AEE465/565 – Rocket Propulsion – Professor Dr. Werner J.A. Dahm
Thrust Performance
Isentropic Nozzle Flow
Total-to-Static Relations:
Normal Shock Relations: (M = Mach before shock)
Thrust Equation:
𝑇 = 𝑚̇𝑝 𝑉𝑒 + (𝑝𝑒 − 𝑃∞ )𝐴𝑒
*First term is jet (momentum) thrust, the second term is
pressure thrust.
𝑇 = 𝑚̇𝑝 𝑉𝑒𝑞
*Valid in all cases
Equivalent Velocity:
Total Relations:
2
𝑝𝑡 2
(𝛾 + 1) 𝑀
=[
]
(𝛾 − 1) 𝑀2 + 2
𝑝𝑡 1
𝜌𝑡
𝛾−1 2
= [1 +
𝑀 ]
𝜌
2
𝛾+1
]
2𝛾𝑀 2 − (𝛾 − 1)
𝛾
𝛾−1
𝑝2 2𝛾𝑀 2 − (𝛾 − 1)
=
𝑝1
𝛾+1
1
𝛾−1
𝑇2 [2𝛾𝑀 2 − (𝛾 − 1)][(𝛾 − 1)𝑀2 + 2]
=
(𝛾 + 1)2 𝑀2
𝑇1
Specific Impulse:
𝐼𝑠𝑝 =
𝑀22 =
1
𝛾+1 2
𝛾−1
2
)
𝑚̇𝑝 =
[𝛾 (
𝛾+1
√𝑅 𝑇𝑡
Thrust Coefficient:
𝑇
(𝑐𝑇 )𝑎𝑐𝑡𝑢𝑎𝑙 =
𝑝𝑡 𝐴∗
(𝛾 − 1)𝑀12 + 2
2𝛾𝑀12 − (𝛾 − 1)
𝛾+1
]
(𝑐𝑇 )𝑖𝑑𝑒𝑎𝑙
]} +
(𝑝𝑒 − 𝑝∞ ) 𝐴𝑒
𝑝𝑡
𝐴∗
𝛾−1
𝛾
2
2 𝛾−1
𝑝𝑒
)(
) [1 − ( )
= 𝛾 {(
𝛾−1 𝛾+1
𝑝𝑡
1
2
]}
*Isentropic and fully expanded nozzle flow (𝑝𝑒 = 𝑝∞ )
1 𝛾+1
1
𝛾+1 2
(𝑐𝑇 )𝑖𝑑𝑒𝑎𝑙 𝑣𝑎𝑐𝑢𝑢𝑚
given Exit Mach number or solve numerically to
find Exit Mach from
1
2
𝛾+1
𝐴
1
2
𝛾 − 1 2 2 𝛾−1
) [1 +
= {(
𝑀 ]}
𝐴∗ 𝑀 𝛾 + 1
2
𝐴∗
𝛾−1
𝛾
Oblique Shock Jump Relations:
Isentropic Area-Mach Number Relation:
𝐴𝑒
(𝑐𝑇 )𝑖𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 =
2
2 𝛾−1
𝑝𝑒
)(
) [1 − ( )
𝛾 {(
𝛾−1 𝛾+1
𝑝𝑡
^This is valid for any choked nozzle.
*Find
𝑉𝑒𝑞
𝑇
=
𝑔0 𝑚̇𝑝 𝑔0
*Equivalent time that 1 lbf of combustion products could
produce 1 lbf of thrust, units of seconds
(𝛾 + 1)𝑀2
𝜌2
=
𝜌1 (𝛾 − 1)𝑀 2 + 2
𝑚̇(𝑥) = 𝑚̇∗ = 𝜌∗ 𝑢∗𝐴∗
(𝑝𝑒 − 𝑝∞ ) 𝐴𝑒
𝑇
=
𝑚̇𝑝
𝑚̇𝑝
*Would be the exit velocity from a fully expanded nozzle
that produces same thrust as actual nozzle.
Shock Jump Relations:
Mass Flow Rate:
𝑝𝑡 𝐴∗
[
𝑉𝑒𝑞 = 𝑉𝑒 +
1
𝛾−1
𝑇𝑡2
=1
𝑇𝑡1
𝑇𝑡
𝛾−1 2
= [1 +
𝑀 ]
𝑇
2
𝑝𝑡
𝛾−1 2
= [1 +
𝑀 ]
𝑝
2
𝛾
𝛾−1
2
2 𝛾−1
)(
) }
= 𝛾 {(
𝛾−1 𝛾+1
𝐴𝑒
𝐴∗
Angle-Mach Relations:
*Once 𝑀𝑒 is known, can solve or 𝑇𝑒 and 𝑝𝑒 using
isentropic relations (above).
