MAE 4261: AIR-BREATHING PROPULSION
TURBOJET ANALYSIS
Overview
Recall that all aircraft engines are heat engines, in that they use the thermal energy
derived from combustion of fossil fuels to produce mechanical energy in the form of
kinetic energy of an exhaust jet. The excess momentum of the exhaust jet over that of the
incoming airflow results in thrust, which is used to propel the aircraft. In class we have
shown that using a control volume approach, the general expression for thrust may be
written as shown in Equation (1):
 eU e  m
 oU o   pe  po Ae
T m
m e  1  f m o
1
f 
m f
m o
We also showed that this expression may be written in a convenient dimensionless form
as shown in Equation (2):
U
p A
T
 1  f  e  1  o e
m oU o
Uo
m oU o
 pe


 1
 po

2
In the modeling of aircraft engines, if we assume that the exhaust pressure is equal to the
ambient pressure1, pe=po, and that f << 1, then the expression becomes:
U

T
 M o  e  1
m o ao
Uo

3
In this expression we have introduced the non-dimensional Mach number, M0=U0/a0,
where a0 is the local speed of sound.
1
It should be noted that the behavior of the nozzle can be much more complex, and that deviation from
ideal expansion becomes important for supersonic flight. This aspect will be looked at in more detail when
we examine the nozzle portion of the engine in detail.
1
We represent a gas turbine engine using a Brayton cycle and are able to derive
expressions for work as functions of temperature (or pressure) at various points in the
cycle. We now seek to perform an ‘ideal cycle analysis’, which is a method for
expressing thrust and thermal efficiency of engines in terms of useful design variables.
The objective of cycle analysis for various propulsion devices (ramjets, turbojets,
turbofans) is to estimate the thrust, T, and the thermal efficiency, thermal (or alternatively
Isp) as a function of (1) typical design limiters, (2) flight conditions, and (3) design
choices so that we can analyze the performance of various engines2. To do so, we will
employ the following methodology:
1. Estimate the ingested mass flow, m o , and the exhaust to inlet velocity ratio, Ue/Uo
in terms of temperature ratios.
2. Use a power balance to relate turbine parameters to compressor parameters.
3. Use an energy balance across the burner (combustor) to relate the combustor
temperature rise to the fuel flow rate and fuel energy content.
We can write the ratio of exhaust to inlet velocity ratio as:
Ue Me

Uo Mo
RTe
RTo

Me
Mo
Te
To
It is most efficient to find the exit Mach number and temperature by keeping track of the
stagnation temperatures and pressures through all engine components. In general, it is the
stagnation properties that most conveniently represent the effect of the components on the
fluid as it flows through the engine. The relations for stagnation temperature and pressure
are given below3:
Tt
 1 2
 1
M
T
2

pt    1 2   1
 1 
M 
p 
2

2
Note that ideal cycle analysis addresses only the thermodynamics of the airflow within the engine and
does not concern itself with the detailed design of the components, such as blading, rotational speed, or any
other geometry. Instead the analysis is focused on the results that the various components produce, such as
temperature and pressure ratios. Later in the course we will look at the detailed geometry and operation of
these components to see how they work to produce given results.
3
The terms stagnation temperature and total pressure are synonyms, meaning exactly the same thing. In
this document stagnation temperatures and pressures are denoted by T t and pt, respectively. In other texts,
such as Mechanics and Thermodynamics of Propulsion, by Hill and Peterson, the stagnation temperature
and pressure are denoted by T 0 and p0, respectively. Both forms of the notation are common in the
literature.
2
Notation and Station Numbering
It is very helpful to define a set of symbols that represent ratios of stagnation properties
as distinguished from static or thermodynamic properties of the gas. Also note that
stagnation properties, Tt and pt, are more easily measured than static properties (T and p).
The table below summarizes this set of useful symbols:
Table 1: Summary of Useful Notation
Symbol
Physical Description
Ratio
of
stagnation
(total)
pressures
across component

