newton's laws as applied to rocket...
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MAE 5930, Rocket Systems Design MAE 5930, Rocket Systems Design
Rocket Science 101: Basic Concepts and Definitions
Newton's Laws as Applied to
"Rocket Science" ... its not just a job ... its an
adventure • How Does a Rocket Work?
Sellers: Chapter 14
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MAE 5930, Rocket Systems Design MAE 5930, Rocket Systems Design 2
What does a rocket “do”?
Rockets take spacecraft to orbit
Move them around in space, and
Slow them down for atmospheric reentry
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MAE 5930, Rocket Systems Design
Basics
3
Rocket’s basic function is to take mass, add energy, and convert that to thrust.
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MAE 5930, Rocket Systems Design
Basics (2)
Combus3on is an exothermic chemical reac3on. O:en an external heat source is required (igniter) to supply the necessary energy to a threshold level where combus3on is self sustaining
Propellants that combust spontaneously are referred to as Hypergolic 4
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MAE 5930, Rocket Systems Design
Basics (2)
• Combustion Produces High temperature gaseous By-products
• Gases Escape Through Nozzle Throat
• Nozzle Throat Chokes (maximum mass flow)
• Since Gases cannot escape as fast as they are produced … Pressure builds up
• As Pressure Builds .. Choking mass flow grows
• Eventually Steady State Condition is reached
5
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MAE 5930, Rocket Systems Design
What is a NOZZLE? (1)
• FUNCTION of rocket nozzle is to convert thermal energy in propellants into kinetic energy as efficiently as possible
• Nozzle is substantial part of the total engine mass.
• Many of the historical data suggest that 50% of solid rocket failures stemmed from nozzle problems.
The design of the nozzle must trade off: 1. Nozzle size (needed to get better performance) against nozzle weight penalty.
2. Complexity of the shape for shock-free performance vs. cost of fabrication
6
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MAE 5930, Rocket Systems Design MAE 5930, Rocket Systems Design
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MAE 5930, Rocket Systems Design MAE 5930, Rocket Systems Design
•
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MAE 5930, Rocket Systems Design MAE 5930, Rocket Systems Design
• Conserva)on of Momentum
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MAE 5930, Rocket Systems Design MAE 5930, Rocket Systems Design
•
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MAE 5930, Rocket Systems Design MAE 5930, Rocket Systems Design
•
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MAE 5930, Rocket Systems Design MAE 5930, Rocket Systems Design
•
F =
Engine massflow
Engine thrust equation
We shrink time As small as possible
Reac)on Force on Rocket
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MAE 5930, Rocket Systems Design
Thrust Components
• is the time rate of change of momentum of the rocket (Newtons)
• is the mass flow rate of the exhaust products (kg/s) • Vexit is the exit velocity of the exhaust products (m/s)
Momentum Thrust:
Pressure Thrust:
• Pressure is identical from all directions except for the Area of the exit nozzle. This pressure difference produces a thrust (which may be negative.)
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MAE 5930, Rocket Systems Design
Collected Rocket Thrust Equation
• Total thrust must be greater than the weight, or the rocket will not fly.
• Vexit and Pexit are related (inversely) • Ideal thrust is achieved when Pexit = Patmosphere
• Thrust + Oxidizer enters combus)on chamber at ~0 velocity, … combus)on adds thermal energy … High Chamber pressure accelerates flow through Nozzle
• Resultant pressure forces produce thrust
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MAE 5930, Rocket Systems Design
Rocket Thrust Equation (2)
What Causes Thrust?
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MAE 5930, Rocket Systems Design
Specific Impulse • Specific Impulse is a scalable characterization of a rocket’s ability to deliver a certain (specific) impulse for a given weight of propellant
• At a constant al)tude, with constant mass flow through engine
• Instantaneous specific impulse
• Total specific impulse
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MAE 5930, Rocket Systems Design
Specific Impulse (2) • Look at total impulse for a rocket
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MAE 5930, Rocket Systems Design
Specific Impulse (2)
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MAE 5930, Rocket Systems Design
Specific Impulse (3)
Effective exhaust velocity
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MAE 5930, Rocket Systems Design
Specific Impulse (4) • Example
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MAE 5930, Rocket Systems Design
Specific Impulse (cont’d)
• Look at instantaneous impulse for a rocket
Instantaneous
• Not necessarily the same
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MAE 5930, Rocket Systems Design
Example 2 • A man is sitting in a rowboat throwing bricks over the stern. Each
brick weighs 5 lbs, he is throwing six bricks per minute, at a velocity of 32 fps. What is his thrust and Isp?