Nozzle Choking Criterion:
𝛾
𝛾−1
𝑝𝑡
𝛾+1
)
=(
𝑝∗
2
𝑀22 sin2(𝜃 − 𝛿) =
𝑎𝑡 𝑀 ∗ = 1
Trajectories
Tsiolkovsky Rocket Equation (velocity increment):
𝑀0
Δ𝑉𝑏𝑢𝑟𝑛 = 𝑉𝑒𝑞 ln ( ) − 𝑔0 𝑡𝑏
𝑀𝑓
(𝛾 − 1)𝑀12 sin2(𝜃) + 2
2𝛾𝑀12 sin2(𝜃) − (𝛾 − 1)
Total Relations:
𝛾
𝑝𝑡 2
*Unchoked if,
𝑝𝑡 1
𝛾
𝑝𝑡
𝛾 + 1 𝛾−1
)
<(
𝑝∗
2
Exit Velocity:
𝑉𝑒 = 𝑎𝑒 𝑀𝑒
*Fully expanded flow in a vacuum (infinite nozzle length)
(𝛾 + 1)𝑀12
cot(𝛿) = tan(𝜃) [
− 1]
2(𝑀12 sin2(𝜃) − 1)
=[
1
𝛾−1
(𝛾 + 1) 𝑀12 sin2 (𝜃) 𝛾−1
𝛾+1
]
[
]
(𝛾 − 1) 𝑀12 sin2 (𝜃) + 2
2𝛾𝑀12 sin2 (𝜃) − (𝛾 − 1)
Shock Jump Relations:
𝑝2 2𝛾𝑀12 sin2(𝜃) − (𝛾 − 1)
=
𝑝1
𝛾+1
𝑤ℎ𝑒𝑟𝑒, 𝑎𝑒 = √𝛾 𝑅 𝑇𝑒
𝛾−1
𝛾
2𝛾
𝑝𝑒
𝑉𝑒 = {
𝑅 𝑇 [1 − ( )
𝛾−1 𝑡
𝑝𝑡
𝑇2 [2𝛾𝑀12 sin2(𝜃) − (𝛾 − 1)][(𝛾 − 1)𝑀12 sin2(𝜃) + 2]
=
(𝛾 + 1)2 𝑀12 sin2(𝜃)
𝑇1
1
2
]}
(𝛾 + 1)𝑀12 sin2(𝜃)
𝜌2
=
𝜌1 (𝛾 − 1)𝑀12 sin2(𝜃) + 2
Where,
𝑘𝐽
(8314
)
𝑅̃
𝑚𝑜𝑙 ∙ 𝐾
=
𝑀𝑊
𝑀𝑊
*Therefore, lower 𝑀𝑊 gives higher 𝑉𝑒 → higher thrust
𝑅=
Isentropic Model of Atmosphere:
Aerodynamic Drag:
1
𝐷 = 𝜌𝑉 2 𝐴𝐶𝐷
2
𝜋
*Circular cross-section with diameter d has area 𝐴 = 𝑑 2
4
Gravitational Force:
𝑅𝑒 2
)
𝑔(𝑧) = 𝑔𝑒 (
𝑅𝑒 + 𝑧
*As a function of altitude 𝑧 in 𝑚,
earth radius 𝑅𝑒 = 6378 (103 ) 𝑚,
and acceleration of gravity at surface 𝑔𝑒 = 9.8 𝑚/𝑠 2
Time and Altitude at Burnout:
𝛾
𝑝(𝑧)
𝛾 − 1 𝑧 𝛾−1
( ∗ )]
≈ [1 −
𝑝𝑠
2
𝑧
*Where surface pressure, 𝑝𝑠 = 101.3 (103 )
Nozzle Expansion:
- Larger velocity increment due to high thrust for short
burn time than due to lower thrust for long burn time.
- Short burn at high thrust reduces energy consumed
lifting propellant (for vertical launch)
ℎ𝑏 = 𝑉𝑒𝑞 {1 −
𝑁
𝑀
*First term is dependent on Mass Ratio 𝑅 = ( 0 )
𝑚2
𝑀𝑓
∗
𝑧 = 8404 𝑚
𝑡𝑏 = (
Summerfeld Separation Criterion:
When nozzle flow is “highly” over expanded, when
ln(𝑅)
1
} 𝑡 − 𝑔 𝑡2
𝑅−1 𝑏 2 𝑒 𝑏
𝑀𝑃
)
𝑚̇𝑝
𝑡𝑏 = (𝑡 − 𝑡0 )
𝑝𝑒
𝑝∞
gets too small, flow in nozzle separates when,
𝑝𝑒
≲𝐾
𝑝∞
for 0.25 ≤ 𝐾 ≤ 0.4
*Boundary layer separation can cause highly turbulent
recirculating flow
Maximum Vehicle Altitude (vertical launch):
1 (Δ𝑉𝑏 )2
ℎ𝑚𝑎𝑥 = ℎ𝑏 +
2 𝑔𝑒
ℎ𝑚𝑎𝑥 =
𝑉𝑒𝑞2 (ln(𝑅))2
𝑅
− 𝑉𝑒𝑞 𝑡𝑏 {
ln(𝑅) − 1}
2𝑔𝑒
𝑅−1
*Minimizing 𝑡𝑏 maximizes ℎ𝑚𝑎𝑥
Rocket Staging
Staging Methods:
Velocity Increments for an Optimally Staged Rocket:
Δ𝑉 due to burnout of the 𝑖 th-stage,
(𝑉𝑏 )𝑖 = (𝑉𝑒𝑞 ) ln(𝑅𝑖 ) − 𝑔0 (𝑡𝑏 )𝑖
𝑖
Final Δ𝑉 imparted on final payload 𝑀𝐿 of stage 𝑁,
𝑁
