01input1.dat &

! prob01input2

! On running the program, you will be prompted to enter the

! names of these two input dat-files that you have created;

! for example, if the input files that you created are of names

! case01input1.dat & case01nput2, you enter only case01input1

! and case01input2 one after the other.Typical input files are

146

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PROGRAM

! the following:

! EXAMPLE 1

! (say under the file names case01input1.dat for propellant

! data)

! &prplnt a=3.51e-05,n=0.36,rhop=1765.,mbar=26.1,cs=1400.,

! cp=2880.,t0=3390.,ts=1000.,ti=300.,beta=60.,mu=1.0e-04,

! pr=0.49,eros=.T./

!(say under the file name case01input2.dat for motor data)

! &motor od=0.21,grainl=1.95,dt=0.07,aebyat=6.,tapangle=.5,

! tailoffend=0.05,deltatime=0.050,idhinitial=0.05 /

! OUTPUTS

! Outputs are to be set in Arial Regular Font Size 9

! in order to keep the column headings and their respective

! output-numbers aligned.

! Outputs are under units 50 & 51 of names xxxxxxout1.out &

! xxxxxxout2.out. xxxxxxout1.out contains detailed output while

! xxxxxxout2.out contains the summary output. xxxxxxout1.out

! lists, for every instant of operation, (i) iteration and

! convergence details, (ii)port characteristics, and

! (iii)port-dynamics. In the unlikely event of the program

! execution error, the partial ouput of xxxxxxout1.out

4.2.

SOURCE

CODE

147

! can be studied for debugging.For easy tracking, the names

! of the output files will have part ’xxxxxx’ in them. For

! example,if your input files are case01input1.dat and

! case01input2.dat the output files created will be

! case01out1.out and case01out2.out.

! *********

! CAUTION

! *********

! Number of incremental stations under the variable name

! "increments" under namelist "motor" has to be less than or

! equal to 200. Number of incremental times under the variable

! name "iiburnout" under namelist "motor" has to be less than

! or equal to 500.

Character*30 :: input1,input2,out1,out2,dat,out

Common/geo/s,ap,d,r,theta,l,idh,s0,p0h,time,od,grainl,dt,

1 tapangle,abi,delx,aebyat,deltatime,error,idhinitial,mp,

2 vci,clamung,vcitotal,vcempty,vpfraction,p0bype,ii,increments

Common/prop/erosn,rgas,gama,capgama,cstar,a,rhop,n,

1 t0,ts,ti,alpha,beta,mu,cp,cs,pr,mbar,eros

Integer :: i,ii,iii,j,jj,jjj,k,kk,kkk

! i is incremental station

148

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PROGRAM

! ii is time step counter

! iii is for incremental-station counter

! iiburnout is the number of time steps

Integer :: ib=1,ib1=1 ! An integer used in the

! subroutine bisection

Integer :: iiburnout =500 ! Maximum number of time steps

Integer :: increments=200,incrementsi

! total number of incremental stations

Integer :: incrementb ! Burnout increment

Logical :: bisect =.false ! When the program sets it true the

! method of bisection is adopted for

! head end Mach number convergence

! Logical :: bisect1 =.false ! When the program sets it true the

! method of bisection is adopted for

! mass flow rate convergence.

Logical :: burnout ! During calculation if burnout occurs,

! the logical variable is set equal to true

Logical :: eros ! When set =.false. in the subroutine

! propellant erosive burning will

! not be considered

Logical :: solution

4.2.

SOURCE

CODE

149

Real*8,parameter :: ru=8314.51

! Universal gas constant (J/kg-mole-K)

Real*8,parameter :: pi=.314159265D+01

Real*8 :: a ! Pre-exponent factor in the

! burning rate equation, r0=ap^n

Real*8 :: abi ! Initial burning area (m^2)

Real*8 :: aebyat ! Nozzle area ratio

Real*8 :: alpha ! The Greek letter Alpha in the

! Lenoir-Robillard erosive

! burning rate equation

Real*8, dimension (500,200) :: ap

! Port areas (m**2)

Real*8 :: apibyat ! Initial-port to throat area ratio

REal*8 :: at !Throat area

Real*8 :: aps1

Real*8 :: aps2

Real*8 :: beta ! The Greek letter Beta in the

! Lenoir-Robillard burning rate equation

Real*8 :: capgama ! A function of ratio of specific heats

Real*8 :: cf0 ! Characteristic thrust coefficient

Real*8 :: cfvac ! Vacuum thrust coefficient

150

CHAPTER

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PROGRAM

Real*8 :: clamung ! Ratio of intial burning-area

! to throat area

Real*8 :: cp ! Specific heat at constant pressure

! for combustion products (J/kg-K)

Real*8 :: cs ! Propellant specific heat (J/kg-K)

Real*8 :: cstar ! Experimental cstar (m/s)

Real*8, dimension (500,200) :: d ! Port diameters (m)

Real*8 :: dt ! Throat diameter (m)

Real*8 :: delM ! Error final on convergence

Real*8 :: delM1 ! Error 1 for the incremental station 1



Real*8 :: delmdot ! Error at convergence:

! Modulus of delmdot1 or delmdot2

Real*8 :: delmdot1 ! Error fraction:

! delmdot1=(mdotp1-mdotn1)/mdotp1


! delmdot2=(mdotp2-mdotn2)/mdotp2


!delmdot3=(mdotp3-mdotn3)/mdotp3

Real*8 :: delmdt1 ! Error in mass convergence (kg/s)

4.2.

SOURCE

CODE

151

Real*8 :: delmdt2 ! Error in mass convergence (kg/s)

Real*8,dimension (100) :: delmdt

! Errors in mass convergence(kg/s)

Real*8 :: deltatime ! Incremental time (s)

Real*8 :: delx ! Incremental distance (m)

Real*8 :: epsilon ! erosive burning ratio

Real*8, dimension (500,200) :: erosn

! Erosive burning ratios at

! the incremental stations

Real*8 :: error=1.e-04 ! Allowable error in massflow rate

! convergence, to be set by namelist input

Real*8 :: error1=5.D-07 ! Allowable error in all other

! convergences

Real*8 :: error2 ! mass balance error =

! (sigmadotprt-sigmadotnzl)/sigmadotprt

Real*8 :: error3 ! mass balance error =

! [(sigmadotnzl+mpsliver2)-mp]/mp

Real*8 :: g ! Mass flux (kg/s-m^2)

REal*8 :: gama ! Ratio of specific heats

Real*8 :: grainl ! Grain length (m)

Real*8, dimension (500) :: idh ! Head end diameters;

152

CHAPTER

4.COMPUTER

PROGRAM

! subscript represents time increments

Real*8 :: idhinitial ! Initial head end diameter (m)

Real*8 :: impulsetot ! Total impulse (N-s)

Real*8 :: ispvac ! Vacuum Specific impulse (N-s/kg)

Real*8 :: ispvactheo ! Theoretical specific impulse c*xCfvac

Real*8, dimension (200) :: l ! Segment station locations;

! subscript represents

! incremental stations (m)

Real*8, dimension (500,200) :: M

! Mach numbers at the incremental stations.

! First and second subscripts respectively

! represent time increments

! and incremental stations

Real*8 :: mbar ! Molar mass (kg/kg-mole)

Real*8, dimension (500,200) :: mdot

Real*8 :: Mbisec ! Bisected trial-value in

! the head-end Mach number

! Mass flow rates (time, location)(kg/s)

Real*8 :: mdotn ! Mass flow rate through the nozzle (kg/s)

Real*8 :: mdotn1 ! Trial nozzle flow-rate(kg/s)

Real*8 :: mdotn2 ! Trial nozzle flow-rate (kg/s)

4.2.

SOURCE

CODE

153

Real*8,dimension (500) :: mdotnzl

! Solved nozzle flow rate(location) (kg/s)

Real*8 :: mdotp1 ! Trial port end mass flow rate(kg/s)

Real*8 :: mdotp2 ! Trial port end mass flow (kg/s)

Real*8 :: mp ! Initial propellant mass (kg)

Real*8 :: mpsliver1 ! Sliver mass at burnout (kg)

Real*8 :: mpsliver2 ! Sliver mass left out at 2% of

! nozzle-end burnout total-pressure

Real*8 :: Mt1 ! Trial Mach number for the

! incremental station 1





Real*8 :: mu ! Viscosity of combustion products(kg/m-s)

Real*8 :: n ! Burning rate index

Real*8 :: od ! Grain outer diameter (m)

Real*8, dimension (500,200) :: p0

! Total pressures at the incremental

! stations(time,location)(Pa)

Real*8 :: p0bype ! Nozzle pressure ratio

154

CHAPTER

4.COMPUTER

PROGRAM

Real*8 :: p0burnout ! Burnout nozzle entry total pressure (Pa)

Real*8, dimension (500) :: p0h

! Solved head-end pressure (time) (Pa)

Real*8, dimension (100) :: p0hh

! Stored head end pressures during

! convergence (Pa)

Real*8 :: p0hbisec ! Trial bisected-value of head-end

! total pressure (Pa)

REal*8 :: p0ht ! Trial head end pressure (Pa)

Real*8 :: p0hteq ! Trial equilibrium pressure (Pa)

Real*8 :: p0ht1 ! Trial head end pressure (Pa)

Real*8 :: p0ht2 ! Trial head end pressure (Pa)

REal*8 :: p0ht3 ! Trial head end pressure (Pa)

Real*8 :: p0s1

Real*8 :: p0s2

Real*8, dimension (500,200) :: p

! Static pressures (time, location)(Pa)

Real*8, dimension (500) :: pe

! Nozzle exit-plane pressures (time)(Pa)

Real*8 :: Pr ! Prandtl number

Real*8 :: ps1

4.2.

SOURCE

CODE

155

Real*8 :: ps2

Real*8, dimension (500,200) :: r

! Total burning rate including

! erosive component (m/s)

Real*8 :: rgas ! Specific gas constant (J/kg-K)

Real*8 :: rhop ! Propellant density (kg/m^3)

Real*8,dimension (500) :: s0

! Head end burning perimeter (time)(m)

Real*8,dimension (500,200) :: s

! Burning perimeters (time,location)(m)

Real*8 :: sigmamdotprt ! Time integral of port exit flows (kg)

Real*8 :: sigmamdotnzl ! Time integral of nozzle flows (kg)

Real*8 :: sliverfraction1 ! Sliver fraction at burnout

Real*8 :: sliverfraction2 ! Sliver fraction at the end of tail-off

Real*8 :: ss1

Real*8 :: ss2

Real*8, dimension (500,200) :: t

! Static temperatures (time,location),(K)

Real*8 :: t0 ! Adiabatic flame temperature (K)

Real*8, dimension (500,200) :: t0c

! Checked total temperatures at the

156

CHAPTER

4.COMPUTER

PROGRAM

! incremental stations (time,location)(K)

Real*8 :: tailoffend ! Fraction of burnout pressure to stop

! tail-off calculation

Real*8 :: tapangle ! Grain port taper angle, initial (deg.)

