# rocket propulsion

```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:
π = πΜπ ππ + (ππ − π∞ )π΄π
*First term is jet (momentum) thrust, the second term is
pressure thrust.
π = πΜπ πππ
*Valid in all cases
Equivalent Velocity:
Total Relations:
2
ππ‘ 2
(πΎ + 1) π
=[
]
(πΎ − 1) π2 + 2
ππ‘ 1
ππ‘
πΎ−1 2
= [1 +
π ]
π
2
πΎ+1
]
2πΎπ 2 − (πΎ − 1)
πΎ
πΎ−1
π2 2πΎπ 2 − (πΎ − 1)
=
π1
πΎ+1
1
πΎ−1
π2 [2πΎπ 2 − (πΎ − 1)][(πΎ − 1)π2 + 2]
=
(πΎ + 1)2 π2
π1
Specific Impulse:
πΌπ π =
π22 =
1
πΎ+1 2
πΎ−1
2
)
πΜπ =
[πΎ (
πΎ+1
√π ππ‘
Thrust Coefficient:
π
(ππ )πππ‘π’ππ =
ππ‘ π΄∗
(πΎ − 1)π12 + 2
2πΎπ12 − (πΎ − 1)
πΎ+1
]
(ππ )πππππ
]} +
(ππ − π∞ ) π΄π
ππ‘
π΄∗
πΎ−1
πΎ
2
2 πΎ−1
ππ
)(
) [1 − ( )
= πΎ {(
πΎ−1 πΎ+1
ππ‘
1
2
]}
*Isentropic and fully expanded nozzle flow (ππ = π∞ )
1 πΎ+1
1
πΎ+1 2
(ππ )πππππ π£πππ’π’π
given Exit Mach number or solve numerically to
find Exit Mach from
1
2
πΎ+1
π΄
1
2
πΎ − 1 2 2 πΎ−1
) [1 +
= {(
π ]}
π΄∗ π πΎ + 1
2
π΄∗
πΎ−1
πΎ
Oblique Shock Jump Relations:
Isentropic Area-Mach Number Relation:
π΄π
(ππ )ππ πππ‘πππππ =
2
2 πΎ−1
ππ
)(
) [1 − ( )
πΎ {(
πΎ−1 πΎ+1
ππ‘
^This is valid for any choked nozzle.
*Find
πππ
π
=
π0 πΜπ π0
*Equivalent time that 1 lbf of combustion products could
produce 1 lbf of thrust, units of seconds
(πΎ + 1)π2
π2
=
π1 (πΎ − 1)π 2 + 2
πΜ(π₯) = πΜ∗ = π∗ π’∗π΄∗
(ππ − π∞ ) π΄π
π
=
πΜπ
πΜπ
*Would be the exit velocity from a fully expanded nozzle
that produces same thrust as actual nozzle.
Shock Jump Relations:
Mass Flow Rate:
ππ‘ π΄∗
[
πππ = ππ +
1
πΎ−1
ππ‘2
=1
ππ‘1
ππ‘
πΎ−1 2
= [1 +
π ]
π
2
ππ‘
πΎ−1 2
= [1 +
π ]
π
2
πΎ
πΎ−1
2
2 πΎ−1
)(
) }
= πΎ {(
πΎ−1 πΎ+1
π΄π
π΄∗
Angle-Mach Relations:
*Once ππ is known, can solve or ππ and ππ using
isentropic relations (above).
