Group 21: Turbine Engine Cycle Design
Dr. Seitzman – AE 4451 Summer 2022
Emma Kate Clark, Min Gyu Kim, Robert Lammens
26 July 2022
1. Executive Summary
This design document details the analysis of four different jet engines for two different vehicles,
a commercial airliner, and a high-performance aircraft. These engines are evenly divided among
the two vehicles, and are evaluated at four different flight conditions, each with its own altitude
and flight speed. The analysis is done on the engine cycle, which is the process of how the
internal temperature, pressure, and energy changes and is distributed across various parts of the
engine. All these variables affect the specific thrust and efficiency of the engine, which is what
this paper sets out to analyze.
The first three engine cycles are performed using a turbofan engine, known for its high
specific thrust at low speeds, while the last engine cycle uses a turbojet engine, a simpler design
used for higher velocity aircraft. The first two engine cycles were for the commercial airliner.
The latter of these two engines used a less powerful compressor, having a lower pressure ratio.
This meant that the thrust of the engine was less in all four conditions, but the engine consumed
less fuel. For the two engine cycles of the high-performance aircraft, a turbofan was used for the
first engine cycle while a turbojet was used for the second. The main difference between these
was that the turbofan had higher specific thrust and lower fuel consumption but had a more
complex design and had more drag innate to the design of the engine compared to the turbojet.
Overall, the two final engines recommended were the first cycle for both engines: the engine
with the most powerful compressor for the commercial airliner, and the turbofan design for the
high-performance aircraft.
2. Introduction
The purpose of this project is to further develop an understanding of turbine engine
performance and design. Cycle analyses were performed using online models of ramjet, turbojet,
and turbofan jet engines – though ramjet diagrams were not used in our final designs. The students
involved were given a set of inlet conditions such as temperature or pressure as well as a specific
thrust parameter and required to create a diagram of functions for each condition. Each function
represents some black box component of a jet engine. This was done to increase the students’
familiarity with the roles each component plays in creating a functioning engine on a fixed-wing
aircraft.
To meet the requirements for 4 flight conditions (2 for a commercial airliner and 2 for a
high-performance aircraft) as listed in Table A, jet engines were designed using the models created
for the cycle analysis. These flight conditions were met by formulating equations for each jet
engine and choosing the design parameters that provide the best performance characteristics for
each mission.
Table A. Standard Flight Conditions.
Vehicle
Flight Condition
Commercial
Airliner
High
Performance
Aircraft
MTO SL Static Thrust
MCR Thrust (High
Altitude Cruise)
MTO Thrust (USAFA
MCR Thrust (High
Altitude Supersonic
Cruise)
Altitude
[km]
𝑇𝑎 [K]
𝑃𝑎
[kPa]
M
Required ST
[kN-s/kg]
0
10.7
298
219
101.3
23.8
0
0.85
2.85
0.875
2.21
15.2
274
216
77.0
11.6
0
1.5
3.00
1.06
The goals of this project were to:
1) Find 4 different engine cycles that provide the best performance for each of the flight
conditions (required ST and low SFC). Include all required output values, including
temperature and pressure values for each component excluding the nozzles and fuel pump.
2) Find for each engine cycle the maximum possible specific thrust (and SFC) for the different
flight conditions.
3) Recommend a single engine cycle for the commercial airliner and the high-performance
aircraft.
Please note that these conditions ordered from top to bottom in Table A will be referred to
as “Condition 1 / 2 / 3 / 4” from here on in this paper for convenience.
These goals, while simple, had some different constraints behind them. The first goal, the
engine cycle analysis, had a condition where the designers had complete freedom of choice when
designing the engine cycle, able to create any type of engine within the limits of the parts. These
limitations included limits on the pressure ratio across the main fan of a turbofan engine, which
could not exceed 1.6, or a limit on how high the compressor bleed ratio could be. The analyses
performed for the second goal, however, were limited to the design choices made for that engine
cycle during the initial cycle analysis for the first goal. The fan bypass ratio and the pressure ratios
across the compressor and fan could not change. Thus, the only design variables able to be changed
were the split ratio, bleed ratio, and fuel-to-air ratios of the main burner and afterburner. The third
goal is much simpler: only a simple paragraph-long description of which engine cycle is more
optimal for the corresponding engine is needed.
As stated in the above paragraph, engine cycles for a ramjet, turbojet, and turbofan were
analyzed considering the following additions to the standard engine model:
•
•
•
•
Bleed air from compressor used to provide cooling to the turbine blades (increasing the
maximum turbine inlet temperature).
Place an afterburner following the core.
Vary the nozzle exhausts by using a combined nozzle that mixes the core and bypass air.
Send part of the bypass air through a fan nozzle (instead of the combinded nozzle) by
using a bypass splitter.
This project was designed to be accomplished by developing equations for each component of the
jet engine in simple terms that vary only across each individual component. Then after obtaining
the data at the exhaust of the engine nozzle the performance characteristics of the engine could be
calculated. The required inputs and outputs for the model are summarized in Table B by which the
equations are formulated using these variables and solving for the desired output values. The
names of the corresponding values can be found in Table S in Appendix C.
Table B. Inputs and Outputs for Model.