Real*8, dimension (500,200) :: theta

! Local taper angle(time,location)(deg)

Real*8,dimension (500) :: thrust

! Thrust (time) (N)

Real*8 :: ti ! Propellant storage temperature (K)

! generally kept at atmospheric temperature

Real*8, dimension (500) :: time

! Time of motor operation (s)

Real*8 :: ts ! Propellant burning-surface

! temperature (K)

Real*8 :: ts1

Real*8 :: ts2

Real*8 :: t0s2 ! Checked total temperature (K)

Real*8, dimension (500,200) :: u

! Velocities (time,location),(m/s)

Real*8 :: us1

4.2.

SOURCE

CODE

157

Real*8 :: us2

Real*8 :: vcempty ! Chamber empty volume including

! nozzle convergence volume(m^3)

Real*8 :: vci ! Initial free volume (m^3)

Real*8 :: vcitotal ! Chamber free-volume, initial (m^3)

Real*8 :: vpfraction ! Propellant volumetric loading fraction

! Real :: x ! equated to p0ht1 while calling

! subroutine FALCI

! Real :: x1,x2 ! non-dimensionalized head end pressures

! p0ht1 and p0ht2 while calling

! subroutine FALCI

Namelist/prplnt/a,n,rhop,mbar,cs,cp,t0,ts,ti,beta,mu,pr,eros

Namelist/motor/od,grainl,dt,aebyat,tapangle,deltatime,

1 increments,idhinitial,error,tailoffend

!

! Opening of input and output data files

Write(*,*)’Enter names of your two input dat-files’,

E ’ one after the other. Two new out-files will be’,

B ’ created based on your input file names. Suppose’,

C ’ your two input dat-file names are case02input1.dat ’,

158

CHAPTER

4.COMPUTER

PROGRAM

D ’and case02input2.dat, enter one after the other ’,

E ’case02input1 and case02input2. The two out-file’,

D ’ names will be case02out1.out and case02out2.out’

Read(*,*)input1,input2

dat=’.dat’

out=’.out’

out1=’out1’

out1=trim(input1(:6))//out1

out1=trim(out1(:10))

out2=’out2’

out2=trim(input1(:6))//out2

out2=trim(out2(:10))

out1=trim(out1)//out

out2=trim(out2)//out

input1=trim(input1)//dat

input2=trim(input2)//dat

open(unit=40,file=input1,status=’old’)

open(unit=41,file=input2,status=’old’)

! Unit 50 out1 is for detailed output; can be used for debugging

open(unit=50,file=out1,status=’replace’)

! Unit 51 is for the summary-output

4.2.

SOURCE

CODE

159

open(unit=51,file=out2,status=’replace’)

!

! Reading and writing inputs

!

Read(40,prplnt)

Write(51,nml=prplnt)

Read(41,motor)

Write(51,nml=motor)

Write(51,190)

190 FORMAT(///,’SOLID PROPELLANT ROCKET MOTOR’,/,

A ’ INTERNAL BALLISTICS PREDICTION’,/,

B ’(Start transients are calculated through Lumped Chamber ’,

C ’Pressure Model or p(t) Model)’,/,

D ’ (Equilibrium operations are calculated throgh quasi ’,

E ’steady flow model or p(x) Model)’,/)

Write(51,191)

191 FORMAT(’NOTE 1. Rocket performance parameters are calculated’,

A ’ for operations in vacuum.’,/’NOTE 2. For non-vacuum’,

B ’ operation, calculated vacuum-thrust has to be corrected.’,

C /’NOTE 3. Under non-vacuum operations nozzle exit plane ’,

160

CHAPTER

4.COMPUTER

PROGRAM

D ’pressure below certain level will lead to nozzle separation.’,/)

incrementsi=increments

impulsetot=0.

sigmamdotprt=0.

sigmamdotnzl = 0.

p0burnout=0.

burnout=.false.

Solution =.false.

delx=grainl/incrementsi

At=pi/4*dt**2

!

! Fixing incremental stations

!

l(1)=delx

Do iii=2,incrementsi

l(iii)=l(iii-1)+delx

End do

!

! Calculate propellant properties

!

Call propellant

4.2.

SOURCE

CODE

161

!

! Nozzle area ratio aebyat is given.

!Calculate pressure ratio p0bype

!

Call prsrratio (p0bype,aebyat,gama)

CF0=capgama*sqrt(2*gama/(gama-1)*(1-(1/p0bype)**((gama-1)/gama)))

CFvac=CF0+aebyat*(1/p0bype)

Ispvactheo=cstar*cfvac

Write(50,930)Ispvactheo

Write(51,930)Ispvactheo

930 Format(’ Vacuum specific impulse theoretical [=c*xCFvac] ’,

A ’(N-s/kg) =’,es13.6,/)

!

! Increment of time by ii

!

Do ii=1,iiburnout

!

!Subroutine geometry fixes the port envelope

!for all times and prints the same

!

Call geometry

162

CHAPTER

4.COMPUTER

PROGRAM

If (ii==1) then

p0hteq=(rhop*a*abi/at*cstar)**(1/(1-n))

p0ht=p0hteq

! Write(51,*)’ p0ht=p0hteq=’,p0ht

p0ht1=0.9*p0ht

p0ht2=1.1*p0ht

Mt1=5.D-03

Mt2=1.1*Mt1

apibyat=(d(1,increments)/dt)**2.

!

! INPUT ERROR CHECK

!

If(d(1,increments)>=od)then

Write(50, 137)d(1,increments),od

Write(51, 137)d(1,increments),od

137 Format(//,’ **** INPUT ERROR **** INPUT ERROR ****’,/,

A ’ INITIAL PORT EXIT DIAMETER =’,D13.6,’ (m) IS GREATER’,

B ’ THAN GRAIN OD =’,D13.6,’ (m)’,/,

C ’ PROGRAM EXECUTION ABORTED’)

GOTO 1011

Else if(d(1,increments)<=dt)then

4.2.

SOURCE

CODE

163

Write(50,138)d(1,increments),dt

Write(51,138)d(1,increments),dt

138 Format(//,’ **** INPUT ERROR **** INPUT ERROR ****’,/,

A ’ INITIAL PORT EXIT DIAMETER =’,D13.6,’ (m) IS LESS’,

B ’ THAN NOZZLE THROAT DIAMETER =’,D13.6,’ (m)’,/,

C ’ PROGRAM EXECUTION ABORTED’)

GOTO 1011

END IF

!

! Initial configuration of the motor is printed

!

Write(51,114)od,idhinitial,grainl,tapangle,mp,dt,

1 aebyat,p0bype,delx,deltatime,d(1,increments),

2 apibyat,abi,vci,vcitotal,vcempty,vpfraction

Write(50,114)od,idhinitial,grainl,tapangle,mp,dt,

1 aebyat,p0bype,delx,deltatime,d(1,increments),

2 apibyat,abi,vci,vcitotal,vcempty, vpfraction

114 Format(’ MOTOR CONFIGUARATION & INCREMENTS’,/,

1 ’ Grain outer diameter (m) =’,Es13.6,/,

B ’ Head end port diameter (m) =’,es13.6,/,

2 ’ Grain length (m) = ’,

164

CHAPTER

4.COMPUTER

PROGRAM

3 Es13.6,/,

5 ’ Grain taper angle (degree) =’,Es13.6,/,

A ’ Initial propellant mass(kg) =’,es13.6,/,

3 ’ Nozzle throat diameter(m) =’,Es13.6,/,

4 ’ Nozzle area ratio =’,

5 Es13.6,/,

8 ’ Nozzle pressure ratio =’,es13.6,/,

6 ’ Incremental distance (m) =’,Es13.6,/,

7 ’ Incremental time (s) = ’,F7.3,/,

8 ’ Port end diameter (m) =’,Es13.6,/,

B ’ Initial-port to throat area ratio =’,Es13.6,/,

C ’ Initial burning area (m^2) =’,es13.6,/,

9 ’ Port volume, initial (m^3) =’,ES13.6,/,

D ’ Chamber free volume, initial (m^3) =’,ES13.6,/,

A ’ Chamber empty-volume (m^3) =’,ES13.6,/,

B ’ Propellant volumetric loadinng fraction=’,Es13.6,

C //)

Write(50,102)(l(i), i=1,increments)

102 Format(/,’ ’,

1 ’ Incremental ’,

2 ’Station Location (m)’,/,20(10(1x,es13.6),/),/)

4.2.

SOURCE

CODE

165

else if (ii==2)then

p0ht1=p0h(1)

p0ht2=1.1*p0ht1

Mt1=M((ii-1),1)

Mt2=Mt1*1.05

else if (ii>2) then

p0ht1=p0h(ii-1)

Mt1=M((ii-1),1)

p0ht2=p0ht1+(p0h(ii-1)-p0h(ii-2))

If (p0ht2<0) then

p0ht2=0.95*p0ht1

End if

if(p0h(ii-1)>p0h(ii-2)) then

Mt2=Mt1*1.05

else

Mt2=0.95*Mt1

end if

End if

!!2001 "If (ii=1) then" ends

!!2000 " If (d(ii,incrementsi)<od) then" starts

166

CHAPTER

4.COMPUTER

PROGRAM

If (d(ii,incrementsi)<od) then ! Burnout check

goto 151

Else if (d(ii,1)>=od)then

write(50,136)

Write(51,136)

136 Format(/,’**** UNEXPECTED COMPLETE-BURNOUT ****’,/,

A ’ CHOOSE FINER TIMESTEP "deltatime" under NAMELIST "motor"’,/,

B ’ PROGRAM-EXECUTION ABORTED ** PROGRAM-EXECUTION ABORTED ’)

Go to 1011

!

! Burnout calculations are done

!