Nozzle Choking Criterion:
πΎ
πΎ−1
ππ‘
πΎ+1
)
=(
π∗
2
π22 sin2(π − πΏ) =
ππ‘ π ∗ = 1
Trajectories
Tsiolkovsky Rocket Equation (velocity increment):
π0
Δπππ’ππ = πππ ln ( ) − π0 π‘π
ππ
(πΎ − 1)π12 sin2(π) + 2
2πΎπ12 sin2(π) − (πΎ − 1)
Total Relations:
πΎ
ππ‘ 2
*Unchoked if,
ππ‘ 1
πΎ
ππ‘
πΎ + 1 πΎ−1
)
&lt;(
π∗
2
Exit Velocity:
ππ = ππ ππ
*Fully expanded flow in a vacuum (infinite nozzle length)
(πΎ + 1)π12
cot(πΏ) = tan(π) [
− 1]
2(π12 sin2(π) − 1)
=[
1
πΎ−1
(πΎ + 1) π12 sin2 (π) πΎ−1
πΎ+1
]
[
]
(πΎ − 1) π12 sin2 (π) + 2
2πΎπ12 sin2 (π) − (πΎ − 1)
Shock Jump Relations:
π2 2πΎπ12 sin2(π) − (πΎ − 1)
=
π1
πΎ+1
π€βπππ, ππ = √πΎ π ππ
πΎ−1
πΎ
2πΎ
ππ
ππ = {
π π [1 − ( )
πΎ−1 π‘
ππ‘
π2 [2πΎπ12 sin2(π) − (πΎ − 1)][(πΎ − 1)π12 sin2(π) + 2]
=
(πΎ + 1)2 π12 sin2(π)
π1
1
2
]}
(πΎ + 1)π12 sin2(π)
π2
=
π1 (πΎ − 1)π12 sin2(π) + 2
Where,
ππ½
(8314
)
πΜ
πππ β πΎ
=
ππ
ππ
*Therefore, lower ππ gives higher ππ → higher thrust
π=
Isentropic Model of Atmosphere:
Aerodynamic Drag:
1
π· = ππ 2 π΄πΆπ·
2
π
*Circular cross-section with diameter d has area π΄ = π 2
4
Gravitational Force:
ππ 2
)
π(π§) = ππ (
ππ + π§
*As a function of altitude π§ in π,
earth radius ππ = 6378 (103 ) π,
and acceleration of gravity at surface ππ = 9.8 π/π  2
Time and Altitude at Burnout:
πΎ
π(π§)
πΎ − 1 π§ πΎ−1
( ∗ )]
≈ [1 −
ππ
2
π§
*Where surface pressure, ππ  = 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)
βπ = πππ {1 −
π
π
*First term is dependent on Mass Ratio π = ( 0 )
π2
ππ
∗
π§ = 8404 π
π‘π = (
Summerfeld Separation Criterion:
When nozzle flow is “highly” over expanded, when
ln(π)
1
} π‘ − π π‘2
π−1 π 2 π π
ππ
)
πΜπ
π‘π = (π‘ − π‘0 )
ππ
π∞
gets too small, flow in nozzle separates when,
ππ
β²πΎ
π∞
for 0.25 ≤ πΎ ≤ 0.4
*Boundary layer separation can cause highly turbulent
recirculating flow
Maximum Vehicle Altitude (vertical launch):
1 (Δππ )2
βπππ₯ = βπ +
2 ππ
βπππ₯ =
πππ2 (ln(π))2
π
− πππ π‘π {
ln(π) − 1}
2ππ
π−1
*Minimizing π‘π maximizes βπππ₯
Rocket Staging
Staging Methods:
Velocity Increments for an Optimally Staged Rocket:
Δπ due to burnout of the π th-stage,
(ππ )π = (πππ ) ln(ππ ) − π0 (π‘π )π
π
Final Δπ imparted on final payload ππΏ of stage π,
π
Circular Orbits: π(π) = π = ππππ π‘πππ‘, π &lt; 0, π = 0
Velocity:
