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# Fundamentals of Aerospace Propulsion

Juan M. Tizón

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Juan M. Tizón. Fundamentals of Aerospace Propulsion. Engineering school. Spain. 2025. [hal-04976945](https://hal.science/hal-04976945v1)

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# FUNDAMENTALS OF AEROSPACE PROPULSION

JUAN M. TIZÓN

E-mail: jm.tizon@upm.es

ORCID-ID: https://orcid.org/0000-0002-8687-6657

DEPARTAMENTO DE MECÁNICA DE FLUIDOS Y PROPULSIÓN AEROESPACIAL, ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA AERONÁUTICA Y DEL ESPACIO (ETSIAE), UNIVERSIDAD POLITÉCNICA DE MADRID (UPM), PZA. DEL CARDENAL CISNEROS 3, 28040 MADRID, SPAIN

<figure><img src="image_2.png" /></figure>

Interstage separation of Apollo 6 during its launch on April 4, 1968. This mission was the second unmanned test of the Saturn V of the Apollo program. Its objectives were to demonstrate trans-lunar injection capability and to verify the Command Module's heat shield during a simulated lunar re-entry. Pogo oscillations during ascent ruptured fuel lines in Rocketdyne J-2 engines, causing two second-stage engines to shut down early. The guidance system compensated by extending burns of the remaining second- and third-stage engines, placing the vehicle into a more elliptical orbit than planned. Despite these anomalies, the mission provided NASA with sufficient confidence to proceed with manned Saturn V flights.

## Abstract

This document introduces the *Principles of Propulsion*, highlighting the most relevant aspects of aerospace propulsion. A simple yet sufficiently accurate model is employed to describe the dynamics of both the vehicle and its propellant. This provides a clear framework for defining and identifying key variables, such as thrust, and for evaluating the system's energy requirements. The findings are then applied to assess the performance of major aerospace propulsion systems. The analysis of a generic system permits comparison between autonomous and non-autonomous configurations and underscores the importance of the dimensionless parameter $x^*$, defined as the square root of the ratio between the energy supplied by the propulsion system and the specific kinetic energy of the vehicle. This parameter proves crucial in selecting the most efficient system type and in evaluating thrust specific consumption under varying flight conditions.
# Basic Concepts and Definitions

Propulsion is the action through which a body undergoes a change in its state of motion. More broadly, it refers to the act of altering or maintaining the speed, trajectory, or attitude of a material entity.

The analysis of the dynamics of a body driven by a propulsive force must be approached using, at least, the principles of Classical Mechanics. The first principle, the law of inertia, states that the velocity of an isolated body remains unchanged, and its trajectory is a straight line. It follows that the need to alter the trajectory or the speed of a body, or to keep at a constant speed the motion of a vehicle against to a resistive medium, are examples of events that require the action of a force. The second principle defines how to evaluate the magnitude and direction of the force required to produce the desired change. This force is proportional to acceleration, with body mass as the proportionality factor. Furthermore, as stated in the third principle, known as the *action-reaction postulate*, the presence of a propulsive force implies the existence of another force of equal magnitude, in the same direction but opposite in sense, acting on a different body. The simplicity of the action-reaction postulate contrasts with the significance of its consequences. The simultaneous existence of two forces and the presence of *two bodies* are mandatory requirements. This complicates the analysis, as it first necessitates a detailed description of the motion of at least two distinct material entities, and it implies the need to supply sufficient energy to sustain the motion of all bodies involved.

The two bodies considered will be the *vehicle* and the *propellant*. The vehicle serves a specific purpose or mission, meaning that the change in its state of motion caused by the propulsive force is predetermined in some way. The propulsive force exerted by the propellant on the vehicle is known as *thrust*.

The nature of the environment surrounding the vehicle is important because it can serve as a propellant. Therefore, a distinction is made between land, marine, air, and space propulsion. In land propulsion, the medium is rigid; in marine and air propulsion, the medium is fluid; and in space propulsion, there is simply no medium.

Finally, since two bodies are set in motion, the relationship established between them is crucial. Based on the nature of this relationship, aerospace propulsion systems are classified into two main categories: *autonomous* and *non-autonomous* systems. The former, also named *rocket engines*, store the propellant inside the vehicle. The latter, commonly referred to as *airbreathing jet engines*, utilize the surrounding air as main propellant. Airbreathing jet engines have the initial advantage of not carrying all their propellant onboard. However, autonomous systems are the only viable option for space propulsion, while for aerial propulsion the performance of each system must be assessed.