Input
Output
𝑇𝑎
𝑃𝑎
𝑇𝑜𝑖
𝑃𝑜𝑖
𝑇𝑒 ,𝑇𝑓𝑒 , 𝑇𝑐𝑒
𝑃𝑓.1
𝑃𝑓.2
𝑤𝑐̇
𝑤𝑝̇
̇
𝑤𝑓𝑡
𝑓𝑚𝑎𝑥
𝑓𝑚𝑎𝑥, 𝑎𝑏
𝑢𝑒 , 𝑢𝑓𝑒 , 𝑢𝑐𝑒
𝑀𝑒 , 𝑀𝑓𝑒 , 𝑀𝑐𝑒
M
Prc
Prf
𝛽
𝜎
b
𝑓
𝑓𝑎𝑏
ST
TSFC
𝜂𝑝
𝜂𝑡ℎ
𝜂𝑜
Lastly, the results of all these cycle analyses are displayed in large tables. For the first goal, creating
cycles that output the temperatures, pressures, and other values listed in the table above, the tables
Commented [LK1]: Figure out how to make the space
between the Table name and the table smaller. Like Table A.
Commented [LK2]: Make this the symbols table. Replace
this table with only the symbols
(no names)
Commented [LK3R2]: This will certainly be re-organized,
lol
have all the parts listed as columns, and have the required outputs listed by row. Therefore, it is
displayed as an array of values each corresponding to a different output and part. The analysis
made for the second goal, where each cycle is placed in a different flight condition and made to
output its maximum thrust in that situation, smaller tables are created which list the specific thrust
and specific fuel consumption for each cycle in each condition. Overall, there are 8 tables; 1 for
each of the four-cycle analyses for the first goal, and 1 for each of the maximum thrust conditions
described, containing a total of 20 cycle analyses. Finally, there are a couple of paragraphs
underneath all the tables describing the logic behind which of the engine cycles created for each
vehicle is chosen to be the best overall for that vehicle.
3. Approach
To analysis the engine cycles, the following steps were taken:
1) Develop equations.
2) Write a code.
3) Run the cycle analysis.
All calculations were performed using SI units. The code was written, and the jet engines
were analyzed using MATLAB. Equations were developed for each component of the engine and
then connected to provide the relationship between the inputs and desired outputs. This allowed
the user to observe things such as the pressure and temperature changes across each component as
well as the overall efficiency and power of the jet engine. The code was formed in such a way that
the user could input the required variables for ambient temperature, pressure, and flow velocity,
and then receive the performance parameters and values. These inputs and outputs were listed
above in Table B. The inputs could then be adjusted as needed to provide the desired outputs for
the specified flight conditions. This model would give the information needed to recommend the
best engine cycle for each flight condition.
To provide the most accurate thermodynamic analysis possible under the given constraints,
the following assumptions were made.
1) All components are adiabatic.
2) Nozzles are perfectly expanded (even at supersonic speeds). 𝑃𝑒 = 𝑃𝑎
3) To model the fan, compressor, and turbines, polytropic efficiencies are used in place of
adiabatic efficiencies.
4) The fuel pressure is stored in pump, 𝑃𝑓.1 a set pressure difference higher, ∆𝑃𝑓.1 than 𝑃𝑎 and
the pump exit pressure 𝑃𝑓.2 needs to be some pressure difference above the main burner
stagnation pressure
5) Ignore mechanical power losses in turbines and shafts (𝜂𝑠ℎ𝑎𝑓𝑡 = 1).
6) Account for stagnation pressure loss due to shock wave formation by including a ram
recovery factor.
7) Bleed air is released from the compressor and renter the core air stream at the turbine
exhaust. It will increase the allowable turbine temperature and its properties can include
the bleed air’s temperature and pressure.
8) The fan is modeled after a forward mounted fan with 𝛽𝑚𝑎𝑥 . The size of the fan impacts the
drag forces and therefore the effective ST. This can be accounted for by a change in drag
term and related to the effective ST.
9) Account for pressure losses in the combined nozzle by adjusting the combined nozzle
mixer exhaust pressure by Prnm.
These assumptions were used to build the equations displayed in Appendix B, which were then
used to analyze the engine cycles. Additionally, a model of the system is provided in Appendix A,
which was used for all the engines regardless of the type of engine. This way, for example, if the
engine being analyzed was an engine without a fan, the code would be formatted so that all the
inputs to the fan component would be the same as the outputs of that component. This model and
these equations were used to build the code used to analyze the engine cycles. The code was
formatted as nothing more than a list of equations separated by which component the equations
pertained to. For example, the code was sectioned into commented parts with the titles of the
component, and each section was filled with the equations for the stagnation temperature, pressure,
and other variables for that one component. One example of this was the turbine work, which was
expressed in the turbine section as the sum of the compressor work and the fuel pump work. As
with any coding language, these equations were formulated sequentially from top to bottom,
defining variables and solving equations as it went until the code output the required specific thrust
value in the MATLAB workspace for the initial engine cycle analysis.
The first step in the engine cycle analysis was to first figure which type of engine was going to be
used. This step ensured the user would know exactly which parts had to be considered in the
equation development for the engine. For example, if a turbojet engine were used, the fan, fan
turbine, nozzle mixer, and the fan and combined nozzles would not need to be analyzed, so the
code could be formatted where the inputs of those components equaled the outputs as mentioned
above. When this engine design was done, the input variables listed in Table B were changed
around to both meet the requirements for the type of engine (only a turbofan engine would have a
fan pressure ratio other than 1) and to meet the required specific thrust value for the final output
section of the code.