π
π=√
π
Period:
ππ = ∑(π‘π )π
π3
π = 2π√
π
π=1
π
1 + ππ
(ππ )π = ∑ [(πππ ) ln (
)] − ππ ππ
π
ππ + ππ
π=1
(Δππ )π = πππ π ln {π + (1 − π) [
1 −1
ππΏ π
π0
] }
Escape from Circular Orbit:
π
Δπ1 ≥ (√2 − 1)√
π
− ππ ππ
Ideal Case (infinitely many stages):
(Δππ )π→∞ = πππ (1 − π) ln [
*Equivalent Velocity πππ
=
π
πΜπ
π0
] − ππ ππ
ππΏ
Parabolic Trajectories: (not orbit) π = 0, π = ∞, π = 1
π∞ = 0
Escape Velocity:
2π
πππ π = √
π
≈ same for all stages
Escape from planet surface:
Orbital Dynamics
2π
πππ π = √
πππππππ‘
Gravitational Parameter:
π β π2
π€βπππ πΊ = 6.67 &times; 10−11
ππ
π = πΊπ ′
Radial Position (from vehicle to center of planet):
π(1 − π 2 )
1 + π cos(π)
π=
Semimajor Axis:
π = ππππ
1
1−π
π=
1
(π + ππ )
2 π
Eccentricity:
π=
ππ − ππ
ππ + ππ
π = √1 −
π2
π2
π =1−
ππππ
π
Circularization Burn:
Circularize at ππ :
2ππ
π
Δπ1 = √ β [1 − √
]
ππ
ππ + ππ
Circularize at ππ :
Vis-Viva:
Stage Mass Relation: π-stage vehicle, the π th-stage has,
π0π = πππ + πππ + ππΏπ
π0(π+1)
ππΏπ
ππΏπ
ππ ≡
=
=
π0π − π0(π+1) π0π − ππΏπ πππ + πππ
*Equal for all similar stages, ππ = π(π+1) = β― = ππ
ππππ‘ππππ =
(ππΏ⁄π )
2 1
π = π( − )
π π
2
0
1⁄
π
Structural Coefficient:
πππ
πππ
πππ
ππ =
=
=
π0π − π0(π+1) π0π − ππΏπ πππ + πππ
*Equal for all similar stages, ππ = π(π+1) = β― = ππ
π0π
π0π
1 + ππ
=
=
π0π − πππ πππ + ππΏπ ππ + ππ
Inclination Change: (circular orbits)
ππ = π(1 − π)
Perigee Velocity:
π 1+π
)
ππ = √ (
π 1−π
π 2ππ
ππ = √
ππ (ππ + ππ )
1
*Both other angles are: π − π
2
π
π
Δπ1 = 2√ sin ( )
π
2
ππ = π(1 + π)
Apogee Velocity:
π 1−π
)
ππ = √ (
π 1+π
2ππ
π
ππ = √
ππ (ππ + ππ )
Orbital Period:
π ππ
ππ = √
π ππ
Orbital Rendezvous:
πΌπΏ =
π (π1 + π2 )3
√
2
2(π2)3
Transfer Time:
π3
π = 2π√
π
Mass Ratio:
ππ =
π
2ππ
Δπ1 = √ β [1 − √
]
ππ
ππ + ππ
1⁄
π
π
(1 − πΏ⁄π )
0
Δ = −π√π 2 − 1
Orbital Maneuvers
*Where b = semi-minor axis
π &lt; 1 → ππππππ π
π = 1 → ππππππππ
π &gt; 1 → βπ¦πππππππ
π = 0 → ππππππ
*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) π &gt; 0
Excess Velocity:
π
π∞ = √2π
π∞ = √
−π
Asymptote Angle:
1
π∞ = π − cos −1 ( )
π
Turning Angle:
1
πΏ = 2 sin−1 ( )
π
Miss Distance:
πππ = ππ»πβππππ =
π (π1 + π2 )3
√
2
2π
Intercept Opportunity Times:
Energy (per unit mass):
π=−
π
2π
1
1
1
=
−
π (ππΆ )1 (ππΆ )2