Circular Orbits: 𝑟(𝜃) = 𝑅 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑒 < 0, 𝜀 = 0
Velocity:
𝜇
𝑉=√
𝑅
Period:
𝑇𝑏 = ∑(𝑡𝑏 )𝑖
𝑅3
𝑇 = 2𝜋√
𝜇
𝑖=1
𝑁
1 + 𝜆𝑖
(𝑉𝑏 )𝑁 = ∑ [(𝑉𝑒𝑞 ) ln (
)] − 𝑔𝑜 𝑇𝑏
𝑖
𝜀𝑖 + 𝜆𝑖
𝑖=1
(Δ𝑉𝑏 )𝑁 = 𝑉𝑒𝑞 𝑁 ln {𝜀 + (1 − 𝜀) [
1 −1
𝑀𝐿 𝑁
𝑀0
] }
Escape from Circular Orbit:
𝜇
Δ𝑉1 ≥ (√2 − 1)√
𝑅
− 𝑔𝑜 𝑇𝑏
Ideal Case (infinitely many stages):
(Δ𝑉𝑏 )𝑁→∞ = 𝑉𝑒𝑞 (1 − 𝜀) ln [
*Equivalent Velocity 𝑉𝑒𝑞
=
𝑇
𝑚̇𝑝
𝑀0
] − 𝑔𝑜 𝑇𝑏
𝑀𝐿
Parabolic Trajectories: (not orbit) 𝑒 = 0, 𝑎 = ∞, 𝜀 = 1
𝑉∞ = 0
Escape Velocity:
2𝜇
𝑉𝑒𝑠𝑐 = √
𝑟
≈ same for all stages
Escape from planet surface:
Orbital Dynamics
2𝜇
𝑉𝑒𝑠𝑐 = √
𝑅𝑝𝑙𝑎𝑛𝑒𝑡
Gravitational Parameter:
𝑁 ∙ 𝑚2
𝑤ℎ𝑒𝑟𝑒 𝐺 = 6.67 × 10−11
𝑘𝑔
𝜇 = 𝐺𝑀 ′
Radial Position (from vehicle to center of planet):
𝑎(1 − 𝜀 2 )
1 + 𝜀 cos(𝜃)
𝑟=
Semimajor Axis:
𝑎 = 𝑟𝑚𝑖𝑛
1
1−𝜀
𝑎=
1
(𝑟 + 𝑟𝑝 )
2 𝑎
Eccentricity:
𝜀=
𝑟𝑎 − 𝑟𝑝
𝑟𝑎 + 𝑟𝑝
𝜀 = √1 −
𝑏2
𝑎2
𝜀 =1−
𝑟𝑚𝑖𝑛
𝑎
Circularization Burn:
Circularize at 𝑟𝑎 :
2𝑟𝑝
𝜇
Δ𝑉1 = √ ∙ [1 − √
]
𝑟𝑎
𝑟𝑎 + 𝑟𝑝
Circularize at 𝑟𝑝 :
Vis-Viva:
Stage Mass Relation: 𝑁-stage vehicle, the 𝑖 th-stage has,
𝑀0𝑖 = 𝑀𝑃𝑖 + 𝑀𝑆𝑖 + 𝑀𝐿𝑖
Payload Ratio:
𝑀0(𝑖+1)
𝑀𝐿𝑖
𝑀𝐿𝑖
𝜆𝑖 ≡
=
=
𝑀0𝑖 − 𝑀0(𝑖+1) 𝑀0𝑖 − 𝑀𝐿𝑖 𝑀𝑃𝑖 + 𝑀𝑆𝑖
*Equal for all similar stages, 𝜆𝑖 = 𝜆(𝑖+1) = ⋯ = 𝜆𝑁
𝜆𝑜𝑝𝑡𝑖𝑚𝑎𝑙 =
(𝑀𝐿⁄𝑀 )
2 1
𝑉 = 𝜇( − )
𝑟 𝑎
2
0
1⁄
𝑁
Structural Coefficient:
𝑀𝑆𝑖
𝑀𝑆𝑖
𝑀𝑆𝑖
𝜀𝑖 =
=
=
𝑀0𝑖 − 𝑀0(𝑖+1) 𝑀0𝑖 − 𝑀𝐿𝑖 𝑀𝑆𝑖 + 𝑀𝑃𝑖
*Equal for all similar stages, 𝜀𝑖 = 𝜀(𝑖+1) = ⋯ = 𝜀𝑁
𝑀0𝑖
𝑀0𝑖
1 + 𝜆𝑖
=
=
𝑀0𝑖 − 𝑀𝑃𝑖 𝑀𝑆𝑖 + 𝑀𝐿𝑖 𝜀𝑖 + 𝜆𝑖
Inclination Change: (circular orbits)
𝑟𝑝 = 𝑎(1 − 𝜀)
Perigee Velocity:
𝜇 1+𝜀
)
𝑉𝑝 = √ (
𝑎 1−𝜀
𝜇 2𝑟𝑎
𝑉𝑝 = √
𝑟𝑝 (𝑟𝑎 + 𝑟𝑝 )
1
*Both other angles are: 𝜋 − 𝜃
2
𝜇
𝜃
Δ𝑉1 = 2√ sin ( )
𝑅
2
Apogee Radius:
𝑟𝑎 = 𝑎(1 + 𝜀)
Apogee Velocity:
𝜇 1−𝜀
)
𝑉𝑎 = √ (
𝑎 1+𝜀
2𝑟𝑝
𝜇
𝑉𝑎 = √
𝑟𝑎 (𝑟𝑎 + 𝑟𝑝 )
Orbital Period:
𝜇 𝑟𝑎
𝑉𝑎 = √
𝑎 𝑟𝑝
Orbital Rendezvous:
Lead Angle:
𝛼𝐿 =
𝜋 (𝑅1 + 𝑅2 )3
√
2
2(𝑅2)3
Transfer Time:
𝑎3
𝑇 = 2𝜋√
𝜇
Mass Ratio:
𝑅𝑖 =
𝜇
2𝑟𝑎
Δ𝑉1 = √ ∙ [1 − √
]
𝑟𝑝
𝑟𝑎 + 𝑟𝑝
Perigee Radius:
1⁄
𝑁
𝑀
(1 − 𝐿⁄𝑀 )
0
Δ = −𝑎√𝜀 2 − 1
Orbital Maneuvers
*Where b = semi-minor axis
𝜀 < 1 → 𝑒𝑙𝑙𝑖𝑝𝑠𝑒
𝜀 = 1 → 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑎
𝜀 > 1 → ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑎
𝜀 = 0 → 𝑐𝑖𝑟𝑐𝑙𝑒
*For Parallel Staging: Mass of 1st stage propellant from
large tank is equal to the total mass of the large tank’s
propellant times the fraction of the 1st stage burn time
over the total burn time of the 1st and 2nd stages.
