50406734 propulsion class note draft 2

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    Critical Ratios for Choking to Occur

    Once the choking condition is prescribed, the evaluation of

    various characteristic parameters becomes so simple that we do not

    need the differential form of the equations (in particular, themomentum equations) to obtain the desired result.

    We have the energy equation for steady one-

    dimensional flow

    Assuming no heat addition, this becomes

    By definition of total conditions, u2 = 0 and T2 = To

    Hence, the above equation becomes

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    The above equation can be written in the following

    form.

    Hence

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    By applying the energy equation to the nozzle throat,

    ct

    t hu

    h =+2

    2

    ct

    t TCpu

    TCp =+2

    2

    1

    2

    1;

    2

    1;;

    +=

    +===

    kk

    t

    c

    t

    ctt

    t

    tt

    k

    P

    Pk

    T

    TkRTu

    RT

    P

    Mass flow through nozzle at the choked flow condition

    may be written as,

    tttn Aum =

    We have

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    1

    2

    1;

    2

    1;;

    +

    +=

    +===

    kk

    t

    c

    t

    ctt

    t

    tt

    k

    P

    Pk

    T

    TkRTu

    RT

    P

    1

    2

    1;

    2

    1;;

    +

    +=+

    ===k

    k

    t

    c

    t

    ctt

    t

    tt

    k

    P

    Pk

    T

    TkRTu

    RT

    P

    tt

    t

    tn

    AkRT

    RT

    Pm =

    Rearranging the above equation we get,

    tc

    t

    c

    c

    t

    c

    n APT

    T

    P

    P

    RT

    km =

    At the chocked flow condition we have,1

    2

    1;

    2

    1;;

    +

    +=

    +===

    kk

    t

    c

    t

    ctt

    t

    tt

    k

    P

    Pk

    T

    TkRTu

    RT

    P

    1

    2

    1;

    2

    1;;

    +=

    +===

    kk

    t

    c

    t

    ctt

    t

    tt

    k

    P

    Pk

    T

    TkRTu

    RT

    P

    Substituting we get

    tc

    t

    c

    c

    t

    c

    n APT

    T

    P

    P

    RT

    km =

    )1(21

    1

    2 +

    +=

    kk

    c

    tcn

    kRT

    kAPm

    ( )11

    1

    2 +

    +=

    kk

    c

    tcn

    kRT

    kAPm

    Mass flow coefficient,

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    )1(21

    1

    2 +

    +=

    kk

    c

    dkRT

    kC

    ( )11

    1

    2 +

    +

    =k

    k

    c

    d

    kRT

    kC

    The reciprocal of mass flow coefficient, Cd has

    dimensions of velocity and is termed as characteristic

    velocity, C*

    Basic Performance Relations

    The basic performance relation is derived from the principle of conservation of matter.

    The propellant mass burned per unit time has to equal the sum of the change in gas mass per

    unit time in the combustion chamber grain cavity and the mass flowing out through the

    exhaust nozzle per unit time (Assuming negligible / nil igniter mass flow).

    ( )11

    1

    2)( +

    ++=

    kk

    c

    tc

    cg

    bpkRT

    kAP

    dt

    VdrA

    tcd

    cgn

    cbp APCdt

    VdaPA +=

    )(

    At steady state condition the above equation

    reduced to,

    tcd

    n

    cbp APCaPA =

    Solving for Pc we get,

    11

    =

    n

    bp

    tdc

    aA

    ACP

    Or

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    n

    td

    bp

    cAC

    aAP

    =

    11

    Rocket motor chamber pressure during the burning time can be evaluated

    approximately using the following equation.

    n

    t

    bp

    cA

    caAP

    =

    11

    where,

    p = density of the propellant

    Ab = Burning surface area of the graina = Constant obtained from the burn rate law )( ncPar= n = Burn rate index obtained from the burn rate law

    )( ncPar=

    c* = Characteristicvelocity (1/Cd)

    At = Nozzle throat area

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    Exit velocity

    Exit velocity assumes importance from the

    point of view of thrust ie., it describe the

    extent to which thermal energy is converted

    to a form that produces useful work.