Else

Do jjj=1,incrementsi

If(d(ii,jjj)>od) then

incrementb=jjj-1

GOTO 150

End if

End do

150 Continue

Do jjj=(incrementb+1),increments

4.2.

SOURCE

CODE

167

theta(ii,jjj)=0.

d(ii,jjj)=od

s(ii,jjj)=0

ap(ii,jjj)=pi/4*od**2

End do

! Fixing BURNOUT condition

If (.not.burnout) then

p0burnout=p0((ii-1),incrementsi)

mpsliver1=rhop*(pi/4.*od**2*l(incrementb)-pi/3/

1 tan(tapangle*pi/180)*((idh(ii)/2+l(incrementb)*

2 tan(tapangle*pi/180))**3-(idh(ii)/2)**3))

sliverfraction1=mpsliver1/mp

Write(50,113)Time(ii),d(ii,incrementb),l(incrementb),

1 mpsliver1,sliverfraction1,p0burnout

Write(51,113)Time(ii),d(ii,incrementb),l(incrementb),

1 mpsliver1,sliverfraction1,p0burnout

113 Format(/,’ BURNOUT HAS OCCURED AT’,es13.6,

1 ’seconds’,/,’ PORT DIAMETER =’,es13.6,

2 ’ AT PORT LOCATION ’,Es13.6,’(m)’,/,’ Sliver mass ’,

3 ’at burnout (kg) ’,

4 ’ =’,

168

CHAPTER

4.COMPUTER

PROGRAM

5 Es13.6,/,’ Sliver fraction at burnout (sliver mass/’,

6 ’initial propellant mass) =’,Es13.6,/,

7 ’ Burnout nozzle-end total-pressure (Pa) ’,

8 ’ =’,Es13.6,/)

Write(51,125) ! Write column heading

125 FORMAT(//,’ TAIL-OFF PERFORMANCE’,

A /,’ p0h=head-end total pressure; p0n=nozzle end total ’,

1 ’pressure;’,/’ dhprt=head-end port dia; dnprt=nozzle-end port ’,

2 ’dia’,/’ Lnprt=exit-port location;delmdot=convergence error’//

3 ’ Time (s) p0h(Pa) p0n(Pa) mdot(kg/s)’,

A ’ epsilon dhprt(m) dnprt(m) pe(Pa) ’,

B ’Fvac(N) Lnprt (m) delmdot’)

burnout=.true.

End if

Incrementsi=incrementb

End if

!2000" If (d(ii,incrementsi)<od) then" ends

! Printing grain configuration for all times

151 Write (50,101)time (ii)

!write(50,*)’ increments=’,increments,’ incrementsi=’,incrementsi

!write(50,*)’ incrementb=’,incrementb

4.2.

SOURCE

CODE

169

101 Format(/,’ ’,

A ’ TIME OF OPERATION (s)=’,

B es13.6, /)

Write(50,100)(d(ii,i),i=1,increments)

Write(50,103)(theta(ii,i),i=1,increments)

Write(50,104)(s(ii,i),i=1,increments)

Write(50,105)(ap(ii,i),i=1,increments)

100 Format(’ ’,

A ’ GRAIN’,

B ’ GEOMETRY’,//,’ Port Diameters at ’,

C ’Incremental Station Locations(m)’,/,

4 20(10(1x,es14.7),/),//)

103 Format(’ ’,

A ’ Taper Angle at ’,

B ’Incremental Station Locations (deg)’,/,

C 20(10(1x,es14.7),/),//)

104 Format(’ ’,

A ’ Burning Perimeter at ’,

B ’Incremental Station Locations (m)’,/,

C 20(10(1x,es14.7),/),//)

105 Format(’ ’,

170

CHAPTER

4.COMPUTER

PROGRAM

A ’ Port Areas at Incremental’,

B ’ Station Locations(m**2)’,/,

C 20(10(1x,es14.7),/),//)

kk=1 ! kk is a counter to check the number

! of iterations through regualar falci

j=1

! with j=1 the first estimate of port-exit flow-rate

! and nozzle flow-rate are calculated

! with j=2 the second estimate of port-exit flow-rate

! and nozzle flow-rate are calculated

! With the j values corresponding to 1 and 2 convergence

! is achieved through regular falci by calling subprogram FALCI.

997 Continue

If (j==1) then

p0ht=p0ht1

else

p0ht=p0ht2

end if

1002 Continue

!

4.2.

SOURCE

CODE

171

! Module to find M(1) Starts ! Module to find M(1) starts

!

k=1

! Gas dynamic properties at the first incremental station

! are calculated. The head end segment is assumed to burn

! with the head-end total-pressure, p0ht

mdot(ii,1)=rhop*a*p0ht**n*(s0(ii)+s(ii,1))/2*delx

999 Continue

delM1=(mdot(ii,1)-SQRT(gama)*(1+(gama-1)/2*Mt1**2)**(-(gama+1)/2/

a (gama-1))*Mt1*p0ht*pi/4*idh(ii)**2/sqrt(rgas*t0))/mdot(ii,1)

delM2=(mdot(ii,1)-SQRT(gama)*(1+(gama-1)/2*Mt2**2)**(-(gama+1)/2/

a (gama-1))*Mt2*p0ht*pi/4*idh(ii)**2/sqrt(rgas*t0))/mdot(ii,1)

! Write(50,*)’ p0ht=’,p0ht,’ Mt1=’,Mt1,’ delM1=’,delm1

! Write(50,*)’ p0ht=’,p0ht,’ Mt2=’,Mt2,’ delM2=’,delM2

If(ABS(delM1)<=error1) then

M(ii,1) =Mt1

delM=delM1

write(50,128)p0ht,M(ii,1),delm

128 FORMAT(’ p0ht=’,D20.13,’M(ii,1)=Mt1=’,D20.13,

A ’ delM1=delM=’,D20.13,/)

ib=1

172

CHAPTER

4.COMPUTER

PROGRAM

goto 1001 ! Print output

else if (ABS(delM2)<=error1) then

M(ii,1)=Mt2

delM=delM2

write(50,129)p0ht,M(ii,1),delm

129 FORMAT(’p0ht = ’,D20.13,’M(ii,1)=Mt2=’,D20.13,

A ’ delM2=delM=’,D20.13,/)

ib=1

goto 1001

End if

Continue

IF((delM1>0..and.delM2>0.)

A .OR.(delM1<0..and.delM2<0.))then

call Falci(Mt1,Mt2,delM1,delM2)

! Write(50,126)Mt1,delm1

126 FORMAT(’ Mt1=’,D20.13,’ delM1=’,D20.13)

! Write(50,127)Mt2,delM2

127 FORMAT(’ Mt2=’,D20.13,’ delM2=’,D20.6,/)

! write(50,*)’***********************’

k=k+1

If (k>=60) then

4.2.

SOURCE

CODE

173

Write(50,*)’ k=’,k, ’ FALCI & BISECTION combinedly have’,

1 ’ been called more than 60 times by the main program for’,

2 ’ the head end Mach number convergence. Calculations’,

3 ’ abandoned’

stop

End if

goto 999

Else

call bisection (Mt1,Mt2,Mt3,delM1,delM2,delM3,ib)

k=k+1

If (k>=60) then

Write(50,*)’ k=’,k, ’ FALCI & BISECTION combinedly have’,

1 ’ been called more than 60 times by the main program for’,

2 ’ the head end Mach number convergence. Calculations’,

3 ’ abandoned’

stop

End if

! Write(50,*)’ Mt1=’,Mt1,’ delM1=’,delm1, ’ ib=’,ib

! Write(50,*)’ Mt2=’,Mt2,’ delM2=’,delM2

! Write(50,*)’ Mt3=’,Mt3,’ delM3=’,delm3

! write(50,*)’***********************’

174

CHAPTER

4.COMPUTER

PROGRAM

GOTO 999

!

END IF

! ! Print output

1001 Continue

i=1

p0(ii,i)=p0ht

p(ii,i) = p0(ii,i)*(1+(gama-1)/2*M(ii,i)**2)**(-gama/(gama-1))

t(ii,i) = t0/(1+(gama-1)/2*M(ii,i)**2)

u(ii,i) =M(ii,i)*sqrt(gama*rgas*t(ii,i))

!

! Calculate Port dynamics

!

DO i=1,(incrementsi-1)

ps1=p(ii,i)

ts1=t(ii,i)

us1=u(ii,i)

p0s1=p0(ii,i)

ss1 =s(ii,i)

aps1=ap(ii,i)

ss2=s(ii,i+1)

4.2.

SOURCE

CODE

175

aps2=ap(ii,i+1)

!

CALL segsteady (ps1,ts1,us1,epsilon,ss1,

A aps1, ss2, aps2, ps2, ts2, us2, p0s2)

!

erosn(ii,i)=epsilon

r(ii,i)=a*p(ii,i)**n*epsilon

t0s2=ts2+us2**2/2/cp

p(ii,i+1)=ps2

t(ii,i+1)=ts2

u(ii,i+1)=us2

p0(ii,i+1)=p0s2

mdot(ii,i+1)=ps2/rgas/ts2*aps2*us2

t0c(ii,i+1)=t0s2

M(ii,i+1)=u(ii,i+1)/Sqrt(gama*rgas*t(ii,i+1))

End do

!

! Relocating "effective" incremental stations after

! burnout. Up to and including incrementsi are the

! effective incremental stations having propellant

! in them. The incremental stations from incrementsi+1

176

CHAPTER

4.COMPUTER

PROGRAM

! to increments are "ineffective" -- there is no

! propellant at those stations.

!

If(incrementsi<increments) then

Do jj=(incrementsi+1), increments

erosn(ii,jj)=0.

r(ii,jj)=0.

p(ii,jj)=ps2

t(ii,jj)=ts2

u(ii,jj)=us2

p0(ii,jj)=p0s2

mdot(ii,jj)=ps2/rgas/ts2*aps2*us2

t0c(ii,jj)=t0s2

M(ii,jj)=us2/sqrt(gama*rgas*ts2)

End do

End if

!

! Port dynamics have been calculated

!

mdotn=p0(ii,increments)*at/cstar

!

4.2.

SOURCE

CODE

177

! Solution = .true. if convergence has been achieved for the time ii

!

