In thermodynamics we represented a gas turbine engine using a Brayton cycle,
as shown in Figure 8.1, and derived expressions for efficiency and work as
functions of the temperature at various points in the cycle. In this section
we will perform “ideal cycle analysis”, which is a method for expressing the
thrust and thermal efficiency of engines in terms of *useful design variables*.
If you take a further course in propulsion, this ideal cycle analysis will
be extended to take account of various inefficiencies in the different components
of the enginethat type of analysis is called non-ideal cycle analysis.

**Figure 8.1** Schematic of a
Brayton cycle.

1. Objective of ideal cycle analysis

**Our objective is to express thrust, T, and thermal efficiency, h
(or alternatively I) as functions of 1) typical design limiters, 2) flight
conditions, and 3) design choices so that we can analyze the performance of
various engines**. The expressions will allow us to define a particular
mission and then determine the optimum component characteristics (e.g. compressor,
combustor, turbine) for an engine for a given mission. Note that ideal cycle
analysis addresses only the thermodynamics of the airflow within the engine.
It does not describe the details of the components (the blading, the rotational
speed, etc.), but only the results the various components produce (e.g. pressure
ratios, temperature ratios). In Section IX we will look in greater detail
at how some of the components (the turbine and the compressor) produce these
effects.

2. Notation and station numbering

**Figure 8.2** Gas turbine engine
station numbering.

Notation:

_{}

_{}
_{}

_{}
()

Stagnation properties, T_{T} & P_{T}, are more easily
measured quantities than static properties (T and p). Thus, it is standard
convention to express the performance of various components in terms of stagnation
pressure and temperature ratios:

p º total or stagnation pressure ratio across component (d, c, b, t, a, n)

t º total or stagnation temperature ratio across component (d, c, b, t, a, n)

where d= diffuser (or inlet), c= compressor, b= burner (or combustor), t= turbine, a= afterburner, and n=nozzle.

3. Ideal Assumptions

1) Inlet/Diffuser: p_{d}
= 1, t_{d} = 1 (adiabatic,
isentropic)

2) Compressor or fan: t_{c}
= p_{c}^{g-1/g} , t_{f} = p_{f}^{g-1/g}

3) Combustor/burner or afterburner: p_{b} = 1, p_{a}
= 1

4) Turbine: t_{t}
= p_{t}^{g-1/g}

5) Nozzle: p_{n}
= 1, p_{n} = 1

GO! |

**Figure 8.3** Schematic with
appropriate component notations added.

__Methodology:__

1) Find
thrust by finding u_{exit}/u_{o }in terms of q_{o}, temperature ratios, etc.

2) Use a power balance to relate turbine parameters to compressor parameters

3) Use an energy balance across the combustor to relate the combustor temperature rise to the fuel flow rate and fuel energy content.

First write-out the expressions for thrust and I:

_{} where f is the fuel/air mass flow ratio

_{} (can neglect fuel)

_{}

That is the easy part! Now we have to do a little algebra to manipulate these expressions into more useful forms.

First we write an expression for the exit velocity:

_{}

Noting that

_{}

We can write

_{}

Thus

_{}
**(*)**

Which expresses the exit temperature as a function of the inlet temperature, the Mach number, and the temperature changes across each component. Since we will use this expression again later we will mark it with an asterisk (*).

We now write the pressure at the exit in a similar manner:

_{}

_{}

_{}

and then equate this to our expression for the temperature (*****)

_{}
**(**)**

and again label it (**) for use later as we do the following expression:

_{}
**(***)**

We now continue on the path to our expression for u_{7}/u_{o}.

_{}

_{}

_{}

Therefore

_{}

Now we have __two steps left__. First we write t_{c} in terms of t_{t},
by noting that they are related by the condition that the power used by the
compressor is equal to the power extracted by the turbine. Second, we put
the burner temperature ratio in terms of the exit temperature of the burner,
(T_{T4} or more specifically q_{t}=T_{T4}/T_{o})
since this is the hottest point in the engine and is a frequent benchmark
used for judging various designs.

The steady flow energy equation tells us that

_{}

Assuming that the compressor and turbine are adiabatic, then

= - rate of shaft work done __by__ the system = rate of shaft work done
__on__ the system

Since the turbine shaft is connected to the compressor shaft

_{} assuming _{}and
C_{p} are the same

This can be rewritten as

_{}
where _{}

so

_{}
or _{}

That was the first step relating the temperature rise across the turbine
to that across the compressor. The remaining step is to write the temperature
rise across the combustor in terms of q_{t}=T_{T4}/T_{o}.

_{}

and for an engine with an afterburner

_{}

Now substituting our expressions for t_{b},
and t_{t} into our expression for u_{7}/u_{o},
and finally into the first expression we wrote for thrust, we get:

_{}
__Specific thrust for a turbojet__

which is what we were seeking, an expression for thrust in terms of important design parameters and flight parameters:

_{}

With algebra _{}

We may write this write in another form which is often used

_{}

Our final step involves writing the __specific impulse and other measures
of efficiency__ in terms of 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.

_{}

and

_{} where again, f is the fuel/air mass flow ratio

So the specific impulse becomes

_{}
__ Specific Impulse for an ideal turbojet__

where I is expressed in terms of typical design parameters, flight conditions, and physical constants

Similarly, we can write our __overall efficiency__, h_{overall}

_{}

_{}
or _{}

The __ideal thermal efficiency__ is

_{}

and the __propulsive efficiency__ can be found from h_{prop}
= h_{overall} / h_{thermal}

We can now use these equations to better understand the performance of a simple turbojet engine. We will use the following parameters (with g=1.4):

Mach number |
Altitude |
Ambient Temp. |
Speed of sound |

0 |
Sea level |
288K |
340m/s |

0.85 |
12km |
217K |
295m/s |

2.0 |
18km |
217K |
295m/s |

GO! |

Note it is more typical to work with the compressor pressure ratio (p_{c}) rather than the temperature ratio
(t_{c}) so we will substitute
the isentropic relationship:

t_{c} = p_{c}^{g-1/g}

into the equations before plotting the results.

**Figure 8.4**
Performance of an ideal turbojet engine as a function of flight Mach number.

** **

**Figure 8.5** Performance of
an ideal turbojet engine as a function of compressor pressure ratio and flight
Mach number.

**Figure 8.6** Performance of
an ideal turbojet engine as a function of compressor pressure ratio and turbine
inlet temperature.

Homework P7 (PDF)

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