Hyperbolic Trajectories: (not orbit) 𝑒 > 0
Excess Velocity:
𝜇
𝑉∞ = √2𝑒
𝑉∞ = √
−𝑎
Asymptote Angle:
1
𝜃∞ = 𝜋 − cos −1 ( )
𝜀
Turning Angle:
1
𝛿 = 2 sin−1 ( )
𝜀
Miss Distance:
𝑇𝑇𝑋 = 𝑇𝐻𝑜ℎ𝑚𝑎𝑛𝑛 =
𝜋 (𝑅1 + 𝑅2 )3
√
2
2𝜇
Intercept Opportunity Times:
Energy (per unit mass):
𝑒=−
𝜇
2𝑎
1
1
1
=
−
𝑇 (𝑇𝐶 )1 (𝑇𝐶 )2
𝑅𝑖
𝑤ℎ𝑒𝑟𝑒 (𝑇𝐶 )𝑖 = 2𝜋√
𝜇
Orbital Transfers
Standard Gravitation Parameters
Hohmann Transfer:
Combustion Tap-Off Systems:
3
Circular Orbit Velocities:
𝜇
(𝑉𝐶 )1 = √
𝑅1
𝜇
(𝑉𝐶 )2 = √
𝑅2
Transfer Orbit Parameters: (Trans time, see Rendezvous)
𝑟𝑝 = 𝑅1
𝑟𝑎 = 𝑅2
𝜇 2𝑅2
𝑉𝑝 = √
𝑅1 𝑅1 + 𝑅2
Body
𝑚
𝜇 = 𝐺𝑀 ′ [ 2 ]
𝑠
Sun
1.327 × 1020
Mercury
2.203 × 1013
Venus
3.249 × 1014
Earth
3.986 × 1014
(Moon)
4.903 × 1012
Mars
4.283 × 1013
Jupiter
1.267 × 1017
Saturn
3.793 × 1016
Uranus
5.794 × 1015
Neptune
Pluto
6.837 × 1015
1.108 × 1012
Propellant Feed Systems
Pressure-Fed Systems:
𝜇 2𝑅1
𝑉𝑎 = √
𝑅2 𝑅1 + 𝑅2
Velocity Increments:
*Uses combustion product gas from combustion chamber
to drive gas turbine that then drives pumps.
Open Systems:
- Turbine exit gas pressure low, dumped overboard
Closed Systems:
- Turbine exit gas dumped into nozzle at low pressure
location to add 𝑚̇𝑝 at nozzle exit
*Moderate/high thrust (Saturn I-B/Blue Origin)
Gas Generator Systems:
𝜇
2𝑅2
Δ𝑉1 = √ [√
− 1]
𝑅1
𝑅1 + 𝑅2
𝜇
2𝑅1
Δ𝑉2 = √ [1 − √
]
𝑅2
𝑅1 + 𝑅2
Bielliptic Transfer:
Deliver Pressure > Tank Pressure,
𝑃ℎ = 𝜌(𝑔 + 𝑎)ℎ
Circular Orbit Velocities:
𝜇
(𝑉𝐶 )1 = √
𝑅1
𝜇
(𝑉𝐶 )2 = √
𝑅2
Transfer Orbit Parameters: (𝑅𝑇𝑋 can be chosen)
𝑅1 = 𝑅𝑝𝑇𝑋1
*Part of fuel and oxidizer are burned in preburner to
produce high-pressure gas that drives turbopumps.
- Preburner operated very fuel-rich to keep
temperature sufficiently low
- Fuel-rich exhaust dumped overboard (open) or
dumped into nozzle (closed)
- Power controlled by adjusting fuel/ox rates into
preburner
- Can achieve higher CC pressure than expander sys.
Staged Combustion Systems:
𝑅𝑇𝑋 = 𝑅𝑎𝑇𝑋1
𝑅2 = 𝑅𝑝𝑇𝑋2
𝑅𝑇𝑋 = 𝑅𝑎𝑇𝑋2
1
𝑎 𝑇𝑋1 = (𝑅1 + 𝑅𝑇𝑋 )
2
1
𝑎 𝑇𝑋2 = (𝑅2 + 𝑅𝑇𝑋 )
2
Expander Systems:
Transit Time:
𝑇𝑇𝑋 = 𝜋
√(𝑎 𝑇𝑋1 )3 + √(𝑎 𝑇𝑋2)3
√𝜇
Velocity Increments:
2
1
𝜇
)−√
Δ𝑉1 = √𝜇 ( −
𝑅1 𝑎 𝑇𝑋1
𝑅1
2
1
2
1
) − √𝜇 (
)
Δ𝑉2 = √𝜇 (
−
−
𝑅𝑇𝑋 𝑎 𝑇𝑋2
𝑅𝑇𝑋 𝑎 𝑇𝑋1
𝜇
2
1
)
Δ𝑉3 = √ − √𝜇 ( −
𝑅2
𝑅2 𝑎 𝑇𝑋2
*Fuel is expanded to gaseous form to drive turbine that
powers pumps which then push propellant into CC.
- Low/moderate thrust applications
- Often used for upper stage engines
*All propellant is burned in two stages (nothing wasted).
First in fuel-rich preburner, then in main CC.
- Allows very high CC pressures (high thrust)
- Can be used with any liquid propellants
- More efficient than gas generator (all fuel burned)
- Requires very high-pressure turbopumps
Propellants and Aerothermochemistry
Propellant Types:
Hydrocarbon Combustion:
Turbines:
Liquid Propellants:
Fuels:
a) Cryogenic
- LH2 – liquid hydrogen (20 K) “deep cryogen”
- LCH4 – liquid methane (111 K)
b) Non-cryogenic
- RP-1 – kerosene [𝐶1 𝐻1.96 ]
- ethyl alcohol – [𝐶2 𝐻5 𝑂𝐻]
c) Hypergolic (𝐸𝑎 ≤ 0) (reacts on contact with oxi.)