# Principles of Propulsion

Let us consider an inertial frame of reference to describe the motion of the material entities shown in Figure 1. During the time interval $dt$, a body of variable mass, $m + \delta m$, separates from the portion $\delta m$, and interacts with a portion of external mass $\delta m_a$. At the beginning of the interval, $\delta m$ moves with $m$ at speed $v$ and $\delta m_a$ is at rest. At $t + dt$ the body, $m$, moves at speed $v + dv$ and both $\delta m$ and $\delta m_a$ move at speed $v - v'$ in the inertial reference frame. Here, $v'$ is the velocity relative to the body, defined as positive in the opposite direction of motion.

The equations of conservation of momentum for the body $m$ and for the masses $\delta m$ and $\delta m_a$ taken together are:

$$m[(v + dv) - v] = fdt \qquad (1)$$
$$ \delta m[(v - v') - v] + \delta m_a[(v - v') - 0] = -f dt \quad (2) $$

where $f dt$ has been used to represent the combined impulse of $\delta m$ and $\delta m_a$ over the body $m$, exerted by means of the internal force $f$.

Thrust is the force exerted by the propellant on the vehicle, and its mathematical definition is given by:

$$ T = f \quad (3) $$

It will not be an easy task to evaluate thrust using Equation (3), because it would be necessary to know the details of the interaction between the vehicle and the propellant. To avoid this inconvenience, it is advisable to calculate the thrust by evaluating the dynamics of the propellants. An expression for $f$ is obtained using Equation (2) and, therefore, a second mathematical expression for the thrust becomes:

$$ T = (\dot{m}_a + \dot{m})v' - \dot{m}_a v \quad (4) $$

where $\dot{m} = \delta m/dt$ and $\dot{m}_a = \delta m_a/dt$ are the net mass flow lost by the vehicle and the mass flow of external propellant, respectively. This expression of thrust is more useful than Equation (3), because it involves integral magnitudes, and does not require detailed knowledge of the local state of the propellant inside the engine. To correctly evaluate the thrust, it is sufficient to know the momentum of the propellant in the capture and expulsion sections.

<figure><img src="image_3.png" /><figcaption>Figure 1: The figure shows a body that, over the course of an infinitesimal interval of time, captures and projects a certain amount of matter.</figcaption></figure>

Writing and applying Equation (4) requires a defined engine model that can be used in analysis and design activities and in determining the magnitudes that directly affect thrust, such as the net mass flow rate. Consequently, the evaluation of thrust evaluating thrust involves adopting a certain degree of approximation according to model characteristics. For the present study, it is sufficient to use the simplest conceptual model, which assumes that the engine and the propellant interact solely through a generic internal force.

Finally, the equations of motion must include how to calculate the mass at each instant; therefore, the following system of equations must be solved:
$$m \frac{dv}{dt} = T \qquad (5)$$

$$\frac{dm}{dt} = -\dot{m} \qquad (6)$$

along with Equation (4), which evaluates the thrust. Equations (5) and Equation (6) describe a non-stationary process that, under certain circumstances, does not affect the operation of the propulsion system. In aerospace propulsion systems, the usual situation is that the characteristic propulsion time, $t_c \sim m/\dot{m}$, is very large compared to the propellant residence time inside the engine, such that, both T as well as $\dot{m}$, may be evaluated with quasi-stationary models in which, at most, time acts as a parameter. If the residence time is noticeably short, the situation becomes more complicated. It is not possible to assume that, neither the thrust nor the mass flow rate are constant, and they must be determined using a non-stationary and, possibly, detailed engine model.

## Meshchersky Equation

The Meshchersky equation<fnref n="1" /> is obtained from the analysis of the dynamics of a system with variable mass and, for a simple system, has the following expression:

$$\frac{d(mv)}{dt} + \frac{dm}{dt} w = 0 \qquad (7)$$

where w is the velocity of the portion of mass gained, dm. To apply Equation (7) to the system in Figure 1, the second term must be generalised considering that the vehicle exchanges mass three times. First, it captures $\delta m_a$ with zero velocity ($w = 0$), so there is no contribution. Then it expels $\delta m_a$ and $\delta m$ with absolute velocity $w = v - v'$, and these two terms must be included. Taking the sign of $dm/dt$ in relation to the defined mass balance, the Meshchersky equation, appropriate for the present model, is:

$$\frac{d(mv)}{dt} + (\dot{m}_a + \dot{m})(v - v') = 0 \qquad (8)$$

which is identical to Equation (5), as it could hardly be otherwise. The clear advantage is that with the help of the Meshchersky equation and carefully defined usage conventions, the vehicle dynamics equation can be derived quickly. However, from the point of view of understanding the fundamentals of propulsion, this development is of limited value, since it is not possible to define thrust, nor does Equation (8) explain how to calculate it. Indeed, in Equation (8) one can only detect a new term on the right-hand side of equality, which could be loosely interpreted as a force, although this is a very risky assumption. Furthermore, there is the risk of leaving the mass of the body inside the time derivative which leads to a misinterpretation of all the other terms.

