Thrust equation

 Let x denote the flight direction.

The thrust of the engine in this
direction is equal to the x
component of Ps integrated over
the complete internal surface,
plus that of P∞ integrated over the
complete external surface.
 Shear stress is ignored here because it is small compared to the pressure.

𝑇 = න 𝑃𝑠 𝑑𝑆 𝑥 + න 𝑃∞ 𝑑𝑆 𝑥 S is surface area

while P∞ is constsnt, න P∞ dS x = P∞ Ai − Ae , thus

𝑇 = න 𝑃𝑠 𝑑𝑆 𝑥 + P∞ Ai − Ae 𝐸𝑞. 𝐼
1
 Thrust equation
 To integrate the equation first term, we can analyze the control volume of internal
zone. The forces on the gas inside the control volume is

𝐹 = න 𝑃𝑠 𝑑𝑆 𝑥 + P∞ Ai − Pe Ae

 Recall Newton’s second law: F = ma.

This can also be written as F = d(mV)/dt;
that is, the force equals the time rate of
change of momentum. This means

𝐹 = 𝑚ሶ 𝑎𝑖𝑟 + 𝑚ሶ 𝑓𝑢𝑒𝑙 𝑉𝑒 − 𝑚ሶ 𝑎𝑖𝑟 𝑉∞

 From the above two equations, we can get:

𝑚ሶ 𝑎𝑖𝑟 + 𝑚ሶ 𝑓𝑢𝑒𝑙 𝑉𝑒 − 𝑚ሶ 𝑎𝑖𝑟 𝑉∞ = න 𝑃𝑠 𝑑𝑆 𝑥 + P∞ Ai − Pe Ae

න 𝑃𝑠 𝑑𝑆 𝑥 = 𝑚ሶ 𝑎𝑖𝑟 + 𝑚ሶ 𝑓𝑢𝑒𝑙 𝑉𝑒 − 𝑚ሶ 𝑎𝑖𝑟 𝑉∞ − P∞ Ai + Pe Ae 2

 Thrust equation
 Go back to Eq. I and institute the term ‫ 𝑥 𝑆𝑑 𝑠𝑃 ׬‬, we will obtain

𝑇 = ‫𝑆𝑑 𝑠𝑃 ׬‬ 𝑥 + P∞ Ai − Ae Eq. I

𝑇 = 𝑚ሶ 𝑎𝑖𝑟 + 𝑚ሶ 𝑓𝑢𝑒𝑙 𝑉𝑒 − 𝑚ሶ 𝑎𝑖𝑟 𝑉∞ − P∞ Ai + Pe Ae + P∞ Ai − Ae

𝑇 = 𝑚ሶ 𝑎𝑖𝑟 + 𝑚ሶ 𝑓𝑢𝑒𝑙 𝑉𝑒 − 𝑚ሶ 𝑎𝑖𝑟 𝑉∞ + Ae Pe − P∞

 This equation is the fundamental thrust equation of jet engines and rockets

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 Turbojet engine
 A photograph of a typical turbojet engine is shown below

4
 Turbojet engine
 From the fundamental thrust equation

𝑇 = 𝑚ሶ 𝑎𝑖𝑟 + 𝑚ሶ 𝑓𝑢𝑒𝑙 𝑉𝑒 − 𝑚ሶ 𝑎𝑖𝑟 𝑉∞ + Ae Pe − P∞

 With neglecting the fuel amounts, the thrust equation of the turbojet engine is

𝑇 = 𝑚ሶ 𝑎𝑖𝑟 𝑉𝑒 − 𝑉∞ + Ae Pe − P∞

 The equation explicitly shows that T can be increased by increasing 𝑉𝑒 − 𝑉∞ .

Thus, the function of a jet engine is to exhaust the gas out the back end faster
than it comes in through the front end.

5
 Turbojet engine
Components:

 1- Inlet (Diffuser): which induces a mass

of air (location 1).

 The flow is reduced to a low subsonic

Mach number, M = 0.2, in a diffuser (point
1 to point 2). For subsonic, the diffuser
must increase the flow area to decelerate
the flow; that is a divergent duct.

 For supersonic, the diffuser must be a

convergent–divergent duct, and the
decrease in flow velocity is accomplished
partly through shock waves, as shown in
the figure .

6
 Turbojet engine
Components:
 Compressor: which is usually series of
alternating rotating and stationary blades to
increase pressure (from P2 to P3). Rotate blades
called rotors and stationary blades called stators.
 The flow passes straight through the blades
without any major deviation in direction; thus
such devices are called axial flow compressors.
 Fuel is injected into the airstream and burned at constant pressure in the combustor
(point 3 to point 4), where the temperature is increased to about 2500 °R.