- hydrazine [𝑁2 𝐻4 ]
- monomethyl hydrazine (MMH)
- unsymmetrical dimethyl hydrazine (UDMH)
Oxidizers:
a) Cryogenic
- LOX – liquid oxygen
- LF2 – liquid fluorine
- FLOX – fluorine-enhanced LOX
- N2O4 – dinitrogen tetroxide
b) Non-cryogenic
- N2O – nitrous oxide
- N2O4 – dinitrogen tetroxide
- ClF5 – chlorine pentafluoride
Overall Stoichiometric Reaction:
𝑚
𝑚
1 𝐶𝑛 𝐻𝑚 + (𝑛 + ) 𝑂2 → 𝑛𝐶𝑂2 + 𝐻2 𝑂
4
2
𝑛 = 𝑐𝑎𝑟𝑏𝑜𝑛 𝑎𝑡𝑜𝑚𝑠, 𝑚 = ℎ𝑦𝑑𝑟𝑜𝑔𝑒𝑛 𝑎𝑡𝑜𝑚𝑠
Application of 1st Law:
(Δℎ𝑡 )1,2 = 𝑞1,2 − 𝑤1,2 = 0
Monopropellants:
* Both fuel and oxidizer are contained in the same
molecule; contact with a catalyst bed initiates reaction
(breaks oxidizer and fuel components apart).
* Usually used for RCS and in-space propulsion at low-mid
thrust applications.
- hydrazine (N2H4) + granular Al w/iridium coating
- hydrogen peroxide (H2O2) + many possible catalysts
- nitromethane (CH3NO2)
- ethylene oxide (C2H4O)
- nitrous oxide (N2O)
- hydroxylammonium nitrate (HAN) (H4N2O4)
Solid Propellants:
* Both fuel and oxidizer are initially in solid form.
Homogenous Propellants:
* Fuel and oxidizer are contained in the same
molecule or in a homogeneous mixture.
- nitroglycerine + nitrocellulose
C3H5(NO2)3 + C6H7O2(NO2)3
- Single/Double/Triple base propellants
Metal powders often added to increase heat of
combustion.
Heterogeneous (composite) Propellants:
* Heterogeneous mixture of oxidizing crystals held in
an organic plastic-like fuel “binder”
a) Fuel Component (common binders)
- hydroxyl-terminated polybutadiene (HTPB)
- rubber/asphalt
b) Oxidizer Component (ground crystals of one or
more of the following)
- ammonium perchlorate (AP)
- ammonium nitrate (AN)
- nitronium perchlorate (NP)
- potassium perchlorate (KP)
- potassium nitrate (PN)
- cyclotrimethylene trinitomine (RDX)
- cyclotetramethylene tetranitramine (HMX)
* The combination of fuel/ox specifies the propellant
(e.g. AP-HTPB, KP-HTPB, …)
Many possible combinations.
→ ℎ𝑡2 − ℎ𝑡1 = 0
→ 𝐶𝑝 (𝑇𝑡2 − 𝑇𝑡1 ) = 0
Mixture Ratio:
𝑀𝑂
𝑟 = ( 2)
𝑀𝑓
∴ 𝑇𝑡2 = 𝑇𝑡1
Turbine Work:
Stoichiometric Mixture Ratio:
32𝑛 + 8𝑚
𝑟𝑠 =
12𝑛 + 𝑚
Equivalence Ratio:
𝑟𝑠
𝜑=
𝑟
𝜑 = 1 → 𝑠𝑡𝑜𝑖𝑐ℎ𝑖𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑐𝑜𝑚𝑏𝑢𝑠𝑡𝑖𝑜𝑛