Our approach for the second goal consisted only of changing the bleed, split, and fuel-to-air ratios
around until the maximum specific thrust value was found. No special optimizers were used for
this process. Instead, the general effect each variable had on the final specific thrust was observed
and considered while adjusting these variables. For instance, increasing the fuel-to-air ratio always
increased the specific thrust, so those variables were kept at their maximum value. Then, although
decreasing the bleed ratio did increase the specific thrust slightly, it also decreased the maximum
possible fuel-to-air ratios in both burners, so both of those effects were considered as well. Overall,
though no significantly complex process was used, the maximum thrust was calculated based on
logical assumptions gathered from observations made on the effects on the specific thrusts by
various adjustments to the allowed input variables. Finally, the primary consideration on which of
the engine cycles were better for the vehicle overall depended mostly on both its ability to meet
the required specific thrust for each condition as well as the efficiency of each engine cycle.
Naturally, the more efficient engine cycle would be beneficial to have on a vehicle in the long
term, but a vehicle unable to meet a manufacturer’s standards would be of no use either. More
details on this subject will be covered in the bottom of the Results section.
4. Results
Initially, the comprehensive engine cycle analysis code was made on MATLAB based on
the equations in Appendix A and B. Multiple variables like ambient conditions, pressure ratio, and
other special ratio values were taken as input. With the function code, several types of engines
were investigated under several conditions. We chose Turbofan and Turbojet as representative
engines for commercial and high-performance engines. Under 4 given conditions, we found 4
identical engine cycles that provide the required thrust with high efficiency. Also, using the
identical engines, we again ran the simulations at all conditions and determined the maximum
specific thrust and corresponding fuel consumption rates.
For the commercial aircraft, the turbofan engine was used. All input variables were put into
the MATLAB function. Both engine cycles allowed the afterburner to increase the thrust. These
cycles were optimized to achieve the required thrust with the best efficiency.
One engine cycle was set with high pressures for fan and compressor, and the other engine
cycle was set with low pressures for fan and compressor. Engine 1 had a compressor pressure ratio
of 36.0 and fan pressure ratio of 1.5 and had split ratio of 0.1, bleed ratio of 0.08, and bypass ratio
of 2.0. For the engine 1, the input variables used are listed in Table C, the cycle analysis is shown
in Table D, and the max thrust and efficiency at all conditions are found in Table E.
Table C. Input Variables that Used for Engine 1 at Condition 1.
Engine 1
Condition 1
𝑇𝑎
𝑃𝑎
[K]
[kPa]
Input
298
101.3
M
Prc
Prf
𝛽
𝜎
b
𝑓
𝑓𝑎𝑏
0
36.0
1.50
2.00
0.100
0.0800
0.0255
0.0008
Table D. Cycle Analysis for Engine 1.
State
1
2
3
4
5.1
5.1m
5.2
6
7
ef
ec
T0
(K)
298
338
961
1840
1310
1230
1110
1390
772
302
331
152
5470
5250
955
955
618
600
2910
137
652
P0
101
(kPa)
Required
W/ma
(kJ/kg)
fmax
Ue
(m/s)
Me
6010
0.507
Output
W/ma
(kJ/kg)
Fuel ratio
Pump
653
0.025
5
0.027
6
137
0.008
00
0.038
3
291
985
.835
2.81
Overal
l
Overall Efficiency [%]
Thermal Efficiency [%]
Propulsive Efficiency
[%]
Specific Thrust
(kN*s/kg)
Specific Fuel
Consumption
(kg/kN*hr)
0.00
94.8
0.00
2.85
42.3
Table E. Maximum Thrust and Corresponding Fuel Consumption Rate of Engine 1 at All Conditions.
Turbofan Engine 1
Maximum Specific Thrust
[kN-s/kg]
TSFC, Specific Fuel
Consumption [kg/kN-hr]
7.09
53.8
4.86
74.1
8.44
50.3
3.86
104.
Condition 1. Max Takeoff
(MTO) Thrust (SL Static
Thrust)
Condition 2. Max Cruise
Thrust (MCR) @ High Alt
Cruise
Condition 3. MTO Thrust @
USAFA
Condition 4. MCR Thrust @
High Alt Supersonic Cruise
On the other hand, Engine 2 had lower pressure ratios, 13.0 for compressor and 1.20 for
fan and used the higher split ratio of 0.4. The input variables, cycle analysis, and the max thrust /
fuel consumption rates at all conditions are presented in Table F, Table G, and Table H,
respectively.
Table F. Input Variables Used for Engine 2 at Condition 2.
Engine 2
Condition 2
𝑇𝑎
𝑃𝑎
[K]
[kPa]
Input
219
23.8
M
Prc
Prf
𝛽
𝜎
b
𝑓
𝑓𝑎𝑏
0.85
13.0
1.20
2.00
0.400
0.0800
0.0145
0.00183
Table G. Cycle analysis for Engine 2.
State
1
T0
251
(K)
P0
37.0
(kPa)
Required
W/ma
(kJ/kg)
2
3
4
5.1
5.1m
5.2
6
7
ef
ec
265
564
1110
816
772
729
797
509
223
256
44.4
599
554
145
145
115
11
338
46.7
312
Pump
1120
0.052
5
Overall
Output
W/ma
(kJ/kg)
312
46.7
0.014
5
0.033
6
Fuel ratio
fmax
0.001
83
0.046
8
Ue
(m/s)
Me
Overall Efficiency [%]
Thermal Efficiency [%]
Propulsive Efficiency
[%]
Specific Thrust
(kN*s/kg)
Specific Fuel
Consumption
(kg/kN*hr)
300
737
1.00
2.31
31.1
76.2
40.8
0.875
67.2
Table H. Maximum Thrust and Corresponding Fuel Consumption Rate of Engine 2 at All Conditions.