    Starting from energy equation again, we

    work for exit conditions,

    cp

    e

    ep TCu

    TC =+2

    2

    =

    c

    ecpe

    T

    TTCU 12

    =

    1

    12c

    ecpe

    P

    PTCu

    Complete expansion to vacuum (Pe =0) will

    give the limiting exhaust velocity. i.e.,

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    cpe TCu 2= dictated solely by the chamber

    temperature, Tc

    The nozzle geometry comes into picture

    through the pressure ratio Pe/Pc .Writing by mass conservation eeettt uAuA = and

    substituting for densities from equation of

    state and for ue and ut from expressions

    derived earlier, i.e.,

    =

    1

    12c

    ecpe

    P

    PTCu

    te TRu =

    One gets,

    2111

    )1(2

    1

    11

    2

    1

    2

    +

    =

    +

    c

    e

    c

    e

    t

    e

    P

    P

    P

    PA

    A

    Therefore if the choked flow prevails, Pe/Pc is

    a unique function of Ae/At. So if one neglects

    the influence of Pc on Tc, the nozzle exit

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    velocity does not change with Pc at all, though

    the mass flow rate increases linearly with Pc.

    2111

    )1(2

    1

    11

    2

    1

    2

    +

    =

    +

    c

    e

    c

    e

    t

    e

    P

    P

    P

    PA

    A

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    Thrust of a Rocket Motor

    eaee APPumF )( +=

    We have,( )1

    1

    1

    2 +

    +

    =

    c

    tc

    RT

    APm

    tc

    t

    e

    c

    a

    c

    ee AP

    A

    A

    P

    P

    P

    PumF

    +=

    =

    1

    12c

    ecpe

    P

    PTCu

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    +

    +

    =

    +

    tc

    c

    a

    c

    e

    t

    e

    c

    ecp

    tc

    c

    APP

    P

    P

    P

    A

    A

    P

    P

    R

    TRCAP

    RTF

    1

    )1(2

    1

    121

    2

    tc

    c

    a

    c

    e

    t

    e

    c

    ecp

    c

    APP

    P

    P

    P

    A

    A

    P

    P

    R

    TRC

    RTF

    +

    +

    =

    +

    1

    )1(2

    1

    121

    2

    We have

    1=

    R

    Cp

    tc

    c

    a

    c

    e

    t

    e

    c

    e APP

    P

    P

    P

    A

    A

    P

    PF

    +

    +=

    +

    1

    )1(2

    1

    11

    2

    1

    2

    F = CF Pc At

    +

    +

    =

    +

    c

    a

    c

    e

    t

    e

    c

    eF

    P

    P

    P

    P

    A

    A

    P

    Pc

    1

    )1(2

    1

    11

    2

    1

    2

    When Pe = Pa , CF = CF0 and it is

    referred to as optimum expansion or

    the optimum nozzle or adapted nozzle

    +=

    +

    1

    )1(2

    1

    0 11

    21

    2c

    eF

    PPC

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    For a set nozzle geometry (i.e., Ae/At), the

    pressure ratio Pc/Pe is unique under choked

    flow condition and hence CF0

    is a constant. Itdoes not depend on chamber pressure (Pc)

    also. The dependence of CF on Pc comes

    through the term (Pa/Pc) only.

    Thrust coefficient CF as a function of Pressure Ratio, Nozzle

    Area Ratio, and Specific Heat Ratio for optimum expansion

    conditions (Pe = Pa)

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    r = 1.3

    Ae/At = 1, 2, 4, 5, 6,10

    Pc/Pe = 20

    CF = 1.2, 1.365, 1.39, 1.37, 1.31,

    +

    +=

    +

    c

    a

    c

    e

    t

    e

    c

    eF

    PP

    PP

    AA

    PPc

    1

    )1(2

    1

    11

    21

    2

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    Flight Performance

    Rocket propulsion systems provide forces to a flight vehicleand cause it to accelerate (or decelerate), overcome drag

    forces, or change flight direction. They are usually applied to

    several different flight regimes:

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    (1) Flight within the atmosphere (air to surface missile

    or sounding rockets)

    (2) Near-space environment (earth satellites)

    (3) Lunar and planetary flights etc.