(d: diffuser (inlet), c: compressor, b: burner (combustor), t: turbine, a: afterburner, n: nozzle)

Ratio of stagnation (total) temperatures across component
(d: diffuser (inlet), c: compressor, b: burner (combustor), t: turbine, a: afterburner, n: nozzle)


Ratio of stagnation (total) pressure to ambient static pressure, p0
Ratio of stagnation (total) temperature to ambient static temperature, T0
The flow upstream of the engine may be written as:
Tt 0
 1 2
 1
M 0  0
T0
2

pt 0    1 2   1
 1 
M0   0
p0 
2


 1
 1

 0  0
0   0
Ideal Assumptions:
Inlet or Diffuser: d=1, d=1 (adiabatic, isentropic)
Combustor or Burner and Afterburner: b=1, a=1
Nozzle: n=1, n=1
For the compressor and for the turbine we can write:
pt 3
 c
pt 2
pt 5
 t
pt 4
Tt 3
c
Tt 2
Tt 5
t
Tt 4

 1
c c

 1
t t
The turbine inlet temperature is given by:
Tt 4
 t
To
This quantity is used so frequently that it gets its own special designation, t. It is also
one of the most important metrics for aircraft engine performance.
3
Turbojet Analysis
A cut-away picture of the General Electric J79 turbojet engine is shown below:
Figure 1: GE J79 Turbojet Engine (http://www.aircraftenginedesign.com). The
engine was used on the Convair B-58 "Hustler", the first US bomber capable of
maintaining speeds in excess of Mach 2. The J79 also was used on some fighters
developed during the late 1950's. The two most famous examples are the McDonnell
Douglas F-4 and Lockheed F-104, both capable of flying at Mach 2.
The schematic for the turbojet from Hill & Peterson is shown below in Figure 2a:
Figure 2a: Turbojet Schematic from Mechanics and Thermodynamics of
Propulsion, 2nd Edition (Figure 5.14)
Another schematic which depicts the same station numbers as well as the nomenclature
used in Table 1 is shown below in Figure 2.b:
Figure 2.b: Schematic of Turbojet Engine using Nomenclature from Table 1.
Note that in Figure 2.b, the nomenclature used for the inlet mass flow rate is m o , the fuel
flow rate is m f , and finally the afterburner fuel flow rate is m a .
4
We can now use the notation from Table 1 to develop expressions for the Thrust, T, and
Specific Impulse, Isp, of the turbojet engine. For this engine, we can write4:
Ue U7 M7


Uo Uo Mo
RT7
RTo

M7
Mo
T7
To
We begin the procedure by tracking the changes of stagnation temperature and pressure
through the engine and recalling that:
Tt 7
 1 2
 1
M7
T7
2
Tracking temperatures through the engine:
 T  T  T  T  T  T  T 
Tt 7  Tto  t1  t 2  t 3  t 4  t 5  t 6  t 7 
 Tt 0  Tt1  Tt 2  Tt 3  Tt 4  Tt 5  Tt 6 
 T  T  T  T  T 
Tt 7  Tt 0  t 2  t 3  t 4  t 5  t 7   Tt 0  d c b t n 
 Tt 0  Tt 2  Tt 3  Tt 4  Tt 5 
  1 2 
Tt 7  T0 1 
M 0  c b t  T0 0 c b t
2


  1 2 
Tt 7  T7 1 
M 7   To o c b t  To t t
2


4
Equation (4) expresses the exit temperature as a function of the inlet temperature, the
Mach number, and the temperature changes across each component. In the same way we
can perform the pressure accounting through the engine:

   1 2   1
pt 7  p7 1 
M 7   po o c b t  n
2


5
If p7=po (assumption of an ideally expanded nozzle) and if b ~ 1 (constant pressure heat
addition), then we can write:

   1 2   1
M 7    0 c  t
1 
2


Now, we equate the expressions from Equations 4 and 5 to give:
4
Note the approximation in this expression. Again, we are dealing with ideal cycle analysis, so we will
assume that specific heat ratios and gas constants remain fixed throughout the engine.
5
 1
  1 2 
M 7    o c  t     o c t
1 
2