If (solution) then

! Write(50,*)’ Sigmamdotprt=’,sigmamdotprt,

! 1 ’ sigmamdotnzl=’,sigmamdotnzl

! Write(50,*)’ Port exit flow =’,mdot(ii,increments),

! 1 ’ Nozzle flow =’,mdotn,’ Delta t=’,deltatime

sigmamdotprt=sigmamdotprt+mdot(ii,incrementsi)*deltatime

sigmamdotnzl=sigmamdotnzl+mdotn*deltatime

! Write(50,*)’ Sigmamdotprt=’,sigmamdotprt,

! 1 ’ sigmamdotnzl=’,sigmamdotnzl

mdotnzl(ii)=mdotn

p0h(ii)=p0ht

g =mdot(ii,incrementsi)/ap(ii,incrementsi)

Call Erosive (p(ii,incrementsi),g,d (ii,incrementsi),

1 r(ii,incrementsi),epsilon)

erosn(ii,incrementsi)=epsilon

Solution =.false.

Goto 1010

End if

! With j=1, the first iteration starts

178

CHAPTER

4.COMPUTER

PROGRAM

If(j==1) then

p0ht1=p0ht

mdotn1=mdotn

mdotp1=mdot(ii,increments)

delmdot1=(mdotp1-mdotn1)/mdotp1

j=2

goto 997 ! up up up

Else

! With j=2, the second iteration starts

p0ht2=p0ht

mdotp2=mdot(ii,increments)

mdotn2=mdotn

delmdot2=(mdotp2-mdotn2)/mdotp2

End if

!

899 Continue

Write(50,*)’_______________________________________’

Write(50,*)’ p0ht1=’,p0ht1,’ delmdot1=’,delmdot1

Write(50,*)’ p0ht2=’,p0ht2,’ delmdot2=’,delmdot2

Write(50,*)’________________________________________’

delmdt1=abs(delmdot1)

4.2.

SOURCE

CODE

179

delmdt2=abs(delmdot2)

! Storing the iterated values of head end total pressures

! and mass flow rate convergence errors

If(delmdt1<delmdt2)then

delmdt(kk)=delmdt1

p0hh(kk)=p0ht1

else

delmdt(kk)=delmdt2

p0hh(kk) =p0ht2

End if

Write(50,*)’kk=’,kk

Write(50,*)’delmdt(kk)=’,delmdt(kk),’p0hh(kk)=’,p0hh(kk)

If(delmdt1<=error) then

solution =.true.

p0ht=p0ht1

delmdot=delmdt1

! write(50,*)’ ’

ib1=1

goto 1002 ! UP UP UP

Else if (delmdt2<= error) then

solution = .true.

180

CHAPTER

4.COMPUTER

PROGRAM

p0ht=p0ht2

delmdot=delmdt2

! write(50,*)’ ’

ib1=1

Goto 1002 ! UP UP UP

End if

Continue

IF((delmdot1>0..and.delmdot2>0.)

A .OR.(delmdot1<0..and.delmdot2<0.))then

Call falci (p0ht1,p0ht2,delmdot1,delmdot2)

Write(50,*)’p0ht1=’,p0ht1,’delmdot1=’,delmdot1

Write(50,*)’p0ht2=’,p0ht2,’delmdot2=’,delmdot2

IF(p0ht1<0..or. p0ht2<0.)then

write(51,*)’p0ht1 or p0ht2 <0 @’,p0ht1,p0ht2

Goto 1003

End if

!

kk=kk+1

If (kk > 100) then

Write(51,*)’ From FALCI kk =’,kk

Goto 1003

4.2.

SOURCE

CODE

181

End if

j=1

goto 997 ! UP UP UP

Else

! write(50,*)’ ib1=’,ib1

call bisection1 (p0ht1,p0ht2,p0ht3,delmdot1,delmdot2,

a delmdot3,ib1)

kk=kk+1

If (kk > 100) then

Write(51,*)’From BISECTION1 kk =’,kk

Go to 1003

End if

j=1

goto 997

END IF

1003 If(delmdt(1)<delmdt(2))then

delmdot=delmdt(1)

p0ht=p0hh(1)

Else

delmdot=delmdt(2)

p0ht=p0hh(2)

182

CHAPTER

4.COMPUTER

PROGRAM

End if

Do kkk=3,(kk-1)

If(delmdt(kkk)<delmdot)then

delmdot=delmdt(kkk)

p0ht=p0hh(kkk)

End if

End do

Solution = .true.

ib1=1

Write(51,*)’ convergence unsuccessful. Solution approximate’

Write(51,*)’P0ht=’,p0ht,’ delmdot=’,delmdot

Go to 1002

!

1010 Continue ! 1010 is reached when solution has been

! obtained for a given time instant ii

pe(ii)=p0(ii,increments)/p0bype

thrust(ii)=cfvac*p0(ii,increments)*at

impulsetot=impulsetot+thrust(ii)*deltatime

If(ii==1) then

Call starttransienteql

Write(51,115)

4.2.

SOURCE

CODE

183

115 FORMAT(/’ ’,

1 ’ EQUILIBRIUM PERFORMANCE’,

2 /’ p0h=head-end total pressure; p0n=nozzle end total ’,

1 ’pressure;’,/’ dhprt=head-end port dia; dnprt=nozzle-end port ’,

2 ’dia’,/’ Lnprt=exit-port location;delmdot=convergence error’//

3 ’ Time (s) p0h(Pa) p0n(Pa) mdot(kg/s)’,

A ’ epsilon dhprt(m) dnprt(m) pe(Pa) ’,

B ’Fvac(N) Lnprt (m) delmdot’)

End if

Write(50,111)time (ii),p0h(ii),p0(ii,increments),t0,

1 t0c(ii,increments),mdot(ii,increments),mdotn,delmdot

111 Format (/,’ CONVERGED SOLUTION’,/,

G ’ Time Instant (s) ’,

H ’ =’,Es13.6,/,

1 ’ Head end total pressure (Pa) =’,

I Es13.6,/,

2 ’ Nozzle end total pressure (Pa) =’,

j ES13.6,/,

3 ’ Adiabatic flame temperature (K) =’,

K Es13.6,/,

4 ’ Nozzle end total temperature (K)"CHECK" =’,Es13.6,/,

184

CHAPTER

4.COMPUTER

PROGRAM

5 ’ Cumulative mass flow rate at the port exit(kg/s) =’,Es13.6,/,

6 ’ Nozzle flow rate (kg/s)"CHECK" =’,

l Es13.6,/,

l Error fraction delmdot at convergence =’,es13.6)

!

Write(50,112)time(ii),(p0(ii,i),i=1,increments)

write(50,131)(M(ii,i),i=1,increments)

Write(50,132)(mdot(ii,i),i=1,increments)

Write(50,133)(erosn(ii,i),i=1,increments)

Write(50,134)(r(ii,i),i=1,increments)

112 Format(//,’ ’,

A ’ PORT DYNAMICS AT TIME INSTANT’,

B Es13.6,’ s’,//,

2 ’ Total Pressure at Incremental ’,

3 ’Stations (Pa)’,/,20(10(1x,es13.6),/),//)

131 Format(’ Mach number at ’,

A ’Incremental Stations’,/,20(10(1x,es13.6),/),//)

132 Format(’ ’,

A ’ Mass flow rate at Incremental Stations’,

C ’(kg/s)’,/,

B 20(10(1x,es13.6),/),//)

4.2.

SOURCE

CODE

185

133 Format(’ ’,

A ’ Erosive Burning Ratio at ’,

C ’Incremental Stations’,/,

b 20(10(1x,es13.6),/),//)

134 Format(’ ’,

A ’ Burning Rate at Incremental Stations ’,

1 ’(m/s)’,/,

B 20(10(1x,es13.6),/))

!

Write(51,116)time(ii),p0h(ii),p0(ii,incrementsi),

1 mdot(ii,incrementsi),erosn(ii,incrementsi),idh(ii),

2 d(ii,incrementsi),pe(ii),thrust(ii),l(incrementsi),delmdot

116 Format(1x,f6.3,1x,2(es11.4,1x),4(es10.3,1x),2(es11.4,1x),

1 2(es10.3,1x))

! Check on complete burnout

! Write(50,*)’ incrementsi=’,incrementsi,’ ii=’,ii,

! a’p0(ii, incrementsi)=’,p0(ii, incrementsi),

! b’ p0burnout=’,p0burnout

If(p0(ii,incrementsi)<tailoffend*p0burnout)then

! Write(50,*)’p0(ii, incrementsi)=’,p0(ii, incrementsi),

! a ’ tailoffend=’,tailoffend,’ p0burnout=’,p0burnout

186

CHAPTER

4.COMPUTER

PROGRAM

mpsliver2=rhop*(pi/4.*od**2*l(incrementsi)-pi/3/

1 tan(tapangle*pi/180)*((idh(ii)/2+l(incrementsi)*

2 tan(tapangle*pi/180))**3-(idh(ii)/2)**3))

sliverfraction2=mpsliver2/mp

Write(51,*) ’ ’

Write(51,121)tailoffend,p0burnout

Write(50,121)tailoffend,p0burnout

121 Format(’ Nozzle entry total-pressure is less than’,

A es13.6,’ of the burnout nozzle-end total-pressure.’,es13.6,

B ’ (PA)’/,’ Tailoff calculation is terminated.’)

Write(51,122)mpsliver2,sliverfraction2

Write(50,122)mpsliver2,sliverfraction2

122 Format(/,’ Final sliver mass = Mass of propellant ’,

A ’left-out at the termination of tailoff (kg) =’,

B es13.6,/,’ Sliver fraction2 = (Final sliver mass)/’,

C ’(initial propellant mass) = ’,

D es13.6)

error2 =(sigmamdotprt-sigmamdotnzl)/sigmamdotprt

error3=((sigmamdotnzl+mpsliver2)-mp)/mp

Goto 118

End if

4.2.