𝜑 > 1 → 𝑓𝑢𝑒𝑙 − 𝑟𝑖𝑐ℎ 𝑐𝑜𝑚𝑏𝑢𝑠𝑡𝑖𝑜𝑛
𝜑 < 1 → 𝑓𝑢𝑒𝑙 − 𝑙𝑒𝑎𝑛 𝑐𝑜𝑚𝑏𝑢𝑠𝑡𝑖𝑜𝑛
Molar Enthalpy of Combustion:
Δℎ̃𝐶 = [∑ 𝐵𝑜𝑛𝑑 𝐸𝑛𝑒𝑟𝑔𝑦 ]
𝑖
− [∑ 𝐵𝑜𝑛𝑑 𝐸𝑛𝑒𝑟𝑔𝑦 ]
𝑟𝑒𝑎𝑐𝑡
𝑖
𝑝𝑟𝑜𝑑
Δℎ̃𝐶 < 0 → 𝑒𝑥𝑜𝑡ℎ𝑒𝑟𝑚𝑖𝑐
Δℎ̃𝐶 > 0 → 𝑒𝑛𝑑𝑜𝑡ℎ𝑒𝑟𝑚𝑖𝑐
Convert to mass-specific enthalpy:
Δℎ̃𝐶
Δℎ𝐶 =
𝑀𝑊𝐻𝐶
𝑀𝑊𝐻𝐶 = 12𝑛 + 1𝑚 [
𝑘𝑐𝑎𝑙
]
𝑔
Convert to [𝑘𝐽⁄𝑘𝑔]:
Δℎ𝐶 [
𝑘𝑐𝑎𝑙
1000 𝑐𝑎𝑙 1000𝑔 4.1814 𝐽
1 𝑘𝐽
)(
)(
)(
)
]∙(
𝑔
𝑘𝑐𝑎𝑙
𝑘𝑔
𝑐𝑎𝑙
1000𝐽
Turbopumps and Turbines
Turbopumps:
Shaft Power from Turbine:
𝑊̇ = 𝜏 ∙ Ω̇
Ideal Pump Work:
|𝑤|𝑖𝑑𝑒𝑎𝑙 =
1
|𝑊|𝑛𝑜𝑛𝑖𝑑𝑒𝑎𝑙 = 𝜂 𝑇 Δ𝑝𝑡
𝜌
Temperature Change Across Turbine:
Δ𝑝𝑡
Δ𝑇 = (1 − 𝜂 𝑇 )
𝜌 ∙ 𝑐𝑉
Turbine Work: (per unit mass)
|𝑤𝑇 | = |𝐶𝑃 (𝑇𝑡𝑒 − 𝑇𝑡𝑖 )|
Turbine Power:
𝑊̇𝑇 = 𝑚̇|𝑤𝑇 |
Turbine Stage Efficiency:
|𝑤|
𝜂𝑆 =
|𝑤|𝑆
Total Turbine Efficiency:
𝑇
1 − ( 𝑡3 )
𝑇𝑡1
𝜂𝑇 =
𝛾−1
𝑝𝑡3 𝛾
1−( )
𝑝𝑡1
Turbine Total Temperature Ratio:
1
(𝑝 − 𝑝𝑡1 )
𝜌 𝑡2
Ideal Pump Power:
1
|𝑤̇ |𝑖𝑑𝑒𝑎𝑙 = 𝑚̇|𝑤|𝑖𝑑𝑒𝑎𝑙 = 𝑚̇ (𝑝𝑡2 − 𝑝𝑡1 )
𝜌
Pump Efficiency:
|𝑤|𝑖𝑑𝑒𝑎𝑙
|𝑤̇ |𝑖𝑑𝑒𝑎𝑙
𝜂𝑃 =
=
|𝑤|𝑎𝑐𝑡𝑢𝑎𝑙 |𝑤̇ |𝑎𝑐𝑡𝑢𝑎𝑙
Non-Ideal Pump Work:
11
(𝑝 − 𝑝𝑡1 )
|𝑤|𝑎𝑐𝑡𝑢𝑎𝑙 =
𝜂𝑃 𝜌 𝑡2
Non-Ideal Pump Power:
1 1
(𝑝 − 𝑝𝑡1 )
|𝑤̇ |𝑎𝑐𝑡𝑢𝑎𝑙 = 𝑚̇|𝑤|𝑎𝑐𝑡𝑢𝑎𝑙 = 𝑚̇
𝜂𝑃 𝜌 𝑡2
*𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
𝜌
Δ𝑝𝑡 = (𝑝𝑡2 − 𝑝𝑡1 ) = 𝑤𝑃 𝜂𝑃
𝑚̇
1st Law for Liquid Flow Through Pump:
1
1
𝑐𝑉 = (𝑇2 − 𝑇1 ) + (𝑝2 − 𝑝1 ) + (𝑉22 − 𝑉12 ) = |𝑤|
𝜌
2
1
𝑐𝑉 = (𝑇2 − 𝑇1 ) + (𝑝𝑡2 − 𝑝𝑡1 ) = |𝑤|
𝜌
Temperature Change Across Pump:
1 − 𝜂𝑃 (𝑝𝑡2 − 𝑝𝑡1 )
(𝑇2 − 𝑇1 ) = (
)
𝜂𝑃
𝜌 ∙ 𝑐𝑉
𝛾−1
𝛾
𝑇𝑡3
𝑝𝑡3
= 1 − 𝜂 𝑇 [1 − ( )
𝑇𝑡1
𝑝𝑡1
]
Turbine Total Pressure Ratio:
𝛾
𝑝𝑡3
1
𝑇𝑡3 𝛾−1
= [1 − (1 − )]
𝑝𝑡1
𝜂𝑇
𝑇𝑡1
Total-to-Static Temperature Relation:
𝛾 − 1 2 −1
𝑇3 = 𝑇𝑡3 [1 +
𝑀3 ]
2
Total-to-Static Pressure Relation:
𝛾
𝛾 − 1 2 −𝛾−1
𝑀3 ]
2
Entropy Change Across Turbine:
𝑇𝑡3
𝑝𝑡3
Δ𝑠 = 𝑠3 − 𝑠1 = 𝐶𝑝 ln ( ) − 𝑅 ln ( )
𝑇𝑡1
𝑝𝑡1
𝛾−1
)
𝑅 = 𝐶𝑝 (
𝛾
𝑅 = 𝐶𝑣 (𝛾 − 1)
Temperature-Velocity Relation:
𝑉2
(𝑇𝑡 − 𝑇) =
2 ∙ 𝐶𝑝
𝑝3 = 𝑝𝑡3 [1 +
Nozzle Design
Simple Conical Nozzles:
Axial Thrust:
1 + cos(𝛼)
𝑚̇𝑝 𝑉𝑒 + (𝑝𝑒 − 𝑝∞ )𝐴𝑒
2
*See notes for Rao and TOP nozzle stuff
𝑇𝑎𝑥𝑖𝑎𝑙 =
Combustion Chamber Performance
DOF and Molecular Energy
Non-Isentropic Nozzle Flow
(𝐷𝑂𝐹)𝑟𝑜𝑡 = 0
Application of 1st Law:
(Δℎ𝑡 )2,𝑒 = 𝑞2,𝑒 − 𝑤2,𝑒 = 0
→ ℎ𝑡2 − ℎ𝑡𝑒 = 0
→ 𝐶𝑝 (𝑇𝑡2 − 𝑇𝑡𝑒 ) = 0
∴ 𝑇𝑡2 = 𝑇𝑡𝑒
→ (𝐷𝑂𝐹)𝑎𝑣𝑎𝑖𝑙 = 3