<footnote n="1">¹ Iván Vsévolodovich Meshchersky (Arkhangelsk, 1859 – Leningrad, 1935) was a scientist and professor of Mechanics. His courses and textbooks in the field were widely recognized both within and beyond Russia, and he is renowned for his pioneering contributions to the study of variable-mass systems. In his work, he not only developed the theoretical foundations for the dynamics of a point with variable mass, but also addressed numerous problems of scientific interest, such as systems of variable mass subjected to central forces, the motion of comets, and the effects of different laws of mass variation. His research laid the theoretical groundwork for rocket dynamics, and his name is inextricably linked to that of K. E. Tsiolkovsky, regarded as the father of cosmonautics.</footnote>

<footnote>For a detailed overview of I. V. Meshchersky's work on variable mass systems, see the following publications:</footnote>

<footnote>* ✓ Dinamika tochki Peremnoj Massy. Akademia Nauk (ed.), Peterburskij Universitet, 1887.</footnote>

<footnote>* ✓ Uravneniya dvizheniya tochki peremennoy massy v obshchem sluchaye. St. Petersburg Polytechnic University News, 1, 77–118, 1904.</footnote>

<footnote>The first is the dissertation with which he earned the equivalent of a PhD at that time, and the second is a scientific article in which he presents and develops an elegant and concise formulation.</footnote>
## Energy Balance

The variation of momentum implies the external supply of a certain amount of mechanical energy. Equation (1) is a classic differential equation describing the motion of mass $m$. By scalar multiplication of the equation by the velocity, $v$, a first integral may be obtained, which evaluates the rate at which its kinetic energy changes. If this were the case for Equation (2), it would be straightforward to evaluate how all bodies use the available mechanical energy. However, this second equation, although constructed by invoking the same principle, has a different structure. Masses $\delta m$ and $\delta m_a$ do not undergo a gradual change in velocity; the capture and ejection processes are more similar to an inelastic collision. The value of $f$ must be finite to adequately exert its action on mass $m$. Therefore, there are two possibilities: on one hand, $\delta m/dt$ and $\delta m_a/dt$ must be finite; or, if $\delta m \ll \delta m_a$, then, at least, the limit $(v - v')/dt$ should exist. This document analyses both situations in detail, beginning with the first, which is relevant in the aerospace propulsion, leaving land propulsion case, which uses the second possibility, for later.

By scalar multiplying Equation (5) by the velocity, the following expression is obtained:

$$m \frac{d}{dt} \left( \frac{1}{2} v^2 \right) = T v \qquad (9)$$

which shows the importance of the work done by the thrust force per unit time in increasing the speed of the vehicle. For this reason, the *useful power* is defined as:

$$\dot{W}_u = T v \qquad (10)$$

Now, using Equation (6), Equation (9) can be rewritten as follows:

$$\frac{d}{dt} \left( \frac{1}{2} m v^2 \right) = \dot{W}_u - \frac{1}{2} \dot{m} v^2 \qquad (11)$$

This expression allows us to evaluate the variation in the mechanical energy of the vehicle. The useful power increases the kinetic energy of the vehicle, but the analysis is not yet complete. At this point, it is not known either where the useful power comes from, or what role is played by the last term on the right side of the equation. Let us call this last term *released power*,

$$\dot{W}_r = \frac{1}{2} \dot{m} v^2 \qquad (12)$$

because it will play a significant role in the analysis of autonomous systems.

Due to the nature of Equation (2), the variation in the mechanical energy of all bodies, $E_m$, during $dt$ must be evaluated by comparing the kinetic energy before and after the time interval:

$$dE_m = \frac{1}{2} m(v + dv)^2 - \frac{1}{2} m v^2 + \frac{1}{2} (\delta m_a + \delta m)(v - v')^2 - \frac{1}{2} \delta m v^2 \qquad (13)$$

Considering that $mdv = Edt$, neglecting second-order terms and using Equation (4), the result is:

$$\frac{dE_m}{dt} = [(\dot{m}_a + \dot{m})v' - \dot{m}_a v]v + \frac{1}{2} \dot{m}_a (v - v')^2 + \frac{1}{2} \dot{m} v'^2 - \dot{m} v v' \qquad (14)$$
and, finally, by performing some operations:

$$ \frac{dE_m}{dt} = \dot{W}_m \qquad (15) $$

where the mechanical power, $\dot{W}_m$, has been defined as:

$$ \dot{W}_m = \frac{1}{2}(\dot{m}_a + \dot{m})v'^2 - \frac{1}{2}\dot{m}_a v^2 \qquad (16) $$

Mechanical power is the increase in net kinetic power, relative to the vehicle, gained by the propellant. This is a classic result in jet propulsion.