ππ
π€βπππ (ππΆ )π = 2π√
π
Orbital Transfers
Standard Gravitation Parameters
Hohmann Transfer:
Combustion Tap-Off Systems:
3
Circular Orbit Velocities:
π
(ππΆ )1 = √
π1
π
(ππΆ )2 = √
π2
Transfer Orbit Parameters: (Trans time, see Rendezvous)
ππ = π1
ππ = π2
π 2π2
ππ = √
π1 π1 + π2
Body
π
π = πΊπ ′ [ 2 ]
π
Sun
1.327 &times; 1020
Mercury
2.203 &times; 1013
Venus
3.249 &times; 1014
Earth
3.986 &times; 1014
(Moon)
4.903 &times; 1012
Mars
4.283 &times; 1013
Jupiter
1.267 &times; 1017
Saturn
3.793 &times; 1016
Uranus
5.794 &times; 1015
Neptune
Pluto
6.837 &times; 1015
1.108 &times; 1012
Propellant Feed Systems
Pressure-Fed Systems:
π 2π1
ππ = √
π2 π1 + π2
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 πΜπ at nozzle exit
*Moderate/high thrust (Saturn I-B/Blue Origin)
Gas Generator Systems:
π
2π2
Δπ1 = √ [√
− 1]
π1
π1 + π2
π
2π1
Δπ2 = √ [1 − √
]
π2
π1 + π2
Bielliptic Transfer:
Deliver Pressure &gt; Tank Pressure,
πβ = π(π + π)β
Circular Orbit Velocities:
π
(ππΆ )1 = √
π1
π
(ππΆ )2 = √
π2
Transfer Orbit Parameters: (πππ can be chosen)
π1 = ππππ1
*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:
πππ = ππππ1
π2 = ππππ2
πππ = ππππ2
1
π ππ1 = (π1 + πππ )
2
1
π ππ2 = (π2 + πππ )
2
Expander Systems:
Transit Time:
πππ = π
√(π ππ1 )3 + √(π ππ2)3
√π
Velocity Increments:
2
1
π
)−√
Δπ1 = √π ( −
π1 π ππ1
π1
2
1
2
1
) − √π (
)
Δπ2 = √π (
−
−
πππ π ππ2
πππ π ππ1
π
2
1
)
Δπ3 = √ − √π ( −
π2
π2 π ππ2
*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 [πΆ1 π»1.96 ]
- ethyl alcohol – [πΆ2 π»5 ππ»]
c) Hypergolic (πΈπ ≤ 0) (reacts on contact with oxi.)
- hydrazine [π2 π»4 ]
- 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:
π
π
1 πΆπ π»π + (π + ) π2 → ππΆπ2 + π»2 π
4
2
π = ππππππ ππ‘πππ , π = βπ¦ππππππ ππ‘πππ
Application of 1st Law:
(Δβπ‘ )1,2 = π1,2 − π€1,2 = 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)
- 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.
→ βπ‘2 − βπ‘1 = 0
→ πΆπ (ππ‘2 − ππ‘1 ) = 0
Mixture Ratio:
ππ
π = ( 2)
ππ
∴ ππ‘2 = ππ‘1
Turbine Work:
Stoichiometric Mixture Ratio:
32π + 8π
ππ  =
12π + π
Equivalence Ratio:
ππ
π=
π
π = 1 → π π‘πππβπππππ‘πππ πππππ’π π‘πππ
π &gt; 1 → ππ’ππ − πππβ πππππ’π π‘πππ
π &lt; 1 → ππ’ππ − ππππ πππππ’π π‘πππ
Molar Enthalpy of Combustion:
ΔβΜπΆ = [∑ π΅πππ πΈπππππ¦ ]
π
− [∑ π΅πππ πΈπππππ¦ ]
πππππ‘
π
ππππ
ΔβΜπΆ &lt; 0 → ππ₯ππ‘βπππππ
ΔβΜπΆ &gt; 0 → πππππ‘βπππππ
Convert to mass-specific enthalpy:
ΔβΜπΆ
ΔβπΆ =
πππ»πΆ
πππ»πΆ = 12π + 1π [
ππππ
]
π
Convert to [ππ½⁄ππ]:
ΔβπΆ [
ππππ
1000 πππ 1000π 4.1814 π½
1 ππ½
)(
)(
)(
)
]β(
π
ππππ
ππ
πππ
1000π½
Turbopumps and Turbines
Turbopumps:
Shaft Power from Turbine:
πΜ = π β ΩΜ
Ideal Pump Work:
|π€|πππππ =
1
|π|ππππππππ = π π Δππ‘
π
Temperature Change Across Turbine:
Δππ‘
Δπ = (1 − π π )
π β ππ
Turbine Work: (per unit mass)
|π€π | = |πΆπ (ππ‘π − ππ‘π )|
Turbine Power:
πΜπ = πΜ|π€π |
Turbine Stage Efficiency:
|π€|
ππ =
|π€|π
Total Turbine Efficiency:
π
1 − ( π‘3 )
ππ‘1
ππ =
πΎ−1
ππ‘3 πΎ
1−( )
ππ‘1
Turbine Total Temperature Ratio:
1
(π − ππ‘1 )
π π‘2
Ideal Pump Power:
1
|π€Μ |πππππ = πΜ|π€|πππππ = πΜ (ππ‘2 − ππ‘1 )
π
Pump Efficiency:
|π€|πππππ
|π€Μ |πππππ
ππ =
=
|π€|πππ‘π’ππ |π€Μ |πππ‘π’ππ
Non-Ideal Pump Work:
11
(π − ππ‘1 )
|π€|πππ‘π’ππ =
ππ π π‘2
Non-Ideal Pump Power:
1 1
(π − ππ‘1 )
|π€Μ |πππ‘π’ππ = πΜ|π€|πππ‘π’ππ = πΜ
ππ π π‘2
*ππππ π‘βπ ππππ£π
π
Δππ‘ = (ππ‘2 − ππ‘1 ) = π€π ππ
πΜ
1st Law for Liquid Flow Through Pump:
1
1
ππ = (π2 − π1 ) + (π2 − π1 ) + (π22 − π12 ) = |π€|
π
2
1
ππ = (π2 − π1 ) + (ππ‘2 − ππ‘1 ) = |π€|
π
Temperature Change Across Pump:
1 − ππ (ππ‘2 − ππ‘1 )
(π2 − π1 ) = (
)
ππ
π β ππ
πΎ−1
πΎ
ππ‘3
ππ‘3
= 1 − π π [1 − ( )
ππ‘1
ππ‘1
]
Turbine Total Pressure Ratio:
πΎ
ππ‘3
1
ππ‘3 πΎ−1
= [1 − (1 − )]
ππ‘1
ππ
ππ‘1
Total-to-Static Temperature Relation:
πΎ − 1 2 −1
π3 = ππ‘3 [1 +
π3 ]
2
Total-to-Static Pressure Relation:
πΎ
πΎ − 1 2 −πΎ−1
π3 ]
2
Entropy Change Across Turbine:
ππ‘3
ππ‘3
Δπ  = π 3 − π 1 = πΆπ ln ( ) − π ln ( )
ππ‘1
ππ‘1
πΎ−1
)
π = πΆπ (
πΎ
π = πΆπ£ (πΎ − 1)
Temperature-Velocity Relation:
π2
(ππ‘ − π) =
2 β πΆπ
π3 = ππ‘3 [1 +
Nozzle Design
Simple Conical Nozzles:
Axial Thrust:
1 + cos(πΌ)
πΜπ ππ + (ππ − π∞ )π΄π
2
*See notes for Rao and TOP nozzle stuff
πππ₯πππ =
Combustion Chamber Performance
DOF and Molecular Energy
Non-Isentropic Nozzle Flow
(π·ππΉ)πππ‘ = 0
Application of 1st Law:
(Δβπ‘ )2,π = π2,π − π€2,π = 0
→ βπ‘2 − βπ‘π = 0
→ πΆπ (ππ‘2 − ππ‘π ) = 0
∴ ππ‘2 = ππ‘π
→ (π·ππΉ)ππ£πππ = 3
Temperature-Velocity Relation:
Monatomic Molecules:
(π·ππΉ)π‘ππππ  = 3