    Gravity-free, Drag-free Space Flight

    This simple rocket flight analysis applies to an outer space

    environment, where there is no air (thus no drag) essentially

    no significant gravitational attraction. The flight direction is

    same as the thrust direction (along the axis of the nozzle) ,

    namely, a one-dimensional, straight-line acceleration path;the propellant mass flow and thrust remain constant for the

    propellant burning duration tp . For a constant propellant

    flow the flow rate is mp/tp , where mp is the total usable

    propellant mass. From Newtons second law and for an

    instantaneous vehicle mass m and a vehicle velocity u,

    F = m du/dtF = mue + (Pe-Pa) Ae

    F/m = ue + (Pe-Pa) Ae / m

    = c (defined as effective exhaust

    velocity)

    F = c m

    Instantaneous vehicle mass m = mo m t

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    After integration we will get the maximum vehicle velocity at

    propellant burnout up in a gravity free environment

    (vacuum).

    up = c ln (Minitial / Mfinal)

    up = c ln (mo / mf)

    Powered Vertical Flight

    Let u be the velocity of the rocket attained after time tfromtake-off (considering the gravitational pull too). The

    corresponding altitude is given by,

    = dtuZ

    =t

    o

    t

    f

    insp dttgdtM

    MIgZ

    0

    ln

    = t

    o

    t

    pin

    insp dttgdt

    tmM

    MIgZ

    0

    ln

    = t

    o

    t

    pinsp

    t

    insp dttgdttmMIgdtMIgZ00

    )(ln

    2

    02

    1)(lnln tgdttmMIgMtIgZ p

    t

    inspinsp =

    Integrating the second term separately,

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    t

    pin

    p

    pint

    pin tdttmM

    m

    tmMdttmM

    0

    0

    )(ln)(ln

    =

    in

    p

    inpin

    p

    in M

    tm

    MttmM

    m

    Mt ln)(ln

    +

    =

    Substituting it in the original equation for altitude, we get

    2

    2

    1

    lnln tgIgtmM

    M

    m

    M

    tmM

    M

    tIgZ sppin

    in

    p

    in

    pin

    in

    sp +

    =

    At the end of powered flight or burn out, t = tp

    p

    fpin

    ZZ

    MtmM

    =

    =

    Therefore above equation reduces to,

    2

    2

    1ln pp

    f

    in

    p

    inpspp tgt

    M

    M

    m

    MtIgZ

    +

    =

    Introducing the propellant mass ratio (zeta) = mp / Min

    ppp

    in

    tmm

    M

    ==

    The above equation yields a simpler relation

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    2

    2

    1ln

    111 p

    f

    inpp tg

    M

    MtCZ

    +=

    in

    f

    M

    Mwhere = 1,

    The above equation shows that the powered flight

    (rocket/missile) altitude is a function of effective jet velocity,

    burning time and propellant mass ratio.

    A missile has a maximum flight speed to jet speed ratio of

    0.2105 and specific impulse equal to 203.88 seconds.

    Determine for a burn out time of 8 seconds.

