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Equation Sheet for AAE 339
Thrust 𝐹𝐹 = 𝑚𝑚̇ (^) 𝑒𝑒 𝑢𝑢𝑒𝑒 − 𝑚𝑚̇ (^) 𝑖𝑖 𝑢𝑢𝑖𝑖 + (𝑝𝑝𝑒𝑒 − 𝑝𝑝𝑎𝑎)𝐴𝐴𝑒𝑒 Specific thrust = (^) 𝑚𝑚̇𝐹𝐹𝑎𝑎
Thrust specific fuel consumption = 𝑚𝑚̇𝐹𝐹𝑓𝑓
Conservation of Energy 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 =^ 𝑄𝑄̇ − 𝑊𝑊̇^ +^ � 𝑚𝑚̇^ 𝑖𝑖𝑖𝑖
Isentropic Relations for Ideal Gas T 2 T 1 =^ �
p 2 p 1 �
𝛾𝛾− 𝛾𝛾 (^) = �𝜌𝜌 2 𝜌𝜌 1 �
𝛾𝛾−
Definitions
𝑀𝑀 =
Ideal Gas Relationships
𝑐𝑐𝑝𝑝 = (^) 𝛾𝛾−1𝛾𝛾𝛾𝛾
𝑝𝑝 = 𝜌𝜌𝑅𝑅𝜌𝜌 𝑎𝑎 = �𝛾𝛾𝑅𝑅𝜌𝜌
Stagnation Properties
𝑇𝑇 = 1 +^
𝛾𝛾− 2 𝑀𝑀^
2
𝑝𝑝 0 𝑝𝑝 =^ �^
𝑇𝑇 0 𝑇𝑇 �
𝛾𝛾 𝛾𝛾−1 (^) = �1 + 𝛾𝛾− 2 𝑀𝑀^
𝛾𝛾 𝛾𝛾−
ℎ = 𝑐𝑐𝑝𝑝𝜌𝜌
Gas Mixtures m = Σ m (^) i n = Σ ni
M y M m mf m i i = i = i
n y ni i =
=
k i n yi^ Mi M m 1
M
R = R
V
V
p y pi i i = =
=
k i
u mfi ui 1
k i i^ i
u yu 1
=
k i i^ i
h mfh 1
=
k i
h yi hi 1
k v (^) i i vi c yc 1 ,
=
k p (^) i i pi c yc 1 ,
Efficiency Relationships ηt =
1 2 (𝑚𝑚̇𝑒𝑒𝑜𝑜𝑒𝑒^2 −𝑚𝑚̇𝑎𝑎𝑜𝑜𝑎𝑎^2 ) 𝑚𝑚̇𝑓𝑓 𝑄𝑄 (^) 𝑅𝑅 𝜂𝜂𝑃𝑃 = 1 𝑇𝑇𝑜𝑜𝑎𝑎 2 (𝑚𝑚̇^ 𝑒𝑒𝑜𝑜𝑒𝑒^2 −𝑚𝑚̇𝑎𝑎𝑜𝑜𝑎𝑎^2 ) 𝜂𝜂 0 = 𝜂𝜂𝑜𝑜𝜂𝜂𝑝𝑝
State Definition
Isentropic Efficiencies 02 02
d^ s^ a a
h h h h
03 02 03 02
c h^ s h h h
04 05 04 05
t s
h h h h
η = − − 05 7 05 7
n s
h h h h
η = − −
Turbofan
β = ṁṁ^ b (^) a
𝑚𝑚̇𝑜𝑜𝑜𝑜𝑜𝑜𝑎𝑎𝑡𝑡 = 𝑚𝑚̇ (^) 𝑎𝑎 + 𝑚𝑚̇ (^) 𝑏𝑏
Pressure ratios
𝑂𝑂𝑃𝑃𝑅𝑅 = 𝑝𝑝𝑝𝑝^0302
Propellers