Turbofan Engine 2
Maximum Specific Thrust
[kN-s/kg]
TSFC, Specific Fuel
Consumption [kg/kN-hr]
5.72
51.7
4.52
73.7
7.07
50.9
3.60
96.1
Condition 1. Max Takeoff
(MTO) Thrust (SL Static
Thrust)
Condition 2. Max Cruise
Thrust (MCR) @ High Alt
Cruise
Condition 3. MTO Thrust @
USAFA
Condition 4. MCR Thrust @
High Alt Supersonic Cruise
Next, for the high-performance aircraft cycles, Conditions 3 and 4 were loaded into the
MATLAB code’s inputs. This aircraft was designed with a high-performance turbofan engine that
had a bypass ratio of 5, a fan pressure ratio of 1.6, and a compressor pressure ratio of 34.4. The
input variables left to be changed were the same as the commercial airliner: the split ratio, the
bleed ratio, and the main and afterburner fuel-to air ratios. The split ratio, which is not listed below,
was set to 1, meaning separate fan and core nozzles were used for the air exit. Below is a table of
the required output values from the engine for Condition 3.
Table I. Input Variables Used for Engine 3 at Condition 3.
Condition 3
𝑇𝑎
[K]
Pa
[kPa]
M
Prc
Prf
𝛽
𝜎
b
𝑓
𝑓𝑎𝑏
Input
274
77.0
0
34.4
1.60
5.50
1.00
0.0120
0.020
0.0420
Table J. Cycle Analysis for Engine 3.
State
1
T0
274.
(K)
P0
77.0
(kPa)
Required
W/ma
(kJ/kg)
2
3
4
5.1
5.1m
5.2
6
7
ef
ec
279
(stat)
1870
(stat)
318
923.
1650
1150
1150
900
2390
2390
123.
4240
4070
727.
741.
263
255.
254.
285.
634.
Overall
4780
.983
Output
W/ma
(kJ/kg)
Fuel ratio
fmax
Ue
Pump
635.
285.
.0193
.0193
.0430
.0432
(m/s)
Me
Overall Efficiency
Thermal Efficiency
Propulsive Efficiency
Specific Thrust
(kN*s/kg)
Specific Fuel
Consumption
(kg/kN*hr)
318.
1170
.949
1.41
0%
37.2%
0%
3.00
74.9
Table K. Maximum Thrust and Corresponding Fuel Consumption Rate of Engine 3 at All Conditions.
Turbofan Engine 3
Condition 1. Max Takeoff
(MTO) Thrust (SL Static
Thrust)
Condition 2. Max Cruise
Thrust (MCR) @ High Alt
Cruise
Condition 3. MTO Thrust @
USAFA
Condition 4. MCR Thrust @
High Alt Supersonic Cruise
Maximum Specific Thrust
[kN-s/kg]
TSFC, Specific Fuel
Consumption [kg/kN-hr]
6.53
34.2
3.27
68.4
7.03
32.1
1.82
116
Next, a table will be shown for the fourth cycle: a turbojet with a compressor ratio of 55.
Since this cycle covers the performance of a turbojet engine, the values for the fan, the fan turbine,
the nozzle mixer, and the fan and combined nozzles are left out, with the nozzle values being
replaced by the core nozzle values. Notice how this engine cycle has a much higher set of
efficiencies than the turbofan engine.
Table L. Input Variables Used for Engine 4 at Condition 4.
Condition 4
Input
𝑇𝑎
[K]
Pa
[kPa]
M
Prc
Prf
𝛽
𝜎
b
𝑓
𝑓𝑎𝑏
216
11.6
1.5
55.0
1.0
0
1.0
0.012
0.01
0.0349
Pump
Overall
Table M. Cycle Analysis for Engine 4.
State
1
T0
313.
(K)
P0
38.3
(kPa)
Required
W/ma
(kJ/kg)
3
4
5.1
5.1m
6
e
1050
1420
779.
782.
2040
1230
2110
2020
130.
132.
128.
725.
.4950
Output
W/ma
(kJ/kg)
Fuel ratio
fmax
Ue
2650
726.
.0100
.0163
.0349
.0454
(m/s)
Me
Overall Efficiency
Thermal Efficiency
Propulsive Efficiency
Specific Thrust
(kN*s/kg)
Specific Fuel
Consumption
(kg/kN*hr)
24.0%
50.2%
47.8%
1.06
153.
Table N. Maximum Thrust and Corresponding Fuel Consumption Rate of Engine 4 at All Conditions.
Turbojet Engine 4
Condition 1. Max Takeoff
(MTO) Thrust (SL Static
Thrust)
Condition 2. Max Cruise
Thrust (MCR) @ High Alt
Cruise
Condition 3. MTO Thrust @
USAFA
Condition 4. MCR Thrust @
High Alt Supersonic Cruise
Maximum Specific Thrust
[kN-s/kg]
TSFC, Specific Fuel
Consumption [kg/kN-hr]
1.51
135
1.49
141
1.57
132
1.34
151
A higher fuel-air ratio increases the velocity at the nozzle and produces more thrust.
However, some extra steps were required to find the maximum thrust of the engine. For this
simulation analysis, a smaller split ratio was used so that more air flows into the core or combined
nozzle. The bleed ratio was also increased to make it possible for the engine to consume more
fuel/propellent. Lastly, it was found that a higher bypass ratio helps to achieve the higher thrust
since it contributes to a greater kinetic energy change within the system.
The next few paragraphs will consist of brief descriptions of the optimal engine for each vehicle
and why that engine was chosen as the optimal engine. For the commercial aircraft, the first
engine (or engine 1) cycle is the best engine which has higher pressure ratios for fan and
compressor. The second engine which has lower pressure ratios showed that it has consumed less
fuel than the first engine, but the extent of the difference was small. Moreover, although their
fuel consumption rates were like each other, the first engine produced more thrust under the
same condition. This means that it is possible for the first engine to spend less fuel for the same
thrust that is achieved by the second engine.
