General Expression for Thrust

MAE 4261: AIR-BREATHING PROPULSION RAMJET 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:

T  m˙ eU e  m˙ oU o  pe  po Ae 1 m˙  1 f m˙ m˙ e o f  f m˙ o

We also showed that this expression may be written in a convenient dimensionless form as shown in Equation 2:

2 T U e po Ae  pe   1 f  1  1 m˙ oU o U o m˙ oU o  po 

In the modeling of aircraft engines, if we assume that the exhaust pressure is equal to the 1 ambient pressure , pe=po, and that f << 1, then the expression becomes:

T U  3  e   M o  1 m˙ o ao U o 

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 (not used in ramjet analysis where there is no compressor or turbine). 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:

U M RT M T e  e e  e e Uo M o RTo M o To

It is most efficient to find the exit Mach number, Me, and temperature, Te, by keeping track of the stagnation temperatures and pressures through several 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:

T  1  t 2  1  1 M p   1 2  T 2 t  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 Tt 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 T0 and p0, respectively. Both forms of the notation are common in the literature. The subscript t is chosen to avoid confusion with the location upstream of the engine designated as 0 (zero).

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 working fluid. 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, station 0, may be written as:

Tt0  1 2  1 M 0   0 T0 2   1 pt0   1 2   1 M 0    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:

p T  t3   t3    1 c c  c   c pt 2 Tt 2 p T  t5   t5    1 t t  t   t pt 4 Tt 4

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 Ramjet Analysis A picture of the BOMARC-A missile which employs a ramjet thruster is shown in Figure 1a, a cut-away picture of a conventional ramjet engine is shown in Figure 1b, and a cutaway drawing of a similar device is shown in Figure 1c:

Figure 1a: Boeing/MARC CIM-10A "BOMARC A" Surface-to-Air Missile (http://www.hill.af.mil/museum/photos/coldwar/bomarca.htm)

Figure 1b: Bristol Siddeley Thor Ramjet (http://www.skomer.u-net.com/projects/thor.htm)

4 Figure 1c: Cut-Away Schematic of a Conventional Ramjet Layout from Mechanics and Thermodynamics of Propulsion, 2nd Edition (Figure 5.7) A schematic representation, taken from Hill & Peterson, is shown below in Figure 2a, and the accompanying T-S diagram is shown in Figure 2b:

Figure 2a: Ramjet Schematic from Mechanics and Thermodynamics of Propulsion, 2nd Edition (Figure 5.6)

Figure 2b: Temperature (or Enthalpy) versus Entropy Diagram for a Ramjet Engine (Figure 5.6, Mechanics and Thermodynamics of Propulsion, 2nd Edition)

The notation in this schematic and T-S diagram may seem somewhat strange, since the combustor is divided into two regions, region 2 and region 3. The reason that this is done is to keep the numbering convention consistent with the turbojet and turbofan engines, which are far more common propulsion devices than the ramjet. In the turbojet and turbofan, the diffuser exit / compressor inlet is always station 2, the compressor exit / combustor inlet is always station 3, and the combustor exit / turbine inlet is always station 4. Station 2 is kept as the exit of the diffuser, but since there is no rotating machinery compressor, station 3 is kept as the inlet to the combustor. We can think of the region between 2 and 3 as the fuel injectors, although that region will not appear, in particular, in our analysis. Station 4 is then kept as the exit of the combustor but again recall that there is no mechanical turbine in the ramjet engine. Examining the T-S diagram, we can make several modeling approximations: 1. The compression and expansion processes are taken to be isentropic, i.e. the process is reversible and there is no heat transfer (adiabatic). On the T-S diagram this corresponds to legs a-02 and 04-6. For all isentropic processes the total temperature and total pressure is a constant. 2. The combustion process, between 02 and 04 (or neglecting the fuel injectors, between 03 and 04) is done at low speed (M<0.3), and is modeled as constant pressure heat addition. Thus the stagnation pressure remains constant, although the stagnation temperature increases due to the heat addition (combustion).

5 Items (1) and (2) imply that the stagnation pressure remains constant throughout the ramjet engine. We will make use of this observation in modeling the ramjet engine. We can now use the notation from Table 1 to develop expressions for the Thrust, T, and Specific Impulse, Isp, of the ramjet engine. For this engine, where the stagnation pressure is a constant throughout the device, we can write4:

Pt0  Pt6 4   1 Pt0   1 2   1 M 0  P0  2    1 Pt6 Pt6   1 2    1 M e  P6 Pe  2 

Where M0 is the flight Mach number and Me is the exit plane Mach number. If we assume that the nozzle is ideally expanded, then Pe=P0, and we can write:

P P t0  t6 P0 Pe

M e  M 0 5

This implies that:

U a T T T 6 e  e  e  t6  t 4 U o a0 To Tt0 Tt3

Now substitute this result into the thrust equation, Equation 3:

T U  3  e   M o  1 m˙ o ao U o  T  T  7  M  t 4 1 ˙ o   mo ao  Tt3 

The ratio Tt4/Tt3 is the total temperature ratio across the combustor, which can be written in shorthand as b. So, the thrust equation becomes:

T 8.1  M o   b 1 m˙ o ao

This equation may also be written as:

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.

6 T    8.2  M  4 1 ˙ o   mo ao   0 

These equations point out some interesting aspects of the ramjet engine: 1. Ramjets (or scramjets) develop no static thrust, they must be moving to develop thrust. This will be in direct contrast to turbojets and turbofans. 2. The device relies on ‘ram’ compression of the air, and has no moving parts (no spinning compressor to compress the air prior to combustion). To have efficient compression of the air, the ramjet requires high flight speeds. 3. The performance of the device relies in the stagnation temperature rise across the burner. Some performance results are summarized in Figure 5.9, which shows that the maximum thrust of the ramjet is developed for flight speeds around Mach 3.

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:

9 m˙ f h  m˙ oc p Tt 4  Tt3  c T  T T  ˙ ˙ p o  t 4 t3  m f  mo    h  T0 T0  c T  T T  ˙ ˙ p o  t3 t0  m f  mo  4   h  Tt0 T0  c T m˙  m˙ p o     f o h 4 3 0 c T 10 m˙  m˙ p o    f o h 4 0

The specific impulse is:

T 11 I sp  m˙ f g

The thrust specific fuel consumption can be written as:

m˙ 12 TSFC  f T

7 These are the desired results. 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.

Lastly, the overall efficiency of the ramjet engine is given by:

TU 0 13 overall  m˙ f h

These results are plotted in Hill and Peterson Figures 5.9 and 5.10. Note that Hill and Peterson use QR for the heating value of the fuel and these expressions simply use h.