𝑢𝑢𝑟𝑟𝑒𝑒𝑡𝑡 = �𝑢𝑢𝑎𝑎^2 + 𝑢𝑢𝑜𝑜^2 𝑢𝑢𝑜𝑜 = Ω𝑟𝑟 𝐽𝐽 = (^) 𝑖𝑖𝑜𝑜𝑛𝑛𝑎𝑎 𝑐𝑐𝐹𝐹 = (^) 𝜌𝜌𝑖𝑖𝑇𝑇 2 𝑝𝑝𝑛𝑛 4 𝑐𝑐𝑃𝑃 = 𝑤𝑤̇𝜌𝜌𝑖𝑖𝑠𝑠ℎ𝑎𝑎 (^3) 𝑛𝑛𝑓𝑓 5 𝑎𝑎 𝜂𝜂𝑝𝑝𝑟𝑟 = (^) 𝑤𝑤̇𝑇𝑇𝑠𝑠ℎ𝑎𝑎𝑝𝑝𝑜𝑜𝑎𝑎𝑓𝑓𝑎𝑎 = 𝑐𝑐𝑐𝑐 (^) 𝑃𝑃^ 𝐹𝐹 J n [=] rev/s 𝜌𝜌 = 𝐴𝐴𝑑𝑑𝑖𝑖𝑑𝑑𝑑𝑑(𝑝𝑝 2 − 𝑝𝑝 1 ) 𝜌𝜌 = 𝐴𝐴𝑑𝑑𝑖𝑖𝑑𝑑𝑑𝑑^12 𝜌𝜌(𝑢𝑢𝑒𝑒^2 − 𝑢𝑢𝑎𝑎^2 ) 𝑢𝑢𝑑𝑑𝑖𝑖𝑑𝑑𝑑𝑑 = (𝑜𝑜𝑒𝑒^ + 2 𝑜𝑜𝑎𝑎) 𝑜𝑜𝑑𝑑𝑑𝑑𝑠𝑠𝑑𝑑 𝑜𝑜𝑎𝑎^ =^
1 2 �^
𝑇𝑇𝑝𝑝 𝐴𝐴𝑑𝑑𝑑𝑑𝑠𝑠𝑑𝑑𝑜𝑜𝑎𝑎^2 𝜌𝜌 2 + 1�
(^1) � 2
2 𝜌𝜌𝑢𝑢𝑎𝑎^ ��^
𝐴𝐴𝑑𝑑𝑖𝑖𝑑𝑑𝑑𝑑 𝜌𝜌𝑢𝑢𝑎𝑎^2 + 1�
1 2
1 (^2) + 1�
1 2 𝜌𝜌𝑢𝑢𝑎𝑎^ ��^
2𝑇𝑇 𝐴𝐴 (^) 𝑑𝑑𝑑𝑑𝑠𝑠𝑑𝑑 𝜌𝜌𝑜𝑜𝑎𝑎^2 + 1�
1 (^2) + 1� = 𝜂𝜂 𝑝𝑝𝑟𝑟𝜂𝜂𝑏𝑏𝑏𝑏𝑃𝑃𝑑𝑑ℎ𝑎𝑎𝑓𝑓𝑜𝑜
Turboprop T = 𝑚𝑚̇(𝑢𝑢𝑒𝑒 − 𝑢𝑢𝑎𝑎) + 𝜂𝜂^ 𝑝𝑝𝑝𝑝^ 𝜂𝜂 𝑜𝑜^ 𝑏𝑏𝑏𝑏𝑎𝑎^ 𝜂𝜂^ 𝑝𝑝𝑎𝑎𝛼𝛼 𝑚𝑚̇Δℎ
Rockets 𝜀𝜀 = 𝐴𝐴𝑒𝑒/𝐴𝐴𝑜𝑜 𝜆𝜆 = (^) 𝑚𝑚𝑝𝑝𝑚𝑚 +𝑚𝑚𝑝𝑝𝑑𝑑
𝑀𝑀𝑅𝑅 = 𝑚𝑚𝑚𝑚^0 𝑓𝑓 = 𝑚𝑚𝑝𝑝𝑝𝑝 𝑚𝑚^ +𝑚𝑚𝑝𝑝𝑝𝑝 +𝑚𝑚𝑝𝑝^ +𝑚𝑚𝑑𝑑 𝑑𝑑
𝑚𝑚 (^) 𝑝𝑝 = 𝑚𝑚 (^) 𝑝𝑝𝑡𝑡𝑀𝑀𝛾𝛾−^ 𝑀𝑀(𝑀𝑀𝛾𝛾𝛾𝛾−1−1)/𝜆𝜆
𝑚𝑚 (^) 𝑖𝑖 = 𝑚𝑚 (^) 𝑝𝑝^ (1−𝜆𝜆 𝜆𝜆) 𝐼𝐼𝑑𝑑𝑝𝑝 = 𝐹𝐹/𝑚𝑚̇𝑔𝑔 𝐼𝐼𝑑𝑑𝑝𝑝 = 𝐼𝐼/𝑚𝑚 (^) 𝑝𝑝 𝐼𝐼𝑑𝑑𝑝𝑝 = 𝑐𝑐𝑓𝑓 𝑐𝑐 ∗/𝑔𝑔
𝐼𝐼𝑑𝑑𝑝𝑝 = 𝐼𝐼𝑑𝑑𝑝𝑝,𝑣𝑣 − 𝑝𝑝𝑚𝑚̇𝑎𝑎^ 𝐴𝐴𝑏𝑏𝑒𝑒
𝑐𝑐 ∗^ = � 𝛾𝛾
�𝑇𝑇𝑑𝑑 𝑀𝑀𝑀𝑀(𝛾𝛾) �^
2 𝛾𝛾+1�
−(𝛾𝛾+1) 2 (𝛾𝛾−1)
𝑚𝑚̇ = 𝑝𝑝𝑑𝑑 𝑐𝑐𝐴𝐴 ∗𝑎𝑎 𝐹𝐹 = 𝑚𝑚̇𝑣𝑣𝑒𝑒 + (𝑝𝑝𝑒𝑒 − 𝑝𝑝𝑎𝑎)𝐴𝐴𝑒𝑒
𝑐𝑐𝑓𝑓 = (^) 𝑝𝑝𝑑𝑑𝐹𝐹𝐴𝐴𝑎𝑎 = � 2𝛾𝛾^
2 𝛾𝛾−1 �^
2 𝛾𝛾+1�
𝛾𝛾+ 𝛾𝛾−1 (^) � 1 −
𝛾𝛾− 𝛾𝛾 (^) ��
1 2
- �𝑝𝑝𝑝𝑝𝑒𝑒𝑑𝑑 − 𝑝𝑝𝑝𝑝𝑎𝑎𝑑𝑑 � 𝜀𝜀
𝑐𝑐𝑓𝑓 = 𝑐𝑐𝑓𝑓,𝑣𝑣𝑎𝑎𝑐𝑐 − 𝑝𝑝𝑝𝑝𝑎𝑎𝑑𝑑 𝜀𝜀
Δ𝑣𝑣 = 𝑔𝑔𝑒𝑒 𝐼𝐼𝑑𝑑𝑝𝑝𝑙𝑙𝑙𝑙 �𝑚𝑚𝑚𝑚𝑜𝑜𝑓𝑓 � − ∫𝑜𝑜 𝑜𝑜 𝑏𝑏𝑔𝑔𝑑𝑑𝑑𝑑− ∫ 0 𝑜𝑜𝑏𝑏� 𝑚𝑚𝑛𝑛� 𝑑𝑑𝑑𝑑
MASS AND DENSITY 1 kg = 2.2046 lb (^) m 1 g/cm 3 = 1000 kg/m 3 1 g/cm 3 = 62.4281b (^) m/ft^3
LENGTH 1 cm = 0.3937 in 1 m = 3.2808 ft
VELOCITY 1 km/h = 0.62137 mile/h
VOLUME 1 cm 3 = 0.061024 in 3 1 m 3 = 35.315 ft^3 1 L = l0 –3^ m 3 1 L = 0.0353 ft^3
FORCE 1 N = 1 kg m/s^2 1 N = 0.22481 lb (^) f
PRESSURE 1 Pa = 1 N/m 2 = 1.4504 × 10–4^ lb (^) f /in 2 1 bar = 10 5 N/m 2 1 atm = 1.01325 bar