    (a) Effective jet velocity

    (b) Mass ratio and propellant mass fraction

    (c) Maximum flight speed, and

    (d) Altitude gain during powered and coasting flights

    Solution:

    (a) C = g Isp = 9.81 x 203.88 = 2000.06 m/s

    (b) Up = C ln (Min/Mf) g tp

    Up/C = ln (Min/Mf) g tp/C

    0.2105 = ln (Min/Mf) 9.81 x 8/2000.06ln (Min/Mf) = 0.2105 + 0.0392 = 0.2497

    Mass ratio = Mf/Min = 0.78

    Propellant mass fraction, = 1 Mf/Min

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    = 1 0.78 = 0.22

    (c ) Up/C = 0.2105

    Up = 0.2105 x 2000.06 = 2121 m/sUp = 1515.65 kmph

    (d )2

    2

    1ln

    111 p

    f

    inpp tg

    M

    MtCZ

    +=

    = 1.594 km

    Zc = 0.5 x 4212 / 9.81 x 1000 = 9.0336 km

    Starting Transient of Solid Rocket Motor

    *

    )(

    c

    APrA

    mmdt

    md

    tcbp

    outinc

    =

    =

    When nozzle is choked, the gaseous mass in the

    chamber per volume can be approximated by the

    perfect gas equation as,

    c

    cccc

    TR

    MVPm =

    If we assume Tc to be a constant during theoperational time (transient or steady state), then

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    dTdP

    TRVM

    dtdV

    dt

    dPV

    dt

    dVP

    RT

    M

    dt

    dm

    c

    c

    cccg

    cc

    cc

    c

    cc

    +=

    +=

    Note that rate of change in free volume is equal to the

    volume rate of propellant consumption.

    dt

    dP

    TR

    VMrA

    dt

    dm c

    c

    ccbg

    c +=

    Substituting it in the mass balance equation, we get

    ( ) = cAP

    rAdt

    dP

    TR

    VM tcbgp

    c

    c

    cc

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    Basic Relations of Motion

    For a vehicle that flies within the proximity of the earth, the gravitational of all other

    heavenly bodies may usually be neglected. Let it be assumed that the vehicle ismoving in rectilinear equilibrium flight and that all control forces, lateral forces, and

    moments that tend to turn the vehicle are zero. The trajectory is two-dimensional and

    is contained in a fixed plane. The vehicle has wings that are inclined to the flight pathat an angle of attack and that give a lift in a direction normal to the flight path. The

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    direction of flight does not coincide with the direction of thrust. Below figure shows

    these conditions schematically.

    Two-dimensional free-body force diagram for

    flying vehicle with wings and fins

    Let be the angle of the flight path with the horizontal

    and (psi) the angle of the direction of thrust with the

    horizontal. In the direction of the flight path the product of

    the mass and acceleration has to equal the sum of all forces,

    namely the propulsive, aerodynamics, and gravitational

    forces.

    From the above figure we have,

    The acceleration

    perpendicular to the flight path is

    dt

    du

    ; for a constant value

    sin)cos( mgDFdt

    dum =

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    of u and the instantaneous radius R of the flight path it isR

    u 2

    .

    The equation of motion in a direction normal to the

    flight velocity is

    cos)sin( mgLFdt

    dmu +=

    We have,2

    21 uACL L =

    2

    21 uACD D =

    The drag D is the aerodynamic force in a direction opposite

    to the flight path due to the resistance of the body to motion

    in a fluid. The lift L is the aerodynamic force acting in a

    direction normal to the flight path. Thhey are expressed as a

    function of the flight speed u, the mass density of the fluid in

    which the vehicle moves , and a typical surface area A. CL

    and CD are lift and drag coefficient, respectively. Forairplanes and winged missiles the area A is understood to

    mean the wing area. For wingless missiles or space launch

    vehicles it is the maximum cross-sectional area normal to the

    missile axis. The lift and drag coefficients are primarily

    functions of the vehicle configuration, flight Mach number,

    and angle of attack, which is the angle between the vehicle

    axis (or the wing plane) and flight direction. The value ofthese coefficients reach a maximum value near a Mach

    number of unity. For wingless vehicle the angle of attack is

    usually very small (0 < < 10 ). The density of the earths

    atmosphere can vary by a factor up to two (for altitude of

    300 to 1200 km) depending on solar activity and night-to-day

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    temperature variations. This introduces a major unknown in

    the drag. The aerodynamic forces are affected by the flow

    and pressure distribution of the rocket exhaust gases.