Relation (15) shows that the propulsion system must supply the mechanical power used to increase the energy of all material elements in the system. It seems evident that the useful power, $\dot{W}_u$, comes from the supplied mechanical power. Considering this, it is imperative to establish a relationship between them. Using the definition (16) of $\dot{W}_m$ and substituting relative velocities for absolute ones, we obtain:

$$ \dot{W}_m = \frac{1}{2}(\dot{m}_a + \dot{m})(v - v')^2 - \frac{1}{2}(\dot{m}_a + \dot{m})v^2 + (\dot{m}_a + \dot{m})vv' - \frac{1}{2}\dot{m}_a v^2 \qquad (17) $$

Rearranging some terms,

$$ \dot{W}_m = \frac{1}{2}(\dot{m}_a + \dot{m})(v - v')^2 - \frac{1}{2}\dot{m}v^2 + [(\dot{m}_a + \dot{m})v' - \dot{m}_a v]v \qquad (18) $$

Finally, using Equation (4), it is possible to write the following identity:

$$ \underbrace{\dot{W}_m}_{\text{Mechanical}} + \underbrace{\frac{1}{2}\dot{m}v^2}_{\text{Released}} = \underbrace{Tv}_{\text{Useful}} + \underbrace{\frac{1}{2}(\dot{m}_a + \dot{m})(v - v')^2}_{\text{Wasted}} \qquad (19) $$

Equation (19) establishes the relationship between useful power and mechanical power. Reading the terms from left to right, it can be said that the mechanical power supplied by the propulsion system, together with the released power, is used to provide useful power for flight and to increase the absolute kinetic power of the propellant. This last term, which had not appeared until now, should be called *wasted power*:

$$ \dot{W}_w = \frac{1}{2}(\dot{m}_a + \dot{m})(v - v')^2 \qquad (20) $$

Wasted power is the kinetic energy per unit time of the propellant once it has left the vehicle, evaluated in the inertial frame. Usually, this term represents a loss, because all the kinetic energy of the jet is lost or dissipated in the atmosphere, and it is not easy to imagine a plausible procedure for its use, at least not in conventional jet propulsion systems.

Using Equation (19), Equation (11) becomes:

$$ \frac{d}{dt}(\frac{1}{2}mv^2) = \dot{W}_m - \dot{W}_w \qquad (21) $$
Equation (21) indicates that the net mechanical power, $\dot{W}_m - \dot{W}_w$, which consists of the power generated by the propulsion system minus the wasted power, is directly used to increase the kinetic energy of the vehicle. Expression (9) shows that the velocity always increases due to the useful power, while (21) shows that this increase does not necessarily result in an increase in kinetic energy.

Using Equation (15) also obtain:

$$ \frac{dE_m}{dt} = \frac{d}{dt}\left(\frac{1}{2}mv^2\right) + \dot{W}_w \qquad (22) $$

which shows that the mechanical energy, contributed by the propulsion system is used to change the mechanical energy of every material entity.

## Simple propulsion system

The simplest or most immediate mechanism of marine or aerial propulsion is a propeller coupled to an engine that provides the required mechanical power. As schematically shown in Figure 2a, consider a system that does not exchange mass with the environment ($\dot{m} = 0$) equipped with a propeller immersed in a fluid of which only a fraction $\dot{m}_a$ interacts mechanically with the propeller. This is a non-autonomous propulsion system, since the propellant and the vehicle are distinct material entities.

Using Equation (4), the thrust is

$$ T = \dot{m}_a(v' - v) \qquad (23) $$

This expression shows that the thrust of the system depends on the flight condition and is positive only if $v' > v$. This feature limits the possibilities of non-autonomous propulsion systems, although the use of a second body other than the vehicle has, as will be seen below, other advantages.