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MAE 5930, Rocket Systems Design
Rocket Equation
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MAE 5930, Rocket Systems Design
Rocket Equation (cont’d)
Re-‐visit
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MAE 5930, Rocket Systems Design
Rocket Equation (2)
• Assuming constant Isp and burn rate …. integra)ng over a burn )me tburn
M -‐> rocket mass
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MAE 5930, Rocket Systems Design
Rocket Equation (3)
• Assuming constant Isp and burn rate …. integra)ng over a burn )me tburn
M -‐> rocket mass
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MAE 5930, Rocket Systems Design MAE 5930, Rocket Systems Design
Anatomy of the Rocket Equation … Its all about ΔV
27
• Consider a rocket burn of dura)on tburn
Ini)al Velocity
Final Velocity
Ini)al Mass
Final Mass
Tsiolkowsky’s Rocket Equa=on
Consumed propellant
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MAE 5930, Rocket Systems Design
Rocket Equation (5)
• Or rewri)ng
Is also called propellant mass Frac)on or “load mass frac)on”
Lmf
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MAE 5930, Rocket Systems Design
"Propellant Mass Fraction"
Load mass fraction Propellant mass fraction
Relating Delta V delivered by a rocket burn to propellant Mass fraction
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MAE 5930, Rocket Systems Design
Propellant Budgeting Equation
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MAE 5930, Rocket Systems Design
Propellant Budgeting Equation (2)
V = g0 ! Isp ! lnMinitial
M final
"
#$%
&'(
M final
Minitial
= e)
*Vg0 !Isp
"
#$
%
&'
Minitial ) Mpropellant
Minitial
= e)
*Vg0 !Isp
"
#$
%
&'1)
Mpropellant
Minitial
= e)
*Vg0 !Isp
"
#$
%
&'
Lmf +Mpropellant
Minitial
=Mpropellant
M final + Minitial
1) e)
*Vg0 !Isp
"
#$
%
&'
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MAE 5930, Rocket Systems Design
Ramifications of "the Rocket Equation"
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MAE 5930, Rocket Systems Design
Specific Impulse (revisited) 450 sec is “best you can get” with chemical rockets
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MAE 5930, Rocket Systems Design
for a Vertically Accelerating Vehicle • Rocket Equation originally derived for straight and level travel • What happens for vertically climbing rocket ? For example .. Look at a hovering vehicle … Lunar Lander During hover, change in velocity is zero .. So according to …
We burn no gas! Of course this result is absurd! Need to account for “gravity losses”
34
!V
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MAE 5930, Rocket Systems Design
for a Vertically Accelerating Vehicle (2)
• Look at 1-D equations of motion for vertically accelerating rocket • Ignore Aerodynamic Drag, assume constant g0
Velocity @ Motor Burnout 35
!V
For constant thrust m(t)
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MAE 5930, Rocket Systems Design
for a Vertically Accelerating Vehicle (3)
• Evaluate Integral
36
!V
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MAE 5930, Rocket Systems Design
for a Vertically Accelerating Vehicle (4)
• Apply earlier fundamental definitions
37
“gravity loss” for Ver3cal accelera3on
!V
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MAE 5930, Rocket Systems Design
Or More Generally …..
38
!V( )tburn =g0 " Isp " ln 1+ Pmf( ) #
g(t) " sin$0
Tburn
% "dt
Applies for rocket accelera)ng along an ARBITRARY path
Propulsive !V loss from acting against gravity....
!V( )gravityloss
= g(t) " sin#0
Tburn
$ "dt
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MAE 5930, Rocket Systems Design
Drag Losses
… and This decay Also applies To launch trajectory
39
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MAE 5930, Rocket Systems Design MAE 5930, Rocket Systems Design
Aerodynamic Forces Acting on Rocket
40
• Lift – acts perpendicular to flight path • Drag – acts along flight path • Thrust – acts along longitudinal axis of rocket
CL =Lift
12! "(h) !V
2
CD =Drag
12! "(h) !V
2
#12! "(h) !V
2 = q #"DyamicPressure"
Aref $ reference area#Typically based on planform or maximum cross section
CL =Lift
12⋅ ρ ⋅V 2 ⋅ Aref
CD =Drag
12⋅ ρ ⋅V 2 ⋅ Aref
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MAE 5930, Rocket Systems Design MAE 5930, Rocket Systems Design
Drag Losses (2)
41
• Lift – acts perpendicular to flight path .. Cannot effect energy level of rocket
• Gravity – acts downward (conservative) … cannot effect energy level of rocket
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MAE 5930, Rocket Systems Design MAE 5930, Rocket Systems Design
Drag Losses (3)
42
For constant CD, M
! =M
CD " Aref# "Ballistic Coefficient"
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MAE 5930, Rocket Systems Design MAE 5930, Rocket Systems Design
Drag Losses (4)
43
• Aerodynamic drag/mass iner)al effects can be incorporated into a single parameter …. Ballis=c Coefficient ( )
• ….. measure of a projectile's ability to coast. … = M/CDAref … M is the projectile's mass and … CDAref is the drag form factor.
• At given velocity and air density drag decelera)on inversely propor)onal to β
Low Ballis)c Coefficients Dissipate More Energy Due to drag
!!
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MAE 5930, Rocket Systems Design MAE 5930, Rocket Systems Design
Summary
44
• Rocket Thrust Equation
• Specific Impulse .. Total …. Instantaneous)
• Rocket Equation
~
Fthrust = !mVexit + Aexit (Pexit ! Patmosphere )
!Vburn = g0 " Isp " lnMinitial
M final
#
$%&
'(
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MAE 5930, Rocket Systems Design MAE 5930, Rocket Systems Design
Summary (2)
• Propellant Mass Budget Equation
Lmf !Mpropellant
Minitial
=Mpropellant
M final + Minitial
1" e"
#Vg0 $Isp
%
&'
(
)*
• Load Mass Fraction
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MAE 5930, Rocket Systems Design
for a Vertically Accelerating Vehicle (7)
Summary (3)
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!V
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MAE 5930, Rocket Systems Design
Summary (4)
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g(t) ! sin"dt0
tburn
#
Path dependent velocity losses
Available Δ V
! =M
CD " Aref# "Ballistic Coefficient"
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MAE 5930, Rocket Systems Design
Why Does this “Turn” Occur at Launch?
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MAE 5930, Rocket Systems Design 49
Questions??