    For space launch vehicles and ballistic missiles the dragloss, when expressed in terms of u, is typically 5 to 10% of

    the final vehicle velocity increment. This relatively low value

    is due to the fact that the air density is low at high altitudes,

    when the velocity is high, and at low altitudes the air density

    is high but the flighyt velocity and thus the dynamic

    pressure are low.

    Gravitational attraction is exerted upon a flying spacevehicle by all planets, stars, the moon, and the sun. Gravity

    force pull the vehicle in the direction of the center of mass of

    the attracting body. Within the immediate vicinity of the

    earth, the attraction of other planets and bodies is negligibly

    small compared to the earths gravitational forces. This is

    the weight.

    If the variation of gravity with the geographical features and

    the oblate shape of the earth are neglected, the acceleration

    of gravity varies inversely as the square of the distance from

    the earths center. If Ro is the radius of the earths surface

    and go the acceleration on the earths surface at the earths

    effective radius Ro , the gravitational attraction g is

    [ ]2

    2

    )/(

    )/(

    hRRg

    RRgg

    ooo

    oo

    +=

    =

    Where h is the altitude. At the equator the earths radius is

    6378.388 km and the standard value of go is 9.80665 m/sec2.

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    At a distance as far away as the moon, the earths gravity

    acceleration is only about 3.3 x 10-4 go

    By substituting the expression for drag and lift in the aboveequations, we get

    sin2

    )cos( 2 gAum

    C

    m

    F

    dt

    du D =

    cos2

    )sin( 2 gum

    C

    m

    F

    dt

    du L +=

    No general solution can be given to these equations, since tp ,

    u, CD, CL, p, or can vary independently with time,

    mission profile, or altitude. Also CD, and CL are functions of

    velocity or Mach number.

    In a more sophisticated analysis other factors may be considered, such as the

    propellant used for nonpropulsive purposes (e.g., altitude control or flight

    stability).

    Rocket Equation for vertical trajectory

    this derivation neglects the effect of air resistance.

    for a rocket drifting in space, -gtis not applicable and can be omitted.

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    LIQUID ROCKET THRUST CHAMBERS

    The thrust chamber is the key subassembly of a rocket engine. Here the liquid propellants are

    metered, injected, atomized, vaporized, mixed, and burned to form hot reaction gas

    products, which in turn are accelerated and ejected at high velocity. A rocket thrust chamber

    assembly has an injector, a combustion chamber, a supersonic nozzle, and mounting provisions.

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    All have to withstand the extreme heat of combustion and the various forces, including the

    transmission of the thrust force to the vehicle. There also is an ignition system if non-

    spontaneously ignitable propellants are used. Some thrust chamber assemblies also have

    integrally mounted propellant valves and sometimes a thrust vector control device.

    LIQUID ROCKET COMBUSTION CHAMBER

    Admittedly, combustion in a liquid rocket is never perfectly smooth; some fluctuations of

    pressure, temperature, and velocity are always present. When these fluctuations interact with the

    natural frequencies of the propellant feed system (with and without vehicle structure) or the

    chamber acoustics, periodic superimposed oscillations, recognized as instability, occur. In

    normal rocket practice smooth combustion occurs when pressure fluctuations during steady

    operation do not exceed about + 5% of the mean chamber pressure.

    Combustion instability

    descriptionFrequency Range (Hz) Cause Relationship

    Low frequency, called

    chugging or feed system

    instability

    10-400

    Linked with pressure

    interactions between propellant

    feed system, if not the entire

    vehicle, and combustion

    chamber

    Intermediate frequency, called

    acoustic, buzzing or entropy

    waves

    400-1000

    Linked with mechanicalvibrations of propulsion

    structure, injector manifold,

    flow eddies, fuel/oxidizer ratio

    fluctuations, and propellant

    feed system resonances

    High frequency, called

    screaming, screeching, or

    squealing

    Above 1000

    Linked with combustion

    process forces (pressure waves)

    and chamber acoustical

    resonance properties

    Use of the word acoustic stems from the fact the frequency of the oscillations is related to

    combustion chamber dimensions and velocity of sound in the combustion gas.