From Equation (16), the required mechanical power is:

$$ \dot{W}_m = \frac{1}{2} \dot{m}_a [v'^2 - v^2] \qquad (24) $$

In this case, the released power is zero, and relationship (19) becomes:

$$ \dot{W}_u = \dot{W}_m - \dot{W}_w \qquad (25) $$

where the wasted power is:

$$ \dot{W}_w = \frac{1}{2} \dot{m}_a (v' - v)^2 \qquad (26) $$

If mechanical power is taken as a reference, the wasted power can be expressed by either of the following two relationships:

$$ \dot{W}_w = \dot{W}_m \frac{(v' - v)}{(v' + v)} = \dot{W}_m \frac{\dot{W}_m}{\frac{1}{2} \dot{m}_a (v' + v)^2} \qquad (27) $$
It is possible to reduce wasted power if the velocities are close to each other, which will be equivalent to using high mass flow rate. Then, if the difference in velocities is considered a first-order infinitesimal $v'/v \sim 1 + \epsilon$, and mechanical power remains finite with $\dot{W}_m/v^2 \sim 1$, the mass flow used is $\dot{m}_a \sim (\dot{W}_m/v^2) \epsilon^{-1}$ and the wasted power becomes:

$$ \dot{W}_w \sim \epsilon \dot{W}_m \qquad (28) $$

Thus, wasted power is an order of magnitude smaller than useful or mechanical power. The efficiency of a simple non-autonomous system is achieved by moving large masses of propellant with a small velocity increment.

<figure><img src="image_4.png" /><figcaption>Figure 2: Schematic of representative propulsion systems. (a) Propeller-engine system (simple); (b) rocket engine (autonomous); (c) automobile with engine (terrestrial); (d) simplified air-breathing jet engine (non-autonomous).</figcaption></figure>

Propellers are efficient propulsion elements; however, the mass of air (or water, in the case of marine propulsion) involved in generating the propulsive force is neither clearly identified, nor is it evident what the induced velocity might be. Therefore, it is common to relate the performance of the propeller to system operating parameters, such as the vehicle speed and the propeller rotation speed, whose ratio produces a dimensionless parameter called the *advance ratio*. The system's performance is well described as a function of the advance ratio alone, and sometimes also considering the effect of the propeller's pitch angle, since this degree of freedom improves its response.

## Autonomous Propulsion

Consider an autonomous propulsion system that uses a net mass flow $\dot{m} > 0$ and does not interact with any other material entity outside the vehicle ($\dot{m}_a \equiv 0$). Under these conditions, the essential process is the continuous ejection of mass by the vehicle, thereby generating thrust. To illustrate this, Figure 2b depicts the typical case of a rocket launched to reach a given altitude and velocity.
Using Equation (4), the thrust generated is:

$$T = \dot{m}v' \qquad (29)$$

This expression shows that the thrust of the system does not depend on flight conditions and is an intrinsic characteristic of the rocket engine.

From Equation (16) the required mechanical power is:

$$\dot{W}_m = \frac{1}{2} \dot{m} v'^2 \qquad (30)$$

while the wasted power is:

$$\dot{W}_w = \frac{1}{2} \dot{m} (v - v')^2 \qquad (31)$$

and Equation (9) becomes:

$$m \frac{d}{dt} \left( \frac{1}{2} v^2 \right) = \dot{W}_u = \dot{m} v' v \qquad (32)$$

The expression shows, first, the monotonic increase in the specific kinetic energy of the vehicle, and second, that the useful power increases with flight speed. This highlights the *Oberth effect*, which states that the higher the vehicle's speed, the greater the increase in specific kinetic energy achieved. This phenomenon is a direct consequence of the released power, which, from an energy perspective, enables the useful power to remain positive.

Finally, note that Equation (21) becomes:

$$\frac{d}{dt} \left( \frac{1}{2} m v^2 \right) = \frac{1}{2} \dot{m} v'^2 \left[ \frac{v}{v'} \left( 2 - \frac{v}{v'} \right) \right] \qquad (33)$$

which indicates that the kinetic energy of the vehicle can increase or decrease. It increases if $v' > v/2$ but it decreases if the exit velocity is less than $v/2$. On the other hand, for a given mechanical energy input, the kinetic energy of the vehicle reaches its maximum growth if $v' = v$, that is, when $\dot{W}_w = 0$. In this case, the mechanical energy is fully invested in increasing the vehicle's mechanical energy, which can serve as a useful design guideline, for example, for a kinetic projectile. In any other situation, the process will not be energetically perfect, but Equation (32) shows that the vehicle's velocity always experiences a net increase.

## Terrestrial Propulsion

The propulsion system of a conventional land vehicle consists of a power engine that, together with a transmission mechanism, drives the traction system, such the wheels of a car (see Figure 2c). In this case, the ground, and ultimately the Earth, represent the material entity with which the vehicle interacts. Since it is a rigid body, no mass flow occurs. Assuming $\dot{m} = 0$ and adopting $\delta m_a = M$, Equation (2) becomes:

$$M(v - v') = -T dt \qquad (34)$$

which implies the necessary existence of the limit,
$$ \lim_{dt \to 0} \frac{v' - v}{dt} = a \qquad (35) $$

where *a* is the acceleration acquired by *M* due to the action force resulting from the thrust applied to the body of mass *m*. Once the acceleration of mass *M* is used in the analysis, it becomes irrelevant that the second body was initially assumed to be at rest.