2
 o c t  1
M 72 
 1
6
It is important to realize that although this expression for the exit Mach number is written
in terms of temperature ratios, it comes from the pressure change in the engine. This is a
general result, namely that the exit Mach number depends on the ratio of jet stagnation
pressure to the ambient pressure, not at all on the temperature. From Equation (4):
T7 T7 Tt 7  0 c b t


b
To Tt 7 T0
 0 c t
U7 M7

Uo M0
0  1 
 1
2
2
 1
M0
T7

T0
 0 c t  1
M 02 
M 02
b
2
 0  1
 1
So far these are general expressions are applicable to any gas stream. We now substitute
them into the expression for the velocity ratio and the thrust:
U7

U0
 0 c t  1 b
0 1
Note that the above expression is identical to Equation 5.45 in Hill & Peterson, with the
proviso that nozzle adiabatic efficiency is unity and that the specific heat ratio within the
nozzle is simply . This expression can be further simplified using the results from
Equations (4) and (5) as:
T7
b 
To
U7

U0
1
 t t
 1
2

M 72
t
 o c

2
 o c t  1  t
 1
  o c
Mo
7



Finally the thrust per unit mass flow (times the speed of sound to make it dimensionless)
is:
6
 
T
2
 o c t  1 t

m o ao
 1
  o c

  M o

8
We now have two steps left. The first step is to write c in terms of t by noting that they
are related by the condition that the power used by the compressor is equal to the power
provided by the turbine, which is called the power balance. The second step is to perform
an energy balance across the combustor to arrive at expressions for specific impulse,
thrust specific fuel consumption, and overall efficiency.
Power Balance Between Turbine and Compressor:
So far we have not made these expressions particular to the turbojet engine, because we
have not included the relationship between the compressor and the turbine. In the turbojet
engine the turbine is connected to the compressor by a shaft, and hence the turbine power
equals the compressor power. We can arrive at this expression starting with the steadyflow energy equation:
m ht  q  w s
 ht = - rate of shaft work
Assuming that the compressor and turbine are adiabatic, then m
done by the system = rate of shaft work done on the system. Since the turbine shaft is
connected to the compressor shaft:
m c p Tt 3  Tt 2   m c p Tt 4  Tt 5 
 Tt 3
T
 T  T 

 1 t 2   t 4 1  t 5 
 Tt 2
 To  To  Tt 4 
 0  c  1   t 1   t 
t  1
o
 c  1
t
9
So, finally for the turbojet engine we have the thrust equation:

T
2 
 t   o  c  1  t

m o ao
 1
 o c
T
2  
1

 t 1 
m o a o
  1    o c

  M o



   0  c  1  M o


10
We are also interested in the fuel consumption. Again, using the definition of specific
impulse:
7
I sp 
T
T

m f g m o fg
Energy (Heat) Balance Across the Burner (Combustor):
The final step involves writing the specific impulse, thrust specific fuel consumption and
other measures of efficiency using these same parameters. We begin by writing the First
Law across the combustor to relate the fuel flow rate and heating value of the fuel to the
total enthalpy rise:
m f h  m o c p Tt 4  Tt 3 
m f  m o
c p To
h
 t   o c 
So, we can write the specific impulse as:
I sp
T


 ha o  m o a o


 gc T     
o c
 p o  t







11
This is the desired result. We have expressed the specific impulse in terms of typical
design parameters, such as the flight Mach number, design variables and fuel and
atmospheric properties.
The overall efficiency of an aircraft engine was given as:
 overall 
TU 0
m f h
Substituting from above, we may show that:
 overall
 T
M 0   1
 m 0 a 0

 t   c 0 
8