    DESIGN CONSIDERATIONS OF LIQUID ROCKET

    COMBUSTION CHAMBER

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    Volume and Shape:

    - Volume large enough for adequate mixing,

    evaporation, and complete combustion

    - Volume depends on speed of reaction of propellants

    and stay time

    - Cylindrical chamber with flat injector is preferred

    Cooling of Thrust Chamber is needed to maintain the combustion chamber and nozzlewall temperatures at acceptable level

    The walls of the Thrust chambers are required to withstand chamber pressure, flight

    loads, ignition pressure surge, thrust loads, and thermal shock due to rapid starting.

    TYPES OF INJECTORS

    Doublet impinging stream pattern

    Triplet impinging stream pattern

    Self impinging stream pattern

    Shower head stream pattern

    Hollow post and sleeve element

    Variable injector area concentric tube injector

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    Fig. Schematic diagrams of several injector types

    FACTORS INFLUENCING INJECTOR BEHAVIOUR

    Propellant combination

    Injector orifice pattern and size

    Transient conditions

    Hydraulic characteristics

    Heat transfer

    Structural design

    The injector hole pattern, impingement pattern, hole distribution and pressure drop have a

    strong influence on combustion stability.

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    PROPELLANT FEED SYSTEMS

    The propellant feed system has two principal functions: to raise the pressure of the propellant

    and to feed them to one or more thrust chambers. The energy for these functions comes either

    from a high-pressure gas, centrigugal pumps, or a combination of the two. The selection of aparticular feed system and its components is governed primarily by the application of the rocket,

    duration, number or type of thrust chambers, past experience, mission, and by general

    requirements of simplicity of design, ease of manufacture, low cost, and minimum inert mass.

    All feed systems have piping, a series of valves, provisions for filling and removing (draining

    and flushing) the liquid propellants, and control devices to initiate, stop, and regulate their flow

    and operation.

    VALVES AND PIPE LINES

    Valves control the flows of liquids and gases. Pipes conduct these fluids to the intended

    components. There are no rocket engines without them. There are different types of valves. All

    have to be reliable, light weight, leak proof, and must withstand intensive vibrations and very

    loud noises. Often the design details, such as clearance, seat materials, or opening time delay

    present development difficulties. Any leakage or valve failure can cause a failure of the rocket

    unit itself. All valves are tested for two qualities prior to installation; they are tested for leaks-

    through the seat and also through the glands and for functional soundness or performance.

    Two valves commonly used in pressuried feed system are isolation valves (when shut,

    they isolate or shut off a portion of the propulsion system) and latch valves; they require power

    for brief period during movements, such as to open or shut, but need no power when latched or

    fastened into position. A simple and very light valve is a burst diaphragm.

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    PROPELLANT TANKS

    Liquid bipropellant rocket engine separate oxidizer and fuel tanks

    Liquid monopropellant rocket engine one propellant tank

    Common tank materials are aluminum, stainless steel, titanium, alloy steel and fibre-

    reinforced plastics

    Optimum shape spherical

    Vehicle integrated tanks are mostly cylindrical with half ellipses at the ends.