Combustion Chambers:
(π·ππΉ)π£ππ = 0
Diatomic Molecules:
(π·ππΉ)π‘ππππ  = 3
(ππ‘ − π) =
Total Pressure Ratio:
Δπ
ππ‘π
= π− π
ππ‘2
*where, ππ‘π &lt; ππ‘2 and Δπ  &gt; 0
(π·ππΉ)π£ππ = 2
(π·ππΉ)πππ‘ = 2
→ (π·ππΉ)ππ£πππ = 7
Total-to-Static Pressure Ratio:
Nozzle Efficiency:
2(3π − 5) ππ ππππππ
(π·ππΉ)πππ‘ = {
2(3π − 6) ππ πππππππππ
5 + 2(3π − 5) ππ ππππππ
→ (π·ππΉ)ππ£πππ = {
6 + 2(3π − 6) ππ πππππππππ
Thrust Coefficient:
π
πΆπ =
ππ‘2 β π΄∗
Thrust:
ππ‘2 β π΄∗
) πΆπ
π = πΜπ (
πΜπ
C* “Characteristic Velocity”:
ππ‘2 β π΄∗
πΆ∗ =
πΜπ
1
πΎ+1 2
∗
πΆπππππ
ππ‘2
2 πΎ−1
) ]
=
√πππ‘2 [πΎ (
ππ‘1
πΎ+1
C* Efficiency:
ππΆ ∗ =
∗
πΆπππ‘π’ππ
∗
πΆπππππ
Specific Heat and Gas Constant Relations:
Specific Heat - Constant Volume:
Specific Impulse:
πΌππ
Energy Stored in One Molecule:
1
π ′ = ππ(π·ππΉ)πππ‘ππ£π
2
πΜ
8.314 π½⁄πππ β πΎ
(π΅πππ‘π§ππππ πΆπππ π‘πππ‘)
π=
=
π΄π£ 6.022 &times; 1023 ππππ‘
πππ
1
πΜ = π΄π£ β π ′ = πΜπ β (π·ππΉ)πππ‘ππ£π (πππ ππππ)
2
1
π = πΜ ⁄ππ = ππ β (π·ππΉ)πππ‘ππ£π (πππ πππ π )
2
*(π·ππΉ)πππ‘ππ£π depends on temperature
- Low T = only trans. active
- Med T = trans. and rot. active
- High T = trans., rot., and vib. are active
1
= πΆ ∗ πΆπ
π
Pressure Ratio:
π2 (1 + πΎπ12 )
=
π1 (1 + πΎπ22 )
Temperature Ratio:
π2
1 + πΎπ12
π2 2
)
=[
2] β (
π1
1 + πΎπ2
π1
Total Temperature Ratio:
πΎ−1 2
2
1+
π2
ππ‘2
1 + πΎπ12
π2 2
2
)
( )
=[
]β(
2
πΎ−1 2
ππ‘1
1
+
πΎπ
π1
2
1+
π1
2
ππ‘2
π1,2
=
+1
ππ‘1 πΆπ ππ‘1
Total Pressure Ratio:
πΎ
πΎ − 1 2 πΎ−1
1+
π2
ππ‘2
1 + πΎπ12
2
)
=[
]
β(
πΎ
−
1
ππ‘1
1 + πΎπ22
1+
π12
2
Combustion Chamber-to-Throat Area Ratio:
1πΎ+1
π΄πΆ
1
2
πΎ − 1 2 2πΎ−1
{(
) [1 +
=
π2 ]}
π΄∗ π2 πΎ + 1
2
Thermal Choking Criterion:
(1 + πΎπ12 )2
ππ‘2
1
≥
ππ‘1 2(1 + πΎ) [1 + πΎ − 1 π 2 ] π 2
1
1
2
*subtract 1 from result to get max allowable ππ‘2
1
πΆΜπ = πΜ (π·ππΉ)πππ‘ππ£π
2
1
πΆπ = π (π·ππΉ)πππ‘ππ£π
2
Specific Heat - Constant Pressure:
1
πΆΜπ = πΜ [1 + (π·ππΉ)πππ‘ππ£π ]
2
1
πΆπ = π [1 + (π·ππΉ)πππ‘ππ£π ]
2
Gas Constant Relations:
π = πΆπ − πΆπ
πΜ = πΆΜπ − πΆΜπ
*given πΆπ and πΎ
π = πΆπ (1 −
πΆπ
1
πΎ−1
) = πΆπ (1 − ) = πΆπ (
)
πΆπ
πΎ
πΎ
*given πΆπ and πΎ
πΆπ
π = πΆπ − πΆπ£ = πΆπ ( − 1) = πΆπ (πΎ − 1)
πΆπ
Specific Heat Ratio:
2
πΎ =1+
(π·ππΉ)πππ‘ππ£π
Monatomic:
(π·ππΉ)πππ‘ππ£π = 3 (independent of temp.)