SOURCE

CODE

187

119 Continue

End do

Continue ! End of time Do loop ii

If(p0((ii-1),incrementsi)>tailoffend*p0burnout)then

Write(50,135)deltatime

Write(51,135)deltatime

135 Format(//,’ **** ERROR MESSAGE **** ERROR MESSAGE’,

F /,’ Complete burnout could not be reached within’,

A /,’ the assigned 500 incremental time steps "iiburnout".’,/,

B ’ Given "deltatime", the incremental time’,D13.6,’ seconds’,

C ’ in the Namelist "motor" is to be increased.’/,

D ’ **** EXECUTION OF PROGRAM ABORTED **** EXECUTION OF’,

E ’ PROGRAM ABORTED ***’)

End If

Goto 1011

118 Write(50,120)mp,sigmamdotprt,sigmamdotnzl,mpsliver2,error2,error3

Write(51,120)mp,sigmamdotprt,sigmamdotnzl,mpsliver2,error2,error3

120 Format(/,’ ’,

J ’ CHECK ON MASS BALANCE’,/,

A ’ Initial propellant mass (kg) ’,

B ’ ’,

188

CHAPTER

4.COMPUTER

PROGRAM

C ’ = ’,es13.6,/,

1 ’ Cumulative port flows (kg) ’,

2 ’ ’,

3 ’ =’,Es13.6,/,

2 ’ Cumulative nozzle flows (kg) ’,

3 ’ ’,

4 ’ =’,Es13.6,/,

3 ’ Final sliver mass ’,

4 ’ ’,

5 ’ =’,es13.6,/,

4 ’ Mass balance error2=[(cumulative port-flow)-’,

5 ’(cumulative nozzle-flow)]/cumulative port-flow =’es13.6,/,

6 ’ Mass balance error3={[(cumulative nozzle-flow)+’,

7 ’(final sliver)]-(initial propellant-mass)}’,/,

h ’ /(initial propellant-mass) ’,

I ’ ’,

J ’ =’,es13.6)

ispvac=impulsetot/mp

Write(51,139)impulsetot,ispvac,ispvactheo

Write(50,139)impulsetot,ispvac,ispvactheo

139 Format(//,’ ’,

4.2.

SOURCE

CODE

189

A ’**** MOTOR PERFORMANCE **** MOTOR PERFORMANCE ****’,/,

b ’ TOTAL IMPULSE IN VACUUM (N-s) ’,

c ’ ’,

d ’ =’, es13.6,/,’ VACUUM’,

e ’ SPECIFIC-IMPULSE CALCULATED[Total impulse in vacuum divided’,

f ’ by propellant mass] (N-s/kg) =’,es13.6,/’ CHECK: ’,

g ’Vacuum specific impulse theoretical [c*xCFvac] (N-s/kg) ’,

h ’ ’,

I ’ =’,Es13.6)

1011 Continue

Close (50)

Close(51)

Stop

End

SUBROUTINE falci (x1,x2,y1,y2)

! This subprogram uses the method of Regula Falci to help solve the

! equation of the type f(x) =0. Final solution is not reached by this

! subprogram. But, for the received pair of trial points (x1,y1) and (x2,y2)

! the subprogram returns to the calling program one of the received points

! (x1,y1) or (x2,y2) and an improved new value of x to calculate and

190

CHAPTER

4.COMPUTER

PROGRAM

! check the related y, which has to be within the desired error-band around 0.

! If not, again this subprogram is called with a new pair of trial points

! (x1,y1) and (x2,y2).

! The subprogram falsi is called by main program, and subprograms

! prsratio and erosive

IMPLICIT NONE

REAL*8 :: x1,x2,y1,y2 ! y1 and y2 correspond to x1 and x2

REAL*8 :: xdsh ! Temporary x value

! Write(50,*)’ Input values’

! WRITE(50,110)x1,y1,x2,y2

IF(x1<x2)then

IF((y1>0. .and.y2>0.).and.(y2>y1))then

Goto 102

Else if (y1>0..and.y2>0.)then

goto 103

END IF

IF((y1<0. .and.y2<0.).and. (y2>y1))then

GOTO 103

Else if (y1<0..and. y2<0.)then

goto 102

END IF

4.2.

SOURCE

CODE

191

xdsh=x1+(x2-x1)/(y1-y2)*y1

x2=xdsh

y2=0.

GO TO 101

ELSE IF (x1>x2) then

IF ((y1>0. .and.y2>0.).and.(y2>y1))then

goto 104

Else if (y1>0. .and.y2>0.) then

goto 105

END IF

IF ((y1<0..and.y2<0.).and.(y2>y1)) then

GOTO 105

ELSE IF (y1<0..and.y2<0.)then

goto 104

END IF

xdsh=x2+(x2-x1)/(y1-y2)*y2

x1=x2

y1=y2

x2=xdsh

y2=0.

GO TO 101

192

CHAPTER

4.COMPUTER

PROGRAM

END IF

102 xdsh=x1-(x2-x1)/(y2-y1)*y1

x2=x1

y2=y1

x1=xdsh

y1=0

GOTO 101

103 xdsh=x2-(x2-x1)/(y2-y1)*y2

x1=x2

y1=y2

x2=xdsh

y2=0.

Goto 101

104 xdsh=x1-(x2-x1)/(y2-y1)*y1

x2=xdsh

y2=0.

GOTO 101

105 xdsh=x2-(x2-x1)/(y2-y1)*y2

x1=xdsh

y1=0.

GOTO 101

4.2.

SOURCE

CODE

193

101 Continue

! Write(50,*)’Corrected values’

! WRITE(50,110)x1,y1,x2,y2

110 FORMAT(’ x1 =’,ES13.6,/,

1 ’ y1 =’,ES13.6,/,

2 ’ x2 =’,ES13.6,/,

4 ’ y2 =’,ES13.6,//)

RETURN

END SUBROUTINE falci

Subroutine geometry

! This subprogram is for an ends-inhibited tapered circular port grain.

! For 0th time (ii =1) burning area, free volume, propellant mass, and

! propellant volumetric loading fraction are calculated.

! For all time instants, this subprogram calculates the port envelop.

! Included under port envelop are: port diameter, burning perimeter,

! port area, and taper angle at all incremental stations.

194

CHAPTER

4.COMPUTER

PROGRAM

IMPLICIT NONE






Integer :: i ! i represents incremental location

Integer :: ii !(ii-1)*deltatime gives

! the operating time (s)

Integer :: increments ! Number of incremental stations

Real*8,parameter :: pi=.314159265D+01

Real*8 :: a ! Pre-exponent factor in the burning rate

! equation r0=ap^n (m/s); p in Pa

Real*8 :: alpha ! A factor in Lenoir-Robillard equation

Real*8 :: abi ! Approximate initial burning area (m^2)



! Port areas (time increment,location)(m^2)

Real*8 :: beta ! Greek letter Beta in

!Lenoir-Robillard equation


4.2.

SOURCE

CODE

195

Real*8 :: clamung ! Intial ratio of burning area to throat area

Real*8 :: cp ! Specific heat at constant pr.(J/kg-K)

Real*8 :: cs ! Specific heat of propellant (J/kg-K)

Real*8 :: cstar ! Characteristic-velocity experimental(m/s)

Real*8, dimension (500,200) :: d

! Port diameters (time-increment, location)(m)

Real*8 :: deltatime ! Incremental time(s)to be specified

! by input its value comes from main


Real*8 :: dt ! Throat diameter specified by input (m)

Logical :: eros ! When set=.false. in the subroutine

! propellant erosive burning will

! not be considered

Real*8 :: error ! Allowable Convergence error for mass

! flow rates to be specified by input

Real*8,dimension (500,200) :: erosn

! Erosive burning ratios (time, location)

Real*8 :: grainl ! grain length (m)

Real*8 :: gama ! Ratio of specific heats

Real*8 :: heighth ! cone height corresponding to

! the idhinitial(m)

196

CHAPTER

4.COMPUTER

PROGRAM

Real*8 :: heightn ! Cone height corresponding to

! the idninital (m)

Real*8 :: heightc ! Nozzle Cone height corresponding to

! the grain OD (m)

Real*8 :: heightt ! Nozzle cone height corresponding to

! throat dia dt (m)

Real*8 :: idhinitial ! Initial head-end diameter

! specified by input(m)

Real*8 :: idninitial ! Intial nozzle end port diameter (m)

Real*8, dimension (500) :: idh ! Head end diameters (time)

Real*8, dimension (200) :: l ! Segment station locations



Real*8 :: mu ! Viscosity of combustion gases (kg/m-s)

Real*8 :: n ! Burning rate index in

! the burning rate equation r0=ap^n

Real*8 :: od ! Outer diameter of the grain (m)


Real*8, dimension (500) :: p0h ! Solved head-end pressure (Pa)

Real*8 :: Pr ! Prandtl number of the combustion gases

Real*8,dimension (500,200) :: r ! Propellant burning rate (time, location)(m/s)

4.2.

SOURCE

CODE

197


Real*8 :: rhop ! Propellant density (kg/m**3)

Real*8,dimension (500,200) :: s

! Burning perimeters (time, location)(m)

Real*8 :: slantlh ! Cone slant length corresponding

!to the idhinitial

Real*8 :: slantln ! Cone slant length corresponding

! to the idninitial

Real*8,dimension (500) :: s0

! Head end burning perimeter (time),m


Real*8 :: ts ! Propellant burning-surface temperature(K)

Real*8 :: ti ! Propellant initial temperature (K)

Real*8 :: tapangle ! grain port taper angle


! taper angle at incremental

! locations (time,location) (deg)


! Time of operation (s)



198

CHAPTER

4.COMPUTER

PROGRAM

Real*8 :: vci ! Port free-volume, inital(m^3)



Real*8 :: vnozzle ! Nozzle convergence volume (m^3)

!

If (ii==1) then !********ii=1 means zero time

!Finding nozzle convergence volume

heightc=OD/2./tan(45.*pi/180.)

heightt=dt/2./tan(45.*pi/180.)

! slantlc=sqrt((OD/2.)**2.+heightc**2.)

! slantlt=sqrt((dt/2.)**2.+heightt**2.)

vnozzle=1./3.*pi*((od/2.)**2.*heightc-(dt/2.)**2.*heightt)

! Fixing initial grain configuration (for time = 0)

If(tapangle>0) then

idninitial = idhinitial+2*tan(tapangle*pi/180.)*grainl

heighth=idhinitial/2./tan(tapangle*pi/180)

heightn=idninitial/2./tan(tapangle*pi/180)

slantln=sqrt((idninitial/2.)**2.+heightn**2.)

slantlh=sqrt((idhinitial/2.)**2.+heighth**2.)

abi=pi*(idninitial/2.*slantln-idhinitial/2.*slantlh)

vci=1./3.*pi*((idninitial/2.)**2.*heightn-

4.2.