Consequently, the equations of motion of the masses of the system are:

$$ m \frac{dv}{dt} = T \qquad (36) $$

$$ Ma = -T \qquad (37) $$

In this case, Equation (37) is used to determine the acceleration of the second body, while the thrust can be determined by analysing the power engine and the mechanism that connects it to the wheels.

The equations that express the variation in the mechanical energy of the system are obtained by multiplying Equation (36) by *v* and Equation (37) by *u*, where

$$ u = \int a \, dt \qquad (38) $$

obtaining

$$ \frac{d}{dt} \left( \frac{1}{2} m v^2 \right) = T v \qquad (39) $$

$$ \frac{d}{dt} \left( \frac{1}{2} M u^2 \right) = -T u \qquad (40) $$

The mechanical power that must be supplied to the system is:

$$ \dot{W}_m = T(v - u) \qquad (41) $$

And, as the useful power is $\dot{W}_u = T v$, the power used by the second body is:

$$ \dot{W}_w = -T u \qquad (42) $$

If it is assumed that $M \gg m$, and a small parameter $\epsilon = m/M$ is defined, then eliminating the thrust between Equation (36) and Equation (37) gives:

$$ \epsilon dv = -adt \qquad (43) $$

As expected, this result indicates that $u \sim \epsilon v$ and, consequently, the wasted mechanical energy is $\dot{W}_w \sim \epsilon \dot{W}_m$, which approaches propulsive perfection. Indeed, if the mass corresponds to that of the Earth (5.9722 $\cdot$ 10$^{24}$ kg) and the vehicle has a human-scale mass, however large, the value of $\epsilon$ becomes extraordinarily small, and the wasted power is negligible.

This simplified analysis assumes a rigid Earth that responds to the vehicle's motion with its own movement, with no energy loss in any dissipative medium and no slippage in the mechanical
propulsion chain. While such assumptions and other context-specific factors must be carefully
analysed, the high efficiency of terrestrial propulsion remains evident.

Aerospace Propulsion

A propulsion system is a mechanism that generates the thrust required for a vehicle to accomplish its intended mission. It typically provides the structural framework where action-reaction forces are established and incorporates the essential components needed to generate or supply the necessary power. An example of such a system is the air-breathing jet engine, whose operation is schematically shown in Figure 2d. The system captures air from the surroundings and expels it at a higher speed in the opposite direction of motion. Energy is obtained from the combustion process between the fuel carried by the vehicle and the oxygen present in the captured air. As will be seen in this section, the coordinated operation of its elements combines release and transformation of energy into a highly compact propulsion system.

From a purely conceptual perspective, an aerospace propulsion system must provide a certain amount of mechanical energy, which is used to produce thrust and the energetic manifestation of thrust is the useful power delivered when propelling the vehicle. In this context, Figure 3 presents a conceptual diagram of power transformation and distribution within a propulsion system. The main elements in the diagram are a power engine E and a propulsor system P. Arrows indicate the direction of energy flow through the elements. At the junctions, represented by circles, an energy balance must be established to ensure that the sum of the incoming arrows (rectangles) matches the sum of the outgoing ones.

<figure><img src="image_5.png" /><figcaption>Figure 3: Conceptual diagram of power distribution within a propulsion system. The label E inside the circle represents a power engine, while the label P identifies an element that distributes the mechanical power among categories relevant for propulsion analysis.</figcaption></figure>

In general, a thermal engine extracts energy from a "Source or Collector", producing mechanical power, $\dot{W}_m$, while also dissipating some heat to the surroundings. This heat dissipation is represented in Figure 3 by the element labeled "Radiator". Following the energy flow, the propulsor system P converts the available power, both the liberated and mechanical power, into useful power and residual losses.

It is important to note that P is not itself a power engine; rather, it is a real or virtual mechanism that performs a transformation process in accordance with Equation (19). This transformation inherently incurs additional losses, likely related to the specific mechanisms employed, though these
losses are not explicitly considered in this analysis. Even without accounting for such inefficiencies, the process is not ideal: the input power is not fully converted into useful power. In all realistic systems, part of the mechanical power is transformed into wasted power.