For the high-performance vehicles, it is safe to say that the 3rd engine cycle with the
turbofan design is best overall for this vehicle. Although the turbojet engine designed for the 4th
engine cycle has better thermal and overall efficiency values for the Max Cruise Thrust
conditions than the turbofan engine does, which was verified by comparing the workspaces of
the two MATLAB scripts, the turbojet does not meet the required thrust values for the takeoff
conditions at either altitude. This means that despite the efficiency advantages, the turbojet
engine simply cannot be used to meet the requirements given for the vehicle. One thing to note is
that the propulsive efficiency values dropped significantly from the conditions analyzed in the
turbofan engine to the turbojet engine, giving another advantage to the 3rd engine cycle as the
optimal engine cycle to use for the High-Performance Aircraft.
5. Appendices
Appendix A – Engine Model Diagram
A diagram of the model is shown in Figure A with the corresponding states and engine components
in Tables O and P, respectively. The mass flow rates across each state and some important engine
component relations are listed in Table Q.
Figure A. Turbofan Diagram.
Commented [LK4]: If something looks wrong in this, let
me know.
I will also probably adjust the margins so this looks bigger.
Commented [KM5R4]: how I understood the nozzle
section was that combined nozzle and core nozzle are
exactly the same nozzle, but they are called differently
depending on the split ratio. When split ratio = 0, all bypass
air goes to the core (or combined) nozzle, but when split
ratio is not zero, some of the bypass air goes to both fan
nozzle and combined (or core) nozzle.
Table O. State Numbering.
State Number
a
1
2
3
4
5.1
5.1m
5.2
6
7
e
ef
ec
f.1
f.2
State Description
Ambient
Diffuser Exhaust
Fan Exhaust
Compressor Exhaust
Main Burner Exhaust
Turbine Exhaust
Turbine Mixer Exhaust
Fan Turbine Exhaust