 After combustion, the hot gas expands through the turbine (point 4 to point 5 – P5 <
P4). The turbine is a series of rotating/stationary blades that extract work from the
flowing gas to drive the compressor.

 The flow is also expanded through a nozzle (point 5 to point 6) and is exhausted to the
atmosphere at a high velocity Ve and at pressure P6 = Pe. For subsonic flight applications,
the nozzle is convergent and Ve is subsonic, or at most sonic. For supersonic aircraft, the
7
exhaust nozzle is usually convergent–divergent and Ve is supersonic.
 Turbojet engine
 The thermodynamic process in an ideal turbojet engine is shown in the p–v diagram

8
Turbojet engine

9
Turbojet engine

10
 Turbojet engine
Thrust buildup for a turbojet engine
Contribution of static pressure to produce thrust.

 The internal duct surfaces of the

diffuser and compressor faces
the thrust direction.

 The increasing high pressure in

the diffuser and especially in the
compressor, acting on this
forward-facing area, creates a
large force in the thrust direction.

 This high pressure also acts on

the component of the forward-
facing area in the burner, so that
F increases in burner (3–4).

 Turbine and nozzle have

negative thrust, but the engine
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net thrust is positive
 Turbojet engine
Thrust buildup for a turbojet engine

 The high exit velocity is the

effect of the production of
thrust, not the fundamental
cause of thrust.

 The gas inside the engine

exerts the pressure distribution
on the solid surface of the
engine, creating Thrust. From
Newton’s third law, the
opposite reaction on the gas,
creating high exit velocity.

12
Turbojet engine

13
Turbojet engine

14
Turbojet engine

15
Turbojet engine

2185 Ib

16
Turbojet engine

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Turbojet engine

18
 Turbofan engine
 A photograph of a typical turbojet engine is shown below

19
 Turbofan engine
It is the same as turbojet engine with an addition to fan and
bypass flow to improve the thrust

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 Turbofan engine
In the case of the turbojet, the kinetic energy left in the jet
exhaust is also a loss, and the high exhaust velocities
produced by a jet engine just exacerbate the situation. This
is why a piston engine–propeller combination is basically a
more efficient device than a turbojet . Note that jet engines
produce more thrust.

Large amount of air passes through the bypass of turbofan

which improve the thrust and efficiency.

The efficiency is denoted by the thrust-specific fuel

consumption, TSFC. For a typical turbojet, TSFC = 1.0 lb of
fuel per pound of thrust per hour; for a typical turbofan,
TSFC = 0.6 lb of fuel per pound of thrust per hour.
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 Turbofan engine
Thrust equation (with ignoring the fuel amounts)

𝑇 = 𝑚ሶ 𝑎𝑖𝑟 𝑉𝑒 − 𝑉∞ + Ae Pe − P∞ 𝑐𝑜𝑟𝑒 + 𝑚ሶ 𝑎𝑖𝑟 𝑉𝑒 − 𝑉∞ + Ae Pe − P∞ 𝑏𝑦𝑝𝑎𝑠𝑠

𝑚ሶ 𝑎𝑖𝑟=𝑚ሶ 𝑎𝑖𝑟,𝑐𝑜𝑟𝑒 + 𝑚ሶ 𝑎𝑖𝑟,𝑏𝑦𝑝𝑎𝑠𝑠

Bypass ratio of a turbofan engine is the ratio between the mass flow rate of
the bypass stream to the mass flow rate entering the core

𝑚ሶ 𝑎𝑖𝑟,𝑏𝑦𝑝𝑎𝑠𝑠
𝛼=
𝑚ሶ 𝑎𝑖𝑟,𝑐𝑜𝑟𝑒

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 Ramjet
Ramjet is similar to turbojet but without compressor and
turbine. It has inlet, subsonic burner, and exit nozzle. The
inlet minimizes the air velocity by shock waves.

It is used for supersonic flight.

Thrust of ramjet T = 𝑚ሶ 𝑎𝑖𝑟 𝑉𝑒 − 𝑉∞ + Ae Pe − P∞

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Rockets

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 Rockets
The rocket is the ultimate high-thrust propulsive mechanism.
With it people have gone to the moon, and space vehicles
weighing many tons have been orbited about the earth or
sent to other planets in the solar system.

the rocket engine carries both its fuel and oxidizer and is
completely independent of the atmosphere for its
combustion. Thus the rocket can operate in the vacuum of
space, where obviously the air-breathing engines cannot.

Fuel and oxidizer are sprayed into the combustion chamber,

where they burn, creating a high-temperature mixture of
combustion products.

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 Rockets
The thrust of a rocket engine is obtained from


m = mfuel + moxidizer 26