ENERGY AND SPECIFIC ENERGY 1 J = 1 N m = 0.73756 ft⋅lb (^) f 1 kJ = 737.56 ft lb (^) f 1 kJ = 0.9478 Btu 1 kJ/kg = 0.42992 Btu/lb (^) m
ENERGY TRANSFER RATE 1 W = 1 J/s = 3.413 Btu/h 1 kW = 1.34l hp
SPECIFIC HEAT 1 kJ/kg K = 0.238846 Btu/lbm⋅°R 1 kcal/kg K = 1 Btu/lb (^) m⋅°R
UNIVERSAL GAS CONSTANT 𝑅𝑅̄ = 8.314 kJ/kmol⋅K 𝑅𝑅̄ =1545 ft lb f /lbmol⋅°R 𝑅𝑅̄ =1.986 Btu/lbmol⋅°R
STD ACCELERATION OF GRAVITY g = 9.80665 m/s^2 g = 32.174 ft/s^2
g c = 32.1740 (^2) f
m lb s
lb ft ⋅
N s^2
kg m ⋅
1 lb (^) m = 0.4536 kg 1 lb (^) m = 0.031081 slug 1 lb (^) m/ft^3 = 0.016018 g/cm 3 1 lb (^) m/ft^3 = 16.018 kg/m 3 1 in = 2.54 cm 1 ft = 0.3048 m 1 mile/h = 1.6093 km/h 1 mile/h = 1.4667 ft/s 1 in 3 = 16.387 cm 3 1 ft^3 = 0.028317 m 3 1 gal = 0.13368 ft^3 1 gal = 3.7854 × l0 –3^ m 3
1 lb (^) f = 1 slug ft/s^2 1 lb (^) f = 32.174 lb (^) m ft/s^2 1 lb (^) f = 4.4482 N 1 lb (^) f /in 2 = 6894.8 Pa 1 lb (^) f /in 2 = 144 lb (^) f /ft^2 1 atm = 14.696 lb (^) f /in 2
1 ft⋅lb (^) f = 1.35582 J 1 Btu = 778.17 ft⋅lb (^) f 1 Btu = 1.0551 kJ 1 Btu/lb (^) m = 2.326 kJ/kg 1 kcal = 4.1868 kJ 1 Btu/h = 0.293 W 1 hp = 2545 Btu/h 1 hp = 550 ft lb (^) f /s 1 hp = 0.7457 kW 1 Btu/lb (^) m⋅°R = 4.1868 kJ/kg⋅K
STANDARD ATMOSPHERIC PRESSURE 1 atm = 1.01325 bar 1 atm = 14.696 lb (^) f /in 2
TEMPERATURE RELATIONS T(°R) = 1.8 T(K) T(°C) = T(K) – 273. T(°F) = T(°R) – 459.