Temperature-Velocity Relation:
Monatomic Molecules:
(𝐷𝑂𝐹)𝑡𝑟𝑎𝑛𝑠 = 3
Combustion Chambers:
(𝐷𝑂𝐹)𝑣𝑖𝑏 = 0
Diatomic Molecules:
(𝐷𝑂𝐹)𝑡𝑟𝑎𝑛𝑠 = 3
(𝑇𝑡 − 𝑇) =
Total Pressure Ratio:
Δ𝑠
𝑝𝑡𝑒
= 𝑒− 𝑅
𝑝𝑡2
*where, 𝑝𝑡𝑒 < 𝑝𝑡2 and Δ𝑠 > 0
(𝐷𝑂𝐹)𝑣𝑖𝑏 = 2
(𝐷𝑂𝐹)𝑟𝑜𝑡 = 2
→ (𝐷𝑂𝐹)𝑎𝑣𝑎𝑖𝑙 = 7
Total-to-Static Pressure Ratio:
Nozzle Efficiency:
2(3𝑛 − 5) 𝑖𝑓 𝑙𝑖𝑛𝑒𝑎𝑟
(𝐷𝑂𝐹)𝑟𝑜𝑡 = {
2(3𝑛 − 6) 𝑖𝑓 𝑛𝑜𝑛𝑙𝑖𝑛𝑒𝑎𝑟
5 + 2(3𝑛 − 5) 𝑖𝑓 𝑙𝑖𝑛𝑒𝑎𝑟
→ (𝐷𝑂𝐹)𝑎𝑣𝑎𝑖𝑙 = {
6 + 2(3𝑛 − 6) 𝑖𝑓 𝑛𝑜𝑛𝑙𝑖𝑛𝑒𝑎𝑟
Thrust Coefficient:
𝑇
𝐶𝑇 =
𝑝𝑡2 ∙ 𝐴∗
Thrust:
𝑝𝑡2 ∙ 𝐴∗
) 𝐶𝑇
𝑇 = 𝑚̇𝑝 (
𝑚̇𝑝
C* “Characteristic Velocity”:
𝑝𝑡2 ∙ 𝐴∗
𝐶∗ =
𝑚̇𝑝
1
𝛾+1 2
∗
𝐶𝑖𝑑𝑒𝑎𝑙
𝑝𝑡2
2 𝛾−1
) ]
=
√𝑅𝑇𝑡2 [𝛾 (
𝑝𝑡1
𝛾+1
C* Efficiency:
𝜂𝐶 ∗ =
∗
𝐶𝑎𝑐𝑡𝑢𝑎𝑙
∗
𝐶𝑖𝑑𝑒𝑎𝑙
Specific Heat and Gas Constant Relations:
Specific Heat - Constant Volume:
Specific Impulse:
𝐼𝑆𝑃
Energy Stored in One Molecule:
1
𝑒 ′ = 𝑘𝑇(𝐷𝑂𝐹)𝑎𝑐𝑡𝑖𝑣𝑒
2
𝑅̃
8.314 𝐽⁄𝑚𝑜𝑙 ∙ 𝐾
(𝐵𝑜𝑙𝑡𝑧𝑚𝑎𝑛𝑛 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑘=
=
𝐴𝑣 6.022 × 1023 𝑝𝑎𝑟𝑡
𝑚𝑜𝑙
1
𝑒̃ = 𝐴𝑣 ∙ 𝑒 ′ = 𝑅̃𝑇 ∙ (𝐷𝑂𝐹)𝑎𝑐𝑡𝑖𝑣𝑒 (𝑝𝑒𝑟 𝑚𝑜𝑙𝑒)
2
1
𝑒 = 𝑒̃ ⁄𝑀𝑊 = 𝑅𝑇 ∙ (𝐷𝑂𝐹)𝑎𝑐𝑡𝑖𝑣𝑒 (𝑝𝑒𝑟 𝑚𝑎𝑠𝑠)
2
*(𝐷𝑂𝐹)𝑎𝑐𝑡𝑖𝑣𝑒 depends on temperature
- Low T = only trans. active
- Med T = trans. and rot. active
- High T = trans., rot., and vib. are active
1
= 𝐶 ∗ 𝐶𝑇
𝑔
Pressure Ratio:
𝑝2 (1 + 𝛾𝑀12 )
=
𝑝1 (1 + 𝛾𝑀22 )
Temperature Ratio:
𝑇2
1 + 𝛾𝑀12
𝑀2 2
)
=[
2] ∙ (
𝑇1
1 + 𝛾𝑀2
𝑀1
Total Temperature Ratio:
𝛾−1 2
2
1+
𝑀2
𝑇𝑡2
1 + 𝛾𝑀12
𝑀2 2
2
)
( )
=[
]∙(
2
𝛾−1 2
𝑇𝑡1
1
+
𝛾𝑀
𝑀1
2
1+
𝑀1
2
𝑇𝑡2
𝑞1,2
=
+1
𝑇𝑡1 𝐶𝑝 𝑇𝑡1
Total Pressure Ratio:
𝛾
𝛾 − 1 2 𝛾−1
1+
𝑀2
𝑝𝑡2
1 + 𝛾𝑀12
2
)
=[
]
∙(
𝛾
−
1
𝑝𝑡1
1 + 𝛾𝑀22
1+
𝑀12
2
Combustion Chamber-to-Throat Area Ratio:
1𝛾+1
𝐴𝐶
1
2
𝛾 − 1 2 2𝛾−1
{(
) [1 +
=
𝑀2 ]}
𝐴∗ 𝑀2 𝛾 + 1
2
Thermal Choking Criterion:
(1 + 𝛾𝑀12 )2
𝑇𝑡2
1
≥
𝑇𝑡1 2(1 + 𝛾) [1 + 𝛾 − 1 𝑀 2 ] 𝑀 2
1
1
2
*subtract 1 from result to get max allowable 𝑇𝑡2
1
𝐶̃𝑉 = 𝑅̃ (𝐷𝑂𝐹)𝑎𝑐𝑡𝑖𝑣𝑒
2
1
𝐶𝑉 = 𝑅 (𝐷𝑂𝐹)𝑎𝑐𝑡𝑖𝑣𝑒
2
Specific Heat - Constant Pressure:
1
𝐶̃𝑃 = 𝑅̃ [1 + (𝐷𝑂𝐹)𝑎𝑐𝑡𝑖𝑣𝑒 ]
2
1
𝐶𝑃 = 𝑅 [1 + (𝐷𝑂𝐹)𝑎𝑐𝑡𝑖𝑣𝑒 ]
2
Gas Constant Relations:
𝑅 = 𝐶𝑃 − 𝐶𝑉
𝑅̃ = 𝐶̃𝑃 − 𝐶̃𝑉
*given 𝐶𝑃 and 𝛾
𝑅 = 𝐶𝑃 (1 −
𝐶𝑉
1
𝛾−1
) = 𝐶𝑃 (1 − ) = 𝐶𝑃 (
)
𝐶𝑃
𝛾
𝛾
*given 𝐶𝑉 and 𝛾
𝐶𝑃
𝑅 = 𝐶𝑃 − 𝐶𝑣 = 𝐶𝑉 ( − 1) = 𝐶𝑉 (𝛾 − 1)
𝐶𝑉
Specific Heat Ratio:
2
𝛾 =1+
(𝐷𝑂𝐹)𝑎𝑐𝑡𝑖𝑣𝑒
Monatomic:
(𝐷𝑂𝐹)𝑎𝑐𝑡𝑖𝑣𝑒 = 3 (independent of temp.)