Afterburner Exhaust
Nozzle Mixer Exhaust
Core Mixer Exhaust
Fan Nozzle Exhaust
Combinded Nozzle Exhaust
Fuel Storage
Fuel Pump Exhaust
Table P. Engine Component References.
Letter Reference
d
f
c
p
Engine Component
Diffuser
Bypass Fan
Compressor
Fuel Pump
Commented [LK6]: I need to come up with a better name
b
t
ft
ab
n
fn
cn
tm
nm
Main Combustor/Burner
Turbine (runs compressor and fuel pump)
Fan Turbine (runs fan)
Afterburner
Core Nozzle
Fan Nozzle
Combined Nozzle
Turbine Mixer
Nozzle Mixer
Table Q. Component Relations.
Component;
Exhaust State
Diffuser, 1
Fan; 2
Compressor; 3
Main Burner; 4
Turbine; 5.1
Turbine Mixer; 5.1m
Fan Turbine; 5.2
Afterburner; 6
Exit Mass Flow Rate
𝑚̇ 1 = (1 − 𝛽)𝑚̇𝑎
𝑚̇ 2 = 𝑚̇𝑎
𝑚̇ 3 = (1 − 𝑏)𝑚̇𝑎
𝑚̇ 4 = (1 + 𝑓)(1 − 𝑏)𝑚̇𝑎
̇ = (1 + 𝑓)(1 − 𝑏)𝑚̇𝑎
𝑚5.1
̇ = (1 + 𝑓(1 − 𝑏))𝑚̇𝑎
𝑚5.1𝑚
̇ = (1 + 𝑓(1 − 𝑏))𝑚̇𝑎
𝑚5.2
𝑚̇ 6 = (1 + 𝑓(1 − 𝑏))(1 + 𝑓𝑎𝑏 )𝑚̇𝑎
Special Relations
𝑇𝑜𝛽 = 𝑇𝑜2 ; 𝑃𝑜𝛽 = 𝑃𝑜2
𝑇𝑜𝑏 = 𝑇𝑜3 ; 𝑃𝑜𝑏 = 𝑃𝑜5.1
Where 𝑚𝑓̇ = 𝑓𝑚̇ 3
𝑃𝑜𝑓 = 𝑃𝑜𝑓.2 ; 𝑇𝑜𝑓 = 𝑇𝑜𝑓.2
𝑊𝑡 = 𝑊𝑐 + 𝑊𝑝
Mixes bleed air, 𝑏𝑚̇ 𝑎 , and
̇
turbine exhaust, 𝑚5.1
𝑊𝑓𝑡 = 𝑊𝑓
̇
Where 𝑚𝑓̇ 𝑎𝑏 = 𝑓𝑎𝑏 𝑚5.2
𝑃𝑜𝑓𝑎𝑏 = 𝑃𝑜𝑓.2 ; 𝑇𝑜𝑓𝑎𝑏 = 𝑇𝑜𝑓.2
𝑚̇ 7 = ((1 + 𝑓(1 − 𝑏))(1 + 𝑓𝑎𝑏 )
Nozzle Mixer; 7
Core Nozzle; e
Fan Nozzle; fe
Combinded Nozzle; ce
+ (1 − 𝜎)𝛽 ) 𝑚̇𝑎
𝑇𝑜𝜎 = 𝑇𝑜𝛽 = 𝑇𝑜2
𝑃𝑜𝜎 = 𝑃𝑜𝛽 = 𝑃𝑜2
Mixes hot core air, 𝑚̇ 6 , with
fraction of BP air, (1 −
𝜎)𝛽𝑚̇ 𝑎
̇
𝑚̇ 𝑒 = ((1 + 𝑓(1 − 𝑏))(1 + 𝑓𝑎𝑏 )𝑚̇𝑎 ) 𝑃𝑒 = 𝑃𝑎 ;𝑚̇ 7 for 𝜎 = 1
̇ = 𝜎𝛽𝑚̇𝑎
𝑚𝑓𝑒
𝑃𝑒 = 𝑃𝑎
̇ = ((1 + 𝑓(1 − 𝑏))(1 + 𝑓𝑎𝑏 ) + (1 − 𝜎)𝛽 ) 𝑚̇𝑎
𝑚𝑐𝑒
𝑃𝑒 = 𝑃𝑎 ; 0 < 𝜎 < 1
Appendix B – Equations
Ambient Condition
𝐶𝑝 =
𝛾
∗𝑅
𝛾−1
𝑢𝑎 = 𝑀√𝛾𝑅𝑇𝑎
Diffuser
𝑟𝑑 = 1 For 𝑀 ≤ 1
𝑟𝑑 = 1 − 0.075(𝑀 − 1)1.35
For 1 < 𝑀 < 5
𝛾𝑑𝑖𝑓𝑓
𝑃𝑜1
𝛾𝑑𝑖𝑓𝑓 −1
𝛾𝑑𝑖𝑓𝑓 − 1
= 𝑃𝑎 ∗ 𝑟𝑑 ∗ (1 + 𝜂𝑑 ∗
∗ 𝑀2 )
2
𝑇𝑜1 = 𝑇𝑎 (1 +
𝛾𝑑 − 1 2
𝑀 )
2
Fan
𝑃𝑜2 = Prf ∗ 𝑃𝑜1
𝑇𝑜2 = 𝑇𝑜1 ∗ (1 +
1
𝜂𝑓𝑎𝑛
(𝑃𝑟𝑓
𝛾𝑓 −1
𝛾𝑓
− 1))
𝑤̇𝑓 = (1 + 𝛽) ∗ 𝐶𝑝𝑓 ∗ (𝑇𝑜2 − 𝑇𝑜1 )
Δ𝑑 = 𝐶𝛽1 ∗ 𝑀2 ∗ (
𝑃𝑎
) ∗ 𝛽1.5
𝑃𝑎𝑡𝑚
Compressor
𝑃𝑜3 = 𝑃𝑟𝑐 ∗ 𝑃𝑜2
𝛾 −1
𝑇𝑜3 = 𝑇𝑜2 ∗ (1 +
𝑐
1
𝛾
(𝑃𝑟𝑐 𝑐 − 1))
𝜂𝑐
𝑤̇𝑐 = 𝐶𝑝𝑐 ∗ (𝑇𝑜3 − 𝑇𝑜2 )
Fuel Pump