Based on Figure 3, the efficiency of a thermal engine depends on its ability to convert the energy from the source into mechanical energy. To quantify this, it is usual to define thermal efficiency, $\eta_T$, as the ratio between the mechanical power output and the energy supplied per unit time:

$$ \eta_T = \frac{\dot{W}_m}{\dot{Q}_s} \qquad (44) $$

The power engine transforms the energy from the source, $\dot{Q}_s dt$, into mechanical energy, $\dot{W}_m dt$, and necessarily, rejects a certain amount of waste heat, $(\dot{Q}_s - \dot{W}_m) dt$, which is presumably carried away by the ejected mass, unless an additional system is incorporated or the internal energy (temperature) of the vehicle is allowed to rise without apparent limit.

In view of Equation (19) or Figure 3, the propulsion efficiency $\eta_P$ must be defined as the ratio between the useful power and the total available power:

$$ \eta_P = \frac{\dot{W}_u}{\dot{W}_m + \frac{1}{2} \dot{m} v^2} \qquad (45) $$

That is also expressed by the following relationship:

$$ \eta_P = \frac{\dot{W}_u}{\dot{W}_u + \dot{W}_w} \qquad (46) $$

Using the definition of thermal efficiency (44), the useful power can be expressed as:

$$ \dot{W}_u = \eta_P \left[ \eta_M \dot{Q}_s + \frac{1}{2} \dot{m} v^2 \right] \qquad (47) $$

Consequently, the useful power is extracted from the source through the concatenation of two processes, the power engine which obtains mechanical energy from the source with efficiency $\eta_T$ and the propulsion system which manages the available mechanical power with efficiency $\eta_P$.

The diagram in Figure 3 accurately represents conventional air and marine propulsion systems that use internal combustion engines as power units and propellers as propulsion systems. In these cases, each system is clearly defined, and they interact through the mechanical power transmitted via the drive shaft connecting the engine to the propeller. However, in jet propulsion engines, where thrust is generated by the complete system, the categories of power engine and the propulsion mechanism become indistinguishable. Although mechanical power can still be evaluated, there is no tangible component, such as the propeller shaft in conventional systems, that clearly defines the boundary between the power unit and the propulsion elements.

Figure 4 presents a conceptual diagram of a powerplant, where the distinction between the power unit and the propulsion system is no longer evident. The diagram follows the structure of Figure 3 but removes the explicit identification of the two systems, as they merge into a single powerplant, hiding the mechanical power inside the system. Circular junctions, where energy balances must be
established, are retained because they express a balance, but their labels are omitted to indicate that they cannot be assigned to a specific mechanism.

<figure><img src="image_6.png" /><figcaption>Figure 4: Conceptual diagram of a powerplant. This diagram is derived from the one shown in Figure 3 by combining the power engine and the propulsion system into a single category, the powerplant.</figcaption></figure>

The overall performance of the system must be defined considering both the energy supplied by the source and the kinetic power of the released propellant. Consequently, the *overall efficiency* is defined by the following expression:

$$ \eta_o = \frac{\dot{W}_u}{\dot{Q}_s + \frac{1}{2} \dot{m} v^2} \qquad (48) $$

Using Equation (47), it follows:

$$ \eta_o = \eta_p \frac{\eta_T \dot{Q}_s + \frac{1}{2} \dot{m} v^2}{\dot{Q}_s + \frac{1}{2} \dot{m} v^2} \qquad (49) $$

If a simple propulsion system is considered, with $\dot{m} = 0$, the overall efficiency reduced to

$$ \eta_{o,\text{simple}} = \eta_p \eta_T \qquad (50) $$

It is noteworthy that in air-breathing jet engines, the heat released during combustion substantially exceeds the released power, so $\dot{Q}_s \gg \frac{1}{2} \dot{m} v^2$, and Equation (49), as in the case of simple systems, yields the simplified relationship:

$$ \eta_{o,\text{air-breathing}} \sim \eta_p \eta_T \qquad (51) $$

The above expression allows us to affirm that an air-breathing jet propulsion system is essentially a simple system, and its primary optimization will closely follow that of such systems. And in attention of the propulsive counterpart, depending on the vehicle velocity, the most efficient system is one that expels a large mass flow at a velocity close to the flight speed.

In this work, the combination of a rigorous formulation of mechanical energy conservation, as expressed in Equation (13), together with a critical assessment of mechanical power generation in
thermal engines, enables the construction of Figure 3 and Figure 4 and the establishment of a sufficiently robust working framework based on first principles. In many textbooks, it is common to find that no rigorous developments are carried out, relying instead on intuitive arguments or vague references to conservation principles that ultimately result in empty statements and, in some cases, lead to erroneous definitions and conclusions. The rigorous approach—applying the conservation of mass, momentum, and energy to a specific fluid control volume encompassing the different parts of jet engines—was, to the best of our knowledge, first employed by Professor Sánchez-Tarifa<fnref n="2" /> in his 1951 book and later in the lectures he delivered for more than thirty years in the Spanish’s university. Figure 4 is inspired by the notes from these lectures, offering a clear and elegant way to understand the operation of a jet engine.