2
→𝛾 =1+
= 1.667
(3)
Diatomic:
- Medium temp
2
(𝐷𝑂𝐹)𝑎𝑐𝑡𝑖𝑣𝑒 = 5 → 𝛾 = 1 +
= 1.4
(5)
- High temp
2
(𝐷𝑂𝐹)𝑎𝑐𝑡𝑖𝑣𝑒 = 7 → 𝛾 = 1 +
= 1.286
(7)
𝛾
𝛾−1
(𝛾 − 1)𝑀𝑒2
𝑝𝑒
1
= {1 − [
]}
𝑝𝑡2
𝜂𝑁 2 + (𝛾 − 1)𝑀𝑒2
Polyatomic Molecules:
(𝐷𝑂𝐹)𝑡𝑟𝑎𝑛𝑠 = 3
2 𝑖𝑓 𝑙𝑖𝑛𝑒𝑎𝑟 (𝐶𝑂2 )
(𝐷𝑂𝐹)𝑣𝑖𝑏 = {
3 𝑖𝑓 𝑛𝑜𝑛𝑙𝑖𝑛𝑒𝑎𝑟 (𝑎𝑙𝑙 𝑜𝑡ℎ𝑒𝑟𝑠)
𝑉2
2 ∙ 𝐶𝑃
𝜂𝑁 =
𝛾 − 1 2 −1
𝑀𝑒 ]
2
𝛾−1
𝑝 𝛾
1−( 𝑒)
𝑝𝑡2
1 − [1 +
Area-Mach Relation:
𝐴𝑒
=
𝐴∗
1 𝛾+1
𝛾
1
2
𝛾 − 1 2 2 𝛾−1
1 𝛾 − 1 2 −𝛾−1
{(
) [1 +
𝑀𝑒 ]}
∙ {1 + (1 − )
𝑀𝑒 }
𝑀𝑒 𝛾 + 1
2
𝜂𝑁
2
*from resulting 𝑀𝑒 , get:
→ 𝑝𝑒 from total-to-static (above)
𝛾 − 1 2 −1
→ 𝑇𝑒 = 𝑇𝑡𝑒 [1 +
𝑀𝑒 ]
2
𝛾−1
)
→ 𝑎𝑒 = √𝛾𝑅𝑇𝑒 𝑤ℎ𝑒𝑟𝑒, 𝑅 = 𝐶𝑝 (
𝛾
→ 𝑉𝑒 = 𝑀𝑒 ∙ 𝑎𝑒
→ 𝑇 = 𝑚𝑝 𝑉𝑒 + (𝑝𝑒 − 𝑝∞ )𝐴𝑒
Non-Isentropic Thrust Coefficient:
𝛾+1
2
2 𝛾−1
𝑝𝑒
𝐶𝑇 = 𝛾 {𝜂𝑛 (
)(
)
[1 − ( )
𝛾−1 𝛾+1
𝑝𝑡2
𝛾−1
𝛾
1
2
]} +
(𝑝𝑒 − 𝑝∞ ) 𝐴𝑒
𝑝𝑡2
𝐴∗
Injectors and Droplet Vaporization
Droplet Characterization:
Droplet Diameter Probability:
1
𝑃𝑟𝑜𝑏 (𝐷 + 𝑑𝐷)
2
𝑑𝐷
*Sauter Mean Diameter (SMD): the droplet diameter with
same surface-to-volume ratio as the entire spray.
*Weber Number: ratio of inertia (mass) to surface tension
*Ohnesorge Number: ratio of friction to surface tension
𝑆𝑀𝐷
= 𝑐 ∙ 𝑊𝑒 𝑝 ∙ 𝑂ℎ𝑞
ℓ
ℓ
𝑊𝑒𝐶𝐶 𝑝 𝑂ℎ𝐶𝐶
(𝑆𝑀𝐷)𝐶𝐶 = (𝑆𝑀𝐷)𝑙𝑎𝑏 ∙ ( 𝐶𝐶 ) (
) (
)
ℓ𝑙𝑎𝑏 𝑊𝑒𝑙𝑎𝑏
𝑂ℎ𝑙𝑎𝑏
𝑃(𝐷) =
Droplet Vaporization:
Heat of Vaporization: (heat flux into droplet)
𝑚̇𝑉 ∙ Δℎ𝑉 = 𝑄̇𝑖𝑛
𝑤ℎ𝑒𝑟𝑒,
𝑑𝑇
̇
𝑄𝑖𝑛 = −𝑘 |
∙ 4𝜋𝑅2
𝑑𝑟 𝑅(𝑡)
𝑑𝑅
𝑚̇𝑉 = −4𝜋𝜌𝐿 𝑅2 (𝑡)
𝑑𝑡
Droplet Diameter: (as function of time)
8𝑘
) ln(1 + 𝐵) ∙ 𝑡
𝐷 2 (𝑡) = 𝐷0 − (
𝜌𝐿 𝐶𝑃
𝐶𝑃 (𝑇∞ − 𝑇𝑉 )
𝐵≡
ΔhV
Vaporization time:
𝜌𝐿 𝐶𝑃
𝐷02
𝑡𝑉 = 𝐷02
=
8𝑘 ln(1 + 𝐵) 𝑘𝑉