SOURCE

CODE

199

a (idhinitial/2.)**2.*heighth)

mp=rhop*(pi/4.*od**2.*grainl-vci)

! write(51,*)’ Port free volume =’,vci

vcitotal=1.1*vci+vnozzle

vcempty=1.1*pi/4.*od**2.*grainl+vnozzle

vpfraction=mp/rhop/vcempty

! write(51,*)’ abi=’,abi,’ Nozzle convergence volume=’,

! a vnozzle,’ Totalfree volume=’,vcitotal

Else if (tapangle==0) then

abi=pi*idhinitial*grainl

vci=pi/4.*idhinitial**2*grainl

mp=rhop*pi/4*grainl*(od**2-idhinitial**2)

vcitotal=1.1*vci+vnozzle

vpfraction=mp/rhop/vcitotal

End if

idh(ii)=idhinitial

s0(ii)=pi*idh(ii)

1000 Do i=1, increments

if(i==1)then

d(ii,i)=idh(ii)+2*delx*tan(tapangle*pi/180.)

else

200

CHAPTER

4.COMPUTER

PROGRAM

d(ii,i)=d(ii,i-1)+2*delx*tan(tapangle*pi/180.)

end if

s(ii,i)=pi*d(ii,i)

ap(ii,i)=pi/4*d(ii,i)**2

theta(ii,i)=tapangle

End do ! Do loop 1000

goto 109 !ii=1 means zero time effected

Else

! Fixing grain configuration

! for time greater than 0

idh(ii)=idh(ii-1)+2*a*p0h(ii-1)**n*deltatime/cos(tapangle*pi/180)

s0(ii)=pi*idh(ii)

Do i=1,increments

if(i==increments) then

theta(ii,i)=theta(ii,i-1)*pi/180

Else

theta(ii,i)=atan((d(ii-1,i+1)-d(ii-1,i))/2/delx)

End if

theta(ii,i)=theta(ii,i)*180./pi

d(ii,i)=d(ii-1,i)+2*r(ii-1,i)*deltatime/cos(theta(ii-1,i)*

1 pi/180.)

4.2.

SOURCE

CODE

201

s(ii,i)=pi*d(ii,i)

ap(ii,i)=pi/4*d(ii,i)**2

End do

End if

109 Continue

if (ii==1)then

time (ii)=(ii-1)*deltatime

else

time(ii)=time(ii-1)+deltatime

end if

Return

End Subroutine Geometry

Subroutine propellant

! This subroutine calculates the propellant properties: ratio of

! specific heats, experimental characteristic velocity c*,

! beta and alpha in the LenoirRobillard erosive burning equation.

! Calculated and other-specified propellant-properties are printed.

! Experimental characteristic velocity is calculated by

! assuming a c*-efficiency of 0.98.

202

CHAPTER

4.COMPUTER

PROGRAM

IMPLICIT NONE

Real*8, parameter : : ru=8314.51 ! Universal gas constant



Logical :: eros ! When set = .false. erosive burning

!is not considered


! burning rate equation, r0=ap^n (m/s)


! Lenoir-Robillard erosive

! burning rate equation

Real*8 :: beta ! The Greek letter beeta in the

!Lenoir-Robillard erosive burning model



! for combustion products

Real*8 :: cs ! Specific heat of propellant (J/kg-K)

Real*8 :: cstareff=0.98 ! c* efficiency

Real*8 :: cstartheo ! Theoretical c* (m/s)

Real*8 :: cstar ! Experimental cstar

Real*8,dimension (500,200) :: erosn ! Erosive burning ratio

4.2.

SOURCE

CODE

203



Real*8 :: mu ! Viscosity (kg/m-s)

Real*8 :: n ! Burning rate index in the burning

! in the burning rate eqn. r0=ap^n





Real*8 :: ts ! Temperature at the burning-surface

! of propellant(K)

Real*8 :: ti ! Initial temperature (K)

rgas=ru/mbar

gama=cp/(cp-rgas)

capgama=sqrt(gama)*(2/(gama+1))**((gama+1)/2/(gama-1))

cstartheo=sqrt(rgas*t0)/capgama

cstar=cstartheo*cstareff

alpha=0.0288*cp*mu**0.2*Pr**(-0.667)*(t0-ts)/(ts-ti)/rhop/cs

Write(51,101)a,n,rhop, mbar,gama,cstar,t0,beta,alpha,eros

Write(50,101)a,n,rhop, mbar,gama,cstar,t0,beta,alpha,eros

101 Format(’ PROPELLANT’,

204

CHAPTER

4.COMPUTER

PROGRAM

J ’ DATA’,/,

1 Pre-exponent factor a in the burning rate equation ’,

4 ’r0=ap^n =’,es13.6,/,

3 ’ Burning rate index n in the burning rate equation ’,

4 ’r0=ap^n =’,es13.6,/,

5 ’ Propellant density (kg/m**3) ’,

6 ’ =’,Es13.6,/,

7 ’ Molar mass of combustion products (kg/kg-mole) ’,

8 ’ =’,Es13.6,/,

9 ’ Ratio of specific heats ’,

A ’ =’,Es13.6,/,

B ’ Experimental Characteristic velocity (m/s) ’,

C ’ =’,Es13.6,/,

D ’ Adiabatic flame temperature (K) ’,

E ’ =’,Es13.6,/,

F ’ Beta in the Lenoir-Robillard equation ’,

G ’ =’,Es13.6,/,

H ’ Alpha in the Lenoir Robillard equation ’,

I ’ = ’,Es13.6,/,

R ’ Erosive Burning is not considered if eros=.FALSE., ie, F’,/,

T ’ Eros ’,

4.2.

SOURCE

CODE

205

U ’ =’,L2)

Return

End subroutine Propellant

Subroutine segsteady (pj, tj, uj,epsilon, ssj,

a areaj,ssjpls1,areajpls1,pjpls1,tjpls1,ujpls1,p0jpls1)

! The Subroutine Segsteady, called by the main program, calculates

! under steady flow assumption the exit properties of the incremental

! control volume by applying the equations of continuity,momentum,

! energy, and ideal-gas.

! Calculated exit properties transferred to the calling program are:

! velocity, static-temperature and pressure, Mach number, and

! total pressure; also the increase in entropy across the

! incremental control volume is calculated.

! Burning rate for the incremental surface is approximated to the one

! at its entry station. Erosive burning at the entry station is accounted

! through the Lenoir- Robillard model. Erosive burning characteristics are

! calculated by calling subroutine Erosive

IMPLICIT NONE

206

CHAPTER

4.COMPUTER

PROGRAM






Real*8, parameter :: pi=3.14159265

Real*8, parameter :: ru=8314.51 !Universal gas constant (J/kgmol-K)

Integer :: ii ! Counter for time(ii-1)*deltatime

! gives time of operation (s)

Integer :: increments ! Number of incremental stations

Logical :: eros ! When set =.false. erosive burning

! is not considered

Real*8 :: a ! Pre-exponent factor of

! burning rate eqn.r=ap^n (m/s)



Real*8 :: alpha ! Alpha in Lenoir- Robillard model

Real*8, dimension (500,200) :: ap ! Port areas (m**2)

Real*8 :: areaj ! Incremental volume Entry area (m**2)

Real*8 :: areajpls1 ! Incremental volume Exit area (m**2)

Real*8 :: beta ! Beta in Lenoir-Robillard model

4.2.

SOURCE

CODE

207

Real*8 :: capgama ! Function of gama, ratio of specific heats

Real*8 :: clamung ! Ratio of initial burning-area

! to throat area

Real*8 :: cp ! Specific heat at constant pressure (J/kg-K)


Real*8 :: cstar ! Characteristic velocity (m/s)

Real*8 :: deltatime ! Incremental time (s), set in the main

Real*8, dimension (500,200) :: d ! Port diameters (m)

Real*8 :: deltas1 ! Entropy increase by solution 1

Real*8 :: deltas2 ! Entropy increase by solution 2

Real*8 :: deltas ! Final solution for entropy change


Real*8 :: dj ! Hydraulic diameter at the jth station (m)

Real*8 :: dmdotj ! added mass flow rate in the

! incremental volume(kg/s)

Real*8 :: dt ! Throat diameter (m)

Real*8 :: epsilon ! Erosive burning ratio

Real*8,dimension(500,200) :: erosn ! Erosive burning

Real*8 :: error ! Allowable error in mass

! flow rate convergence


208

CHAPTER

4.COMPUTER

PROGRAM

Real*8 :: gj ! mass flux at the jth station (kg/m**2-s)


Real*8, dimension (500) :: idh ! Head end diameters

Real*8 :: idhinitial ! Initial head end diameter (m)

Real*8, dimension (200) :: l ! Incremental station locations (m)

Real*8 :: machj ! Inlet Mach number

Real*8 :: machjpls1 ! Exit Mach number final solution

Real*8 :: machjpls11 ! Exit Mach number solution 1

Real*8 :: machjpls12 ! Exit Mach number solution 2

Real*8 :: mbar ! molar mass kg/kg-mole

Real*8 :: mdotj ! Entry mass flow rate (kg/s)

Real*8 :: mdotjpls1 ! exit mass flow rate (kg/s)

Real*8 :: mp ! Total propellant mass (kg)

Real*8 :: mu ! Viscosity of combustion products (kg/m-s)

Real*8 :: n ! burning rate index in the burning

! rate equation r=ap^n

Real*8 :: od ! Grain outer diameter (m)


Real*8, dimension(500) :: p0h ! Solved head end pressure (Pa)

Real*8 :: p0jpls1 ! Exit total pressure (Pa)

Real*8 :: p0jpls11 ! total pressure at the exit solution 1(Pa)

4.2.