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    TANK PRESSURIZATION

    Typical pressure ranges of pressurizedfeed systems is from 1.3 to 9.0 MPa

    Typical pressure ranges of turbo-pump feed systems is from 0.07 to 0.34 MPa

    Inert gases such as helium or nitrogen are the most common method of pressurization

    Dynamic loads on Liquid Rockets Propellant Slosh

    - Free surface oscillations of Fluid

    - Lateral loads and C.G shift

    - uncover the tank outlet

    Propellant Hammer

    - Similar to Water Hammer due to sudden closure or opening of valves

    Geysering effect and its elimination

    - Caused by rising bubbles in vertical propellant pipes due to heating

    - Introduction of Geysering inhibitor pipe

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    Rockets and Mission Analysis

    Concept Quiz

    It may be shown that the thrust of a rocket scales with area and weight scales with volume, both

    based on the same characteristic length scale, L. This is expressed below as:

    Thrust, T T scales with thethroat area, At

    2

    ~~ LAT t

    Weight,

    W

    W scales with the

    volume, V

    3~~ LVW

    1. If we double T by increasing the characteristic length scale, how does T/W vary?T/W ~ 1/L ~ 1/T ~ T-

    Double T, T/W changes by 1/2

    2. If we double T by increasing the characteristic length scale, how does W vary?W ~ T3/2

    2(3/2) = 2.83

    W goes up by factor of 2.83

    HOME WORK!

    1. Differentiate solid, liquid and hybrid rockets

    2. Derive an expression for coasting time and coasting altitude of a rocket with verticaltakeoff.

    3. What is the purpose of thrust vector control in rockets?

    4. Name two liquid propellants and two solid propellants.

    5. Why is it better to launch a spaceship from near the equator?

    6. Derive rocket equation to evaluate the relative velocity (V) in terms of effectiveexhaust velocity and mass ratio.

    7. Derive an expression for the exit velocity of a rocket nozzle.

    8. Prove that powered rocket altitude is a function of effective jet velocity, burning time

    and mass ratio.

    9. A 5,000 kg spacecraft is in Earth orbit traveling at a velocity of 7,790 m/s. Its engineis burned to accelerate it to a velocity of 12,000 m/s placing it on an escape trajectory.

    The engine expels mass at a rate of 10 kg/s and an effective velocity of 3,000 m/s.

    Calculate the duration of the burn.

    10. Consider two rockets:

    Rocket 1: n=3 stages, each stage with identical payload ratio, =0.25, exit

    velocity, Ue=3,000 m/s, and structural coefficient,

    =0.1. Rocket 2: n=5 stages, each stage with identical payload ratio, =0.5, exit

    velocity, Ue=4,000 m/s, and structural coefficient, =0.2.

    Which of these rockets has a larger overall mass ratio, Mo1/ML?

    11. A rocket has a maximum flight speed to jet speed ratio of 0.2105 and specific

    impulse equal to 204 seconds. Determine for a burn out time of 8 seconds.

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    (i) Effective jet velocity, (ii) Mass ratio and propellant mass fraction,

    (iii) Maximum flight speed, and (iv) Altitude gain during powered and

    coasting flights.12. Assume a rocket of total mass 100 tons, carrying a spacecraft payload of 1 ton. The

    engines develop a constant exhaust velocity of 3,000 m/s. The structural mass is

    assumed to be 10% of the fuel mass.1) Determine the velocity of this configuration as a single stage rocket2) If the rocket is divided into two smaller stages, each with half the fuel, and the

    structural mass also shared equally, and the payload being the same,

    determine the total velocity increment for the two stage configuration.

    13. Repeat part (2), assuming 3 stages. What do you notice about the total velocityincrement as you add more and more stages? As an engineer, how would you

    determine how many stages to use? .

    14. A rocket engine burning liquid oxygen and liquid hydrogen operates at a

    combustion chamber pressure of 75 atmospheres. If the nozzle is expanded to operate

    at sea level, calculate the exhaust gas velocity relative to the rocket. Use the designcharts providing optimum mixture ratio, adiabatic flame temperature, gas molecular

    weight, and specific heat ratio for liquid oxygen and liquid hydrogen.

    15. What are the technological challenges for the design of multistage rocketswith both solid propellant motors and liquid propellant engines? Discuss about the

    selection criteria of propellants, important hardware components, solid propellant grain

    design and liquid engine design considerations, vectoring, cooling rocket systems, andmerits and demerits of solid and liquid rockets.