2
→πΎ =1+
= 1.667
(3)
Diatomic:
- Medium temp
2
(π·ππΉ)πππ‘ππ£π = 5 → πΎ = 1 +
= 1.4
(5)
- High temp
2
(π·ππΉ)πππ‘ππ£π = 7 → πΎ = 1 +
= 1.286
(7)
πΎ
πΎ−1
(πΎ − 1)ππ2
ππ
1
= {1 − [
]}
ππ‘2
ππ 2 + (πΎ − 1)ππ2
Polyatomic Molecules:
(π·ππΉ)π‘ππππ  = 3
2 ππ ππππππ (πΆπ2 )
(π·ππΉ)π£ππ = {
3 ππ πππππππππ (πππ ππ‘βπππ )
π2
2 β πΆπ
ππ =
πΎ − 1 2 −1
ππ ]
2
πΎ−1
π πΎ
1−( π)
ππ‘2
1 − [1 +
Area-Mach Relation:
π΄π
=
π΄∗
1 πΎ+1
πΎ
1
2
πΎ − 1 2 2 πΎ−1
1 πΎ − 1 2 −πΎ−1
{(
) [1 +
ππ ]}
β {1 + (1 − )
ππ }
ππ πΎ + 1
2
ππ
2
*from resulting ππ , get:
→ ππ from total-to-static (above)
πΎ − 1 2 −1
→ ππ = ππ‘π [1 +
ππ ]
2
πΎ−1
)
→ ππ = √πΎπππ π€βπππ, π = πΆπ (
πΎ
→ ππ = ππ β ππ
→ π = ππ ππ + (ππ − π∞ )π΄π
Non-Isentropic Thrust Coefficient:
πΎ+1
2
2 πΎ−1
ππ
πΆπ = πΎ {ππ (
)(
)
[1 − ( )
πΎ−1 πΎ+1
ππ‘2
πΎ−1
πΎ
1
2
]} +
(ππ − π∞ ) π΄π
ππ‘2
π΄∗
Injectors and Droplet Vaporization
Droplet Characterization:
Droplet Diameter Probability:
1
ππππ (π· + ππ·)
2
ππ·
*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
πππ·
= π β ππ π β πβπ
β
β
πππΆπΆ π πβπΆπΆ
(πππ·)πΆπΆ = (πππ·)πππ β ( πΆπΆ ) (
) (
)
βπππ πππππ
πβπππ
π(π·) =
Droplet Vaporization:
Heat of Vaporization: (heat flux into droplet)
πΜπ β Δβπ = πΜππ
π€βπππ,
ππ
Μ
πππ = −π |
β 4ππ2
ππ π(π‘)
ππ
πΜπ = −4πππΏ π2 (π‘)
ππ‘
Droplet Diameter: (as function of time)
8π
) ln(1 + π΅) β π‘
π· 2 (π‘) = π·0 − (
ππΏ πΆπ
πΆπ (π∞ − ππ )
π΅≡
ΔhV
Vaporization time:
ππΏ πΆπ
π·02
π‘π = π·02
=
8π ln(1 + π΅) ππ
```