𝑃𝑓1 = 𝑃𝑠𝑡𝑜𝑟𝑎𝑔𝑒 − Δ𝑃𝑑𝑦𝑛 − Δ𝑃𝑙𝑖𝑛𝑒𝑠 ; 𝑃𝑠𝑡𝑜𝑟𝑎𝑔𝑒 = 𝑃𝑎 + Δ𝑃𝑓1
𝑃𝑓2 = 𝑃𝑐𝑐 + Δ𝑃𝑖𝑛𝑗 + Δ𝑃𝑙𝑖𝑛𝑒𝑠
Δ𝑃𝑝𝑢𝑚𝑝 = 𝑃𝑓2 − 𝑃𝑓1
𝑤̇𝑝𝑢𝑚𝑝 =
Δ𝑝𝑝𝑢𝑚𝑝
𝜂𝑝 ∗ 𝜌𝑓𝑢𝑒𝑙
̇
̇ ⋅(
𝜔𝑝𝑢𝑚𝑝, 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑
= 𝜔𝑝𝑢𝑚𝑝
𝑚̇𝑝
̇ ⋅𝑓
) = 𝜔𝑝𝑢𝑚𝑝
𝑚̇𝑎
Main Burner
𝑃𝑜4 = 𝑃𝑟𝑏 ∗ 𝑃𝑜3
𝜂𝑏 ∗
𝑇𝑜4 = 𝑇𝑜3 ∗ (
Δℎ𝑅
∗ 𝑓 + 𝑇𝑜3
𝐶𝑝𝑏
)
𝑓+1
𝑇𝑚𝑎𝑥 = 𝑇𝑚𝑎𝑥,0 + 𝐶𝑏1 ∗ (
𝑓𝑚𝑎𝑥 =
𝑏
𝑏max
1
2
)
𝑇𝑚𝑎𝑥
−1
𝑇𝑜3
𝜂𝑏 Δℎ𝑅 𝑇𝑚𝑎𝑥
−
𝐶𝑝𝑏 𝑇𝑜3
𝑇𝑜3
Turbine
𝑤̇𝑡 = 𝑤̇𝑐 + 𝑤̇𝑓𝑝
𝑇𝑜51
𝑊̇𝑡
𝑚̇𝑎
= 𝑇𝑜4 −
(1 − 𝑏 + 𝑓)𝐶𝑝𝑡
𝛾𝑡
𝑃𝑜51
1
𝑇𝑜51 𝛾𝑡 −1
= 𝑃𝑜4 (1 − (1 −
))
𝜂𝑡
𝑇𝑜4
Turbine Mixer
𝑃𝑜51𝑚 = 𝑃𝑜51
𝑇𝑜51𝑚 =
𝑏
∗ (𝑇𝑜𝑏𝑦𝑝𝑎𝑠𝑠 − 𝑇𝑜51 ) + 𝑇𝑜51
1+𝑓
Fan Turbine
𝑇𝑜52
𝑊̇𝑓𝑡
𝑚̇𝑎
= 𝑇𝑜51𝑚 −
(1 + 𝑓) ∗ 𝐶𝑝𝑓𝑡
𝑤̇𝑓𝑡 = 𝑤̇𝑓 = (1 + 𝛽) ∗ 𝐶𝑝𝑓 ∗ (𝑇𝑜2 − 𝑇𝑜1 )
𝛾𝑓𝑡
𝑃𝑜52
1
𝑇𝑜52 𝛾𝑓𝑡−1
= 𝑃𝑜52𝑚 ∗ (1 −
(1 −
))
𝜂𝑓𝑡
𝑇𝑜51𝑚
Afterburner
𝑃𝑜6 = 𝑃𝑟𝑎𝑏 ∗ 𝑃𝑜52
Δℎ𝑅
+ + 𝑓)𝑇𝑜52
𝐶𝑝𝑎𝑏 (1
1 + 𝑓 + 𝑓𝑎𝑏
𝜂𝑎𝑏 𝑓𝑎𝑏 ∗
𝑇𝑜6 =
𝑓𝑎𝑏
𝑇𝑜6
𝑇𝑜52 − 1
= (1 + 𝑓) ∗
𝜂𝑎𝑏 Δℎ𝑅
𝑇
− 𝑜6
𝐶𝑝,𝑎𝑏 𝑇𝑜52 𝑇𝑜52
Nozzle Mixer
𝑃𝑜7 = 𝑃𝑟𝑛𝑚 ∗ 𝑃𝑜7,𝑟𝑒𝑣 ; Prnm = 1 − Cnm (1 − 𝜎)𝛽
𝑃𝑜7,𝑟𝑒𝑣 = exp (
𝑛𝑢𝑚 = (1 + 𝑓 + 𝑓𝑎𝑏 ) ∗ (𝐶𝑝 ∗ ln (
𝑛𝑢𝑚
)
𝑑𝑒𝑛
𝑇𝑜6
𝑇𝑜7
) + 𝑅 ∗ 𝑙𝑛(𝑃𝑜6 )) − (1 − 𝜎)𝛽 ∗ (𝐶𝑝 ln ( ) − 𝑅 ∗ 𝑙𝑛(𝑃𝑜2 ))
𝑇𝑜7
𝑇𝑜2
𝑑𝑒𝑛 = ((1 − 𝜎)𝛽 + 1 + 𝑓 + 𝑓𝑎𝑏 ) ∗ 𝑅 ∗ ln(𝑃07 )
𝑇𝑜7 =
𝐶𝑝6 ∗ (1 + 𝑓 + 𝑓𝑎𝑏 ) ∗ 𝑇𝑜6 + (1 − 𝜎) ∗ 𝛽 ∗ 𝑇𝑜𝑏𝑦𝑝𝑎𝑠𝑠
𝐶𝑝7 ∗ (1 + 𝑓 + 𝑓𝑎𝑏 + (1 − 𝜎)𝛽)
Fan Nozzle
𝑇𝑒𝑓 = 𝑇𝑜2 𝑜𝑟 𝑏𝑦𝑝𝑎𝑠𝑠 ∗ (1 − 𝜂𝑓𝑛 (1 − (
𝛾𝑓𝑛 −1
𝛾𝑓𝑛
𝑃𝑒
)
𝑃𝑜2
𝑢𝑒𝑓 = √2 ∗ 𝐶𝑝𝑓𝑛 ∗ (𝑇𝑜2 − 𝑇𝑒𝑓 )
Combined Nozzle
𝛾𝑐𝑛 −1
𝛾𝑐𝑛
𝑃𝑒
𝑇𝑒𝑐 = 𝑇𝑜7 ∗ (1 − 𝜂𝑐𝑛 (1 − ( )
𝑃𝑜7
𝑢𝑒 = √2 ∗ 𝐶𝑝𝑐𝑛 ∗ (𝑇𝑜7 − 𝑇𝑒 )
Performance
))
))
𝑆𝑇 = (1 + 𝑓 + 𝑓𝑎𝑏 + (1 − 𝜎)𝛽) ∗ 𝑢𝑒,𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 + 𝜎𝛽 ∗ 𝑢𝑒𝑓 − (1 + 𝛽) ∗ 𝑢𝑎
𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑆𝑇 = 𝑆𝑇 − 𝑑𝑟𝑎𝑔 ; 𝑑𝑟𝑎𝑔 = Δ𝑑
𝑇𝑆𝐹𝐶 =
𝜂𝑜𝑣𝑒𝑟𝑎𝑙𝑙 =
𝜂𝑡ℎ𝑒𝑟𝑚𝑎𝑙 =
Δ𝐾𝐸
=
𝑓𝑢𝑒𝑙 𝑝𝑜𝑤𝑒𝑟
𝑚̇𝑓
𝑓 + 𝑓𝑎𝑏
=
𝑡ℎ𝑟𝑢𝑠𝑡 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑆𝑇
1
𝑢𝑎
∗
; ℎ𝑜𝑤 𝑚𝑢𝑐ℎ 𝑐ℎ𝑒𝑚 𝐸 𝑜𝑓 𝑓𝑢𝑒𝑙 ≫ 𝑡ℎ𝑟𝑢𝑠𝑡 𝑝𝑜𝑤𝑒𝑟
𝑇𝑆𝐹𝐶 Δℎ𝑅
1 + 𝑓 + 𝑓𝑎𝑏 + (1 − 𝜎)𝛽
𝜎𝛽
1+𝛽
2
2
∗ 𝑢𝑒𝑐
+
∗ 𝑢𝑒𝑓
−
∗ 𝑢𝑎2
2
2
2
; 𝑐ℎ𝑒𝑚𝑖𝑐𝑎𝑙 𝐸 ≫ 𝐾𝐸
(𝑓 + 𝑓𝑎𝑏 ) ∗ Δℎ𝑅
𝜂𝑝𝑟𝑜𝑝𝑢𝑙𝑠𝑖𝑜𝑛 =
𝜂𝑜𝑣𝑒𝑟𝑎𝑙𝑙
; ℎ𝑜𝑤 𝑚𝑢𝑐ℎ 𝐾𝐸 𝑜𝑓 𝑓𝑙𝑜𝑤 ≫ 𝑡ℎ𝑟𝑢𝑠𝑡 𝑝𝑜𝑤𝑒𝑟
𝜂𝑡ℎ𝑒𝑟𝑚𝑎𝑙
Table R: Given Property Values.