SOURCE

CODE

209

Real*8 :: p0jpls12 ! Total pressure at the exit solution 2(Pa)

Real*8 :: pj ! static pressure at entry (Pa)

Real*8 :: pjpls1 ! Exit static pressure (Pa)

Real*8 :: pjpls11 ! Static pressure at teh exit solution 1(pa)

Real*8 :: pjpls12 ! Static pressure at the exit solution 2(Pa)


Real*8 :: quada !"a" in the quadratic equation for uj+1

Real*8 :: quadb !"b" in the quadratic equation for uj+1

Real*8 :: quadc ! "c" in quadratic equation for uj+1

Real*8 :: quadd ! The argument of the square-root

Real*8, dimension (500,200) :: r ! Total burning rate (m/s)

Real*8 :: rgas ! specific gas constant (J/kg-K)

Real*8 :: rhoj ! gas density at entry (kg/m**3)


Real*8 :: rj ! Total Burning rate at the jthstation(m/s)

Real*8,dimension (500,200) :: s ! Burning perimeters (m)

Real*8, dimension (500) :: s0 ! Head end perimeters

Real*8 :: ssj ! Burning perimeter at the entry station

! of the control volume

Real*8 :: ssjpls1 ! Burning perimeter at the exit station

! of the control volume

210

CHAPTER

4.COMPUTER

PROGRAM

Real*8 :: t0 ! Adiabatic flame temperature

Real*8 :: tapangle ! Grain taper angle (deg)

Real*8, dimension (500,200) :: theta ! Angles at incremental

! stations (deg)

Real*8 :: ti ! Propellant initial temperature

Real*8, dimension (500) :: time ! Time of operation (s)

Real*8 :: tj ! Entry static temperature (K)

Real*8 :: tjpls1 ! Exit static temperature

Real*8 :: tjpls11 ! Exit static temperature solution 1(K)

Real*8 :: tjpls12 ! Exit static temperature solution 2 (K)

Real*8 :: ts ! Propellant burning-surface temperature(K)

Real*8 :: uj ! Entry velocity (m/s

Real*8 :: ujpls1 ! Exit velocity (m/s)

Real*8 :: ujpls11 ! Exit velocity (m/s)solution 1

Real*8 :: ujpls12 ! Exit velocity (m/s)solution 2



Real*8 :: vci ! Initial free volume of chamber (m^3)



!

4.2.

SOURCE

CODE

211

Namelist/propellant/t0,rhop,a,n,mbar,cp,gama,rgas,epsilon

Namelist/inlet2/areaj,mdotj,dmdotj,mdotjpls1,rhoj,tj,uj,machj

!

machj=uj/sqrt(gama*rgas*tj)

rhoj=pj/Rgas/tj

gj=pj/Rgas/tj*uj

dj=4*areaj/ssj

mdotj=rhoj*areaj*uj

!

Call erosive (pj,gj,dj,rj,epsilon)

dmdotj=rhop*(ssj+ssjpls1)/2*delx*rj

mdotjpls1=mdotj+dmdotj

quada=2./(areajpls1+areaj)-(gama-1)/gama/2/areajpls1

quadb=-(pj/mdotjpls1+2.*mdotj*uj/mdotjpls1/(areajpls1+areaj))

quadc=rgas*t0/areajpls1

quadd=quadb**2-4*quada*quadc

If(quadd<0) then

goto 100

End if

ujpls11=(-quadb+SQRT(quadb**2-4*quada*quadc))/2/quada

ujpls12=(-quadb-SQRT(quadb**2-4*quada*quadc))/2/quada

212

CHAPTER

4.COMPUTER

PROGRAM

tjpls11=t0-ujpls11**2/2/cp

tjpls12=t0-ujpls12**2/2/cp

If (tjpls12>0) then

pjpls12=rgas*tjpls12*mdotjpls1/areajpls1/ujpls12

machjpls12=ujpls12/sqrt(gama*rgas*tjpls12)

p0jpls12=pjpls12*(t0/tjpls12)**(gama/(gama-1))

deltas2=cp*log(tjpls12/tj)-rgas*log(pjpls12/pj)

End if

If(tjpls11>0)then

pjpls11=rgas*tjpls11*mdotjpls1/areajpls1/ujpls11

machjpls11=ujpls11/sqrt(gama*rgas*tjpls11)

p0jpls11=pjpls11*(t0/tjpls11)**(gama/(gama-1))

deltas1=cp*log(tjpls11/tj)-rgas*log(pjpls11/pj)

End if

If (tjpls11>0.and.tjpls12>0) then

if(p0jpls11<p0jpls12) then

ujpls1=ujpls11

tjpls1=tjpls11

pjpls1=pjpls11

machjpls1=machjpls11

p0jpls1=p0jpls11

4.2.

SOURCE

CODE

213

deltas=deltas1

else

ujpls1=ujpls12

tjpls1=tjpls12

pjpls1=pjpls12


p0jpls1=p0jpls12

deltas=deltas2

end if

else if(tjpls12>0) then

ujpls1=ujpls12

tjpls1=tjpls12

pjpls1=pjpls12


p0jpls1=p0jpls12

deltas=deltas2

else

ujpls1=ujpls11

tjpls1=tjpls11

pjpls1=pjpls11


214

CHAPTER

4.COMPUTER

PROGRAM

p0jpls1=p0jpls11

deltas=deltas1

end if

100 Continue

Return

END subroutine segsteady

Subroutine Erosive (p,g,d,r,epsilon)

IMPLICIT NONE

! This program calculates the erosive burning rate using Lenoir-Robillard

! model. The implicit equation in erosive burning rate r is solved

! by Regula-Falci by calling Subroutine FALCI.

! Subroutine Erosive is called by the subroutine Segsteady and also

! directly by the main program.

Integer :: k



Logical :: eros ! When set = .false. in the subroutine

!propellant erosive burning will

!not be considered

Real*8,parameter :: pi=3.14159265

4.2.

SOURCE

CODE

215

Real*8,parameter :: ru= 8314.51 ! Universal gas constant (J/kg-mole-K)


! burning rate equation, r0=ap^n (m/s)

Real*8 :: alpha ! The Greek letter alpha in the

! Lenoir-Robillard

! erosive burning rate equation





Real*8 :: capgama ! Function of ratio of specific heats

! for combustion products

Real*8 :: cstar ! c* (m/s)

Real*8 :: d ! Hydraulic diameter (m)

Real*8 :: delr ! Error at convergence

Real*8 :: delr2 ! Error for the rt2

Real*8 :: delr1 ! Error for rt1

! Real :: delr11,delr21 ! Swapped errors of delr1 and delr2

! Real :: delrt ! Temporary error during bisection

Real*8 :: error1=1.e-05 ! Acceptable error for the convergence

Real*8 :: epsilon ! Erosive burning ratio

216

CHAPTER

4.COMPUTER

PROGRAM

Real*8,dimension (500,200) :: erosn

! Erosive burning ratios

Real*8 :: g ! Massflux (kg/m^2-s)


Real*8 :: mbar ! Molar mass (kg/kgmol)

Real*8 :: mu ! Viscosity of

! combustion products (kg/m-s)

Real*8 :: n ! Burning rate index in the burning

! rate equation, r0=ap^n

Real*8 :: p ! Static pressure (Pa)


Real*8 :: r ! Total burning rate (r=r0+re)


Real*8 :: r0 ! Burning rate under zero crossflow(m/s)

Real*8 :: re ! Erosive burning rate component (m/s)


Real*8 :: rt1 ! Trial burning rate (m/s)

Real*8 :: rt2 ! Trial burning rate (m/s)


Real*8 :: ts ! Burning-propellant

! surface-temperature (K)

4.2.

SOURCE

CODE

217

Real*8 :: ti ! Propellant initial temperature (K)

!

r0=a*p**n

If(.not.eros) then

r=r0

Goto 1001

End if

rt1=r0*1.01

rt2=rt1*1.01

k=1

999 delr1=(r0+alpha*g**0.8/d**0.2/exp(beta*rt1*rhop/g)-rt1)/r0

delr2=(r0+alpha*g**0.8/d**0.2/exp(beta*rt2*rhop/g)-rt2)/r0

If(ABS(delr1)<error1) then

r =rt1

delr=delr1

goto 1001

else if (ABS(delr2)<error1) then

r=rt2

delr=delr2

goto 1001

End if

218

CHAPTER

4.COMPUTER

PROGRAM

!

k=k+1

If (k>30) then

Write(51,*)’ k=’,k,’ FALSI has been been called more’,

1 than 30 times by the Subroutine Erosive.Calculations abandoned.’

stop

End if

call Falci(rt1,rt2,delr1,delr2)

goto 999

1001 epsilon=r/r0

re=r-r0

1003 Continue

Return

End subroutine erosive

Subroutine prsrratio (p0bype,aebyat,gama)

! Subroutine prseratio calculates the nozzle pressure ratio p0bype

! of the nozzle of given area ratio aebyat and

! ratio of specific heats gama. The value of the ratio

! of specific heats gama comes from the main. For convergence the

! subroutine uses the subroutine FALSI.

4.2.

SOURCE

CODE

219

IMPLICIT NONE

Integer :: k=1 ! Used while debugging




Real*8 :: capgama ! Function of gama

Real*8 :: error1 =1. e-05 ! Convergence error

Real*8 :: p0bype1 ! Trial pressure ratio 1

Real*8 :: p0bype2 ! Trial pressure ratio 2

Real*8 :: delp0bype1 ! Trial error 1

Real*8 :: delp0bype2 ! Trial error 2

Real*8 :: delp0bype ! Converged error

capgama=sqrt(gama)*(2/(gama+1))**((gama+1)/2/(gama-1))

p0bype1=10

p0bype2=12

999 Continue

delp0bype1=(capgama/aebyat/sqrt(2*gama/(gama-1)*

1(1-p0bype1**(-(gama-1)/gama))))**gama-(1/p0bype1)

delp0bype2=(capgama/aebyat/sqrt(2*gama/(gama-1)*

1(1-p0bype2**(-(gama-1)/gama))))**gama-(1/p0bype2)

If(ABS(delp0bype1)<error1) then

220

CHAPTER

4.COMPUTER

PROGRAM

p0bype =p0bype1

delp0bype=delp0bype1

goto 1001

else if (ABS(delp0bype2)<error1) then

p0bype=p0bype2

delp0bype=delp0bype2

goto 1001

End if

k=k+1

If (k>30) then

Write(51,*) ’ FALSI has been called more than’,

1 ’30 times by the subroutine prsrratio’,

2 ’ Calculations abandoned’

stop

End if

call Falci(p0bype1,p0bype2,delp0bype1,delp0bype2)

goto 999

1001 Continue

Write(*,*)’Ae/At=’,aebyat, ’p0bype =’, p0bype,

1 ’ delp0bype=’, delp0bype

Return

4.2.

SOURCE

CODE

221

End subroutine prsrratio

Subroutine starttransienteql

! The start transient is calculated assuming the

! "equilibrium pressure analysis", that is assuming that there

! is one uniform pressure for the entire chamber-cavity.

! Furthermore it is assumed that (i) for the duration of start

! transient the burned distance is negligible, that is, the

! burning area is constant and (ii) entire grain surface is

! instantaneously ignited with negligible igniter mass.

! This subprogram prints the calculated start transient.