Appendix C – Nomenclature
Table S: Variable Names.
Variable
Name
Ambient Temperature [K]
Variable
𝑇𝑎
𝑃𝑎
Ambient Pressure [kPa]
𝑃𝑓.2
M
Mach Number
𝑤𝑐̇
Prc
Stagnation Pressure Ratio of
Compressor
𝑃𝑓.1
Name
Fuel Storage Pressure [kPa]
Fuel Pump Exit Pressure
[kPa]
Specific Power of
𝑊̇
Compressor: 𝑤𝑐̇ = 𝑚̇𝑐
𝑎
Prf
Stagnation Pressure Raito of
Fan
𝑤𝑝̇
𝑤𝑓̇
Specific Power of Pump:
𝑤𝑝̇ =
̇
𝑊𝑝
𝑚̇ 𝑎
[kJ/kg]
Specific Power of Fan: 𝑤𝑓̇ =
𝑊̇ 𝑓
𝑚̇ 𝑎
[kJ/kg]
Specific Power of Fan
𝑚̇
̇
𝑊𝑓𝑡
𝛽
Bypass Ratio: 𝛽 = 𝑚̇ 𝑠
̇
𝑤𝑓𝑡
Turbine: 𝑤𝑓𝑡̇ =
𝜎
Split Ratio: 𝜎 = fraction of BP
air through fan nozzle
𝑤̇𝑡
Specific Power of Turbine:
𝑊̇
𝑤̇𝑡 = ̇ 𝑡 [kJ/kg]
𝑚
b
Bleed Ratio: 𝑏 = 𝑚̇ 𝑏
𝑓
Fuel Air Ratio: 𝑓 = 𝑚̇ 𝑓
𝑎
𝑚̇
𝑢𝑒 , 𝑢𝑓𝑒 , 𝑢𝑐𝑒
𝑎
𝑚̇
𝑎
𝑀𝑒 , 𝑀𝑓𝑒 , 𝑀𝑐𝑒
𝑚̇ 𝑎
[kJ/kg]
𝑎
Nozzle Exhaust Velocities
[m/s]
Nozzle Exhaust Mach
Number
Afterburner Fuel Air Ratio:
𝑓𝑎𝑏
𝑇𝑜𝑖
𝑃𝑜𝑖
𝑇𝑒 , 𝑇𝑓𝑒 , 𝑇𝑐𝑒
𝑓𝑚𝑎𝑥
𝑓𝑚𝑎𝑥, 𝑎𝑏
𝑇𝑚𝑎𝑥
𝑇𝑚𝑎𝑥, 𝑎𝑏
𝑓𝑎𝑏 =
𝑚𝑓̇ 𝑎𝑏
𝑚̇𝑎
Stagnation Temperature at
State “i” [K]
Stagnation Pressure at State
“i” [kPa]
Nozzle Exhaust Temperatures
[K]
Max Allowed 𝑓
Max Allowed 𝑓𝑎𝑏
Max Allowed Burner
Temperature [K]
Max Allowed Afterburner
Temperature [K]
ST
Specific Thrust [kN-s/kg]
TSFC
Thrust Specific Fuel
Consumption [kg/kN-hr]
𝜂𝑜
Overall Efficiency
𝜂𝑝
Propulsive Efficiency
𝜂𝑡ℎ
𝑊𝑖
Thermal Efficiency
Average Normalized
Specific Heat Across
Component “i”
Gas Molecular Weight
Across Component “i”
𝜂𝑖
Adiabatic Efficiencies
𝑐𝑝,𝑖
𝑅
𝜂𝑝,𝑖
Polytropic Efficiencies
𝜂𝑏
Burner Efficiency
𝜂𝑎𝑏
Afterburner Efficiency
Fuel Heating Value [MJ/kg]
𝜌𝑓
∆𝑑
Specific Drag Loss [N/(kg/s)]
Prnm
𝑏𝑚𝑎𝑥
Max Allowed b
𝑃𝑒
Fuel Density [kg/m^3]
Pressure Loss for Nozzle
Mixer
Exhaust Pressure [K]
𝑟𝑑
Prb
Prab
∆ℎ𝑅
Ram Recovery Factor
Stagnation Pressure Ratio of
Burner
Stagnation Pressure Ratio of
Afterburner
Other terms:
MCR: Maximum Cruise
MTO: Maximum Takeoff
SFC: Specific Fuel Consumption
SL: Sea Level
USAFA: United States Air Force Academy