IMPLICIT NONE






Real*8 :: a ! Pre-exponent factor in the burning

! rate equation r0=ap0^n (m/s)

Real*8 :: aa

222

CHAPTER

4.COMPUTER

PROGRAM




! Lenoir-Robillard

! erosive burning rate equation


! Port areas (m**2)

Real*8 :: at ! Throat area

Real*8 :: bb



Real*8 :: capgama ! Function of specific heats ratio

Real*8 :: cp ! Specific heat at constant

! pressure (J/kg-K)

Real*8 :: cf0 ! Characteristic thrust coefficient

Real*8 :: cfvac ! Vacuum thrust coefficient

Real*8 :: clamung ! ratio of initial burning

! area to throat area


Real*8 :: cstar ! experimental cstar (m/s)

Real*8, dimension (500,200) :: d

4.2.

SOURCE

CODE

223

! Port diameters (m)

Real*8,dimension (200) :: deltign ! Incremental time (s)

Real*8 :: deltatime ! Incremental time (s)


Real*8 :: dt ! throat diameter (m)

Real*8 :: delp ! Incremental pressure

Real*8, dimension (500,200) :: erosn

! Erosive burning ratios at

! the incremental stations

Logical :: eros ! When set =.false. in the subroutine

! propellant erosive burning

! will not be considered

Real*8 :: error ! Allowable error in massflow rate



Real*8 :: impulseign ! Time integral of thrust

Real*8, dimension (200) :: l ! Segment station locations;

! subscript represents

! incremental stations (m)

Real*8, dimension (200) :: mdotign

! Mass flow rates during

224

CHAPTER

4.COMPUTER

PROGRAM

! start transient(time)kg/s)

Real*8, dimension (500) :: idh ! Head end diameters;

! subscript represents time increments

Real*8 :: idhinitial ! head end initial port diameter (m)

Integer :: ii ! Time counter

Integer :: increments ! total number of incremental stations

Real*8:: mbar ! Molar mass of combustion products

Integer :: m

Integer :: mm


Real*8 :: mu ! Viscosity of combustion products(kg/m-s)

Real*8 :: n ! burning rate index in the burning

Real*8 :: od ! grain outer diameter (m)


Real*8 :: pr ! Prandtl number

Real*8, dimension (500) :: p0h ! Solved head-end

! equilibrium-pressure (time) (Pa)

Real*8, dimension (200) :: p0hign ! Head-end pressure

! during start transient(time) (Pa)

Real*8 :: p0ht1st ! first head-end

! equilibrium-pressure (Pa)

4.2.

SOURCE

CODE

225

Real*8 :: p0htini ! initial start transient pressure (Pa)

Real*8, dimension (500) :: pe ! Nozzle exit-plane pressures (time) (Pa)

Real*8, dimension (200) :: peign ! Nozzle exit-plane pressures

! during start transient (time) (Pa)

Real*8,parameter :: pi=.314159265E+01

Real*8, dimension (500,200) :: r

! Total burning rate including

! erosive component (m/s)



Real*8, parameter :: ru = 8314.51

! Universal gas constant (J/kg-mole-K)

Real*8,dimension (500,200) :: s ! Burning perimeters (time,location)(m)

Real*8,dimension (500) :: s0 ! Head end burning perimeter (time)(m)

Real*8 :: sigmamdotign

! time integral of mass consumed during ignition


Real*8 :: tapangle ! Port taper angle (deg)


! Local taper angle(time,location)(deg)

Real*8,dimension (500) :: thrust

226

CHAPTER

4.COMPUTER

PROGRAM

!Thrust during equilibrium operation (N)

Real*8, dimension (200) :: thrustign

! Thrust during start transient (N)

Real*8 :: ti ! Propellant storage temperature (K)

! generally kept at atmospheric temperature


! Time during equilibrium and tail-off operation (s)

Real*8 :: ts ! Propellant burning-surface

! temperature (K)

Real*8, dimension (200) :: timeign ! Time during start transient (s)



Real*8 :: vci ! Initial free volume of chamber (m^3)



namelist/input1/abi,at,clamung,rgas,gama,capgama,cstar,cf0,

1 cfvac,p0htini,delp

At=pi/4*dt**2

clamung=abi/at

Cf0=capgama*sqrt(2*gama/(gama-1)*(1-(1/p0bype)**((gama-1)/gama)))

4.2.

SOURCE

CODE

227

Cfvac=CF0+aebyat*(1/p0bype)

p0htini=(gama+1.)/2.*1.01325e+05

delp= (0.975*p0h(1)-p0htini)/100.

! write(51,*)’ p0h(1)=’,p0h(1), ’ delp (Pa)=’,delp

! write(51,*)’ vcitotal=’,vcitotal,’ capgama=’,capgama

aa = vcitotal/capgama**2/(1.-n)/cstar/at

bb=p0h(1)**(1.-n)

! write(51,*) ’ aa=’,aa,’ bb=’,bb

timeign(1) =0.

sigmamdotign=0.

impulseign=0.

p0hign(1)=p0htini

peign(1)=p0hign(1)/p0bype

mdotign(1)=abi*rhop*a*p0hign(1)**n

thrustign(1)=cfvac*p0hign(1)*at

! write(51,nml=input1)

Write(51,115)

Write(50,115)

115 FORMAT(//,’START TRANSIENT CALCULATED ASSUMING "EQUILIBRIUM’,

1 ’ PRESSURE ANALYSIS"’,//,’ Time (s) p0h(Pa)’,

2 ’ mdot(kg/s) pe(Pa) ’,

228

CHAPTER

4.COMPUTER

PROGRAM

3 ’Vac-Thrust(N)’)

Do mm = 2,101

p0hign(mm)=p0hign(mm-1)+delp

deltign(mm)=aa*log((bb-p0hign(mm-1)**(1.-n))/

a (bb- p0hign(mm)**(1.-n)))

! write(51,*)’ mm=’,mm, ’ deltign(mm)=’,deltign(mm)

! Write(51,*)’p0hign(mm-1)=’,p0hign(mm-1),’ p0hign(mm)=’,p0hign(mm)

timeign(mm)=timeign(mm-1)+deltign(mm)

! write(51,*)’ timeign(mm)=’,timeign(mm),’ deltign(mm)=’,deltign(mm)

peign(mm)=p0hign(mm)/p0bype

mdotign(mm)=abi*rhop*a*p0hign(mm)**n

sigmamdotign=sigmamdotign+

a (mdotign(mm-1)+mdotign(mm))/2*deltign(mm)

thrustign(mm)=cfvac*p0hign(mm)*at

impulseign=impulseign+

a (thrustign(mm-1)+thrustign(mm))/2.*deltign(mm)

! write(51,*)’ mm=’,mm, ’ time(mm) (s)=’,timeign(mm),

! a ’ impulseign (N-s) =’,impulseign

End Do

Continue

Do m=1,101

4.2.

SOURCE

CODE

229

Write(51,116)timeign(m),p0hign(m),mdotign(m),

A peign(m),thrustign(m)

Write(50,116)timeign(m),p0hign(m),mdotign(m),

A peign(m),thrustign(m)

116 Format(1x,es13.6,1x,4(es15.6,2x))

End do

Write(51,117)sigmamdotign,impulseign

Write(50,117)sigmamdotign,impulseign

117 format(/,’Time integral of propellant mass consumed during’,

A ’ start transient (kg) =’, es15.6,/,

B ’ Impulse = time integral of thrust during start transient ’,

C ’(N-s) =’, es15.6,/)

time(1)=timeign(101)+2.*deltign(101)

Return

End subroutine starttransienteql

subroutine bisection (x1,x2,x3,y1,y2,y3,ib)

IMPLICIT NONE

Real*8 :: x1,x2,x3 ! variables

Real*8 :: y1,y2,y3 ! Errors

230

CHAPTER

4.COMPUTER

PROGRAM

Integer :: ib

If (ib==1) then

x3=x2

y3=y2

x2=(x1+x2)/2.

y2=0.

ib=2

goto 1010

else if ((y1>0.and. y2<0).OR.(y1<0.and.y2>0))then

x3=x2

y3=y2

x2=(x1+x2)/2.

y2=0.

goto 1010

else

x1=x2

y1=y2

x2=(x2+x3)/2.

y2=0.

end if

1010 continue

4.2.

SOURCE

CODE

231

Return

End subroutine bisection

subroutine bisection1 (x1,x2,x3,y1,y2,y3,ib)

IMPLICIT NONE

Real*8 :: x1,x2,x3 ! variables

Real*8 :: y1,y2,y3 ! Errors

Integer :: ib

If (ib==1) then

x3=x2

y3=y2

x2=(x1+x2)/2.

y2=0.

ib=2

goto 1010

else if ((y1>0.and. y2<0).OR.(y1<0.and.y2>0))then

x3=x2

y3=y2

x2=(x1+x2)/2.

y2=0.

232

CHAPTER

4.COMPUTER

PROGRAM

goto 1010

else

x1=x2

y1=y2

x2=(x2+x3)/2.

y2=0.

end if

1010 continue

Return

End subroutine bisection1

Bibliography

[1] Gordon, S. and McBride, B. J., Computer Program for Calculation of

Complex Chemical Equilibrium Compositions, Rocket Performance,

Incident and Reflected Shocks, and Chapman-Jouget Detonations,

NASA SP-273, March 1976.

[2] www.grc.nasa.gov/WWW/CEAWeb/ceaguiDownload-win.htm - as-

sessed on 6th May 2016.

[3] Gordon, S. and McBride, B. J., Computer Program for Calculation

of Complex Chemical Equilibrium Compositions and Applications: I.

Analysis, NASA Reference Publication 1311, Lewis Research Center,

Cleveland, Ohio, October 1994.

[4] Lenoir, J. M. and Robillard, G., “A Mathematical Model to Predict

the Effects of Erosive Burning of Solid Propellant Rockets,” Proceed-

ings of the Sixth International Symposium on Combustion, 1957, pp.

663-672.

[5] Anon., Internal Ballistic Performance of Solid Propellant Rocket Mo-

tors, Vol. 1: Program Manual, D2-125286-1, The Boeing Company,

Aerospace Division in Seattle, USA, 1966.

[6] Barrere, M., Jaumotte, A., De Veubeke, B. F., and Vandenkerckhove,

J., Rocket Propulsion, Elsevier Publishing Company, New York, 1960.

233


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Solid Propellant Rocket Motors - Professor … Starting from the preliminary design stage to the development of designed hardware and also beyond, performance prediction of propulsion