## Thrust Specific Consumption

Aerospace propulsion systems power all types of vehicles flying both in the atmosphere and in the vacuum of outer space. In general, the most versatile and efficient propulsion system is jet propulsion, where the power system and the propulsor system are indistinguishable. Many conventional jet propulsion systems use thermal energy released by combustion, with all or part of the reactants initially stored onboard the vehicle, making the *net propellant mass flow*, $\dot{m}$, play an essential role. Typically, the stored propellant has an energy capacity $q_f$ (with SI units of [J/kg]), so the available thermal power is $\dot{Q}_s = \dot{m}q_f$. It is readily apparent that the efficiency with which $\dot{Q}_s$ is converted into mechanical power, as well as the amount of propellant initially carried at the start of the flight, are critical to the vehicle's performance.

In general, the most important technological parameter determining the performance of aerospace propulsion systems is the *thrust specific consumption*<fnref n="3" />, defined by the expression:

$$S = \dot{m}/T \qquad (52)$$

where $\dot{m}$ is the net propellant mass flow, and $T$ is the thrust.

To highlight the importance of this parameter, the following paragraphs qualitatively derives the relationships that enable the evaluation of the effect of thrust specific consumption on the performance of aerospace vehicles.

Let us consider an airplane in balanced, horizontal, and uniform flight. The weight of the vehicle must be compensated by the aerodynamic lift force, which is

$$L = mg \qquad (53)$$

<footnote n="2">² Carlos Sánchez Tarifa (1921–2012) was a Spanish aeronautical engineer, a pioneer in jet engine research, and a driving force in the development of Spain’s aerospace propulsion industry, whose career spanned a broad range of technological and scientific disciplines. He began his career at the Instituto Nacional de Técnica Aeroespacial (INTA) in its Combustion Group, established by Gregorio Millán under the mentorship of Theodore von Kármán. During this period, he led the development and construction of the INI-11, the first turbojet engine built in Spain. In 1951, he authored the pioneering volume *Motores de reacción y turbinas de gas* (Vol. I), published by INTA, in which he applied first-principles analysis to the study of jet engines—an approach uncommon at the time and still not universal today. This work emerged in an era when very few such books existed and when the number of true experts in the field was equally limited. Over the following decades, he contributed to a wide array of projects in combustion research, microgravity experimentation, wind energy, fluid mechanics, and advanced aerospace propulsion systems; among these, his role as Chief Engineer at the company SENER in the development of the EJ200 engine for the Eurofighter stands out. As Professor of Propulsion at the Universidad Politécnica de Madrid (Spain) for over three decades, he left a lasting mark on generations of aerospace engineers—many of whom were first captivated by his spirited lectures and later worked alongside him on significant projects.</footnote>

<footnote n="3">³ In non-autonomous systems (air-breathing engines), this parameter is conventionally termed "thrust-specific fuel consumption" since the mass flow corresponds to fuel consumption, and this designation is standard nomenclature for such systems. In autonomous systems (rockets), the inverse of this parameter—specific impulse—is typically employed. Since this text encompasses both types of propulsion systems without distinction, the general term "thrust-specific consumption" has been adopted.</footnote>
where *L* is the aerodynamic lift and *g* is the local gravitational acceleration. Similarly, aerodynamic drag *D* should be compensated by the thrust:

$$D = T \qquad (54)$$

Dividing Equation (53) by Equation (54) and considering that $T = \dot{m}S^{-1}$, we obtain, after rearranging

$$gdt = -\frac{1}{S}(L/D)\frac{dm}{m} \qquad (55)$$

where the *L/D* is the lift to drag ratio. If the above expression is integrated over the time during which the vehicle is propelled, if all parameters remain constant except mass, we obtain:

$$\frac{Sg(R/v)}{L/D} = \ln\left(1 - \frac{\Delta m}{m}\right) \qquad (56)$$

where *R* is the range, *v* the vehicle's velocity, and $\Delta m$ is the net propellant mass expelled (negative) by the propulsion system. This equation is a simplified form of *Breguet's equation*, named after Louis C. Breguet (1880-1955), a French aviation pioneer, engineer, and inventor who made significant contributions to both military and commercial aircraft design. As is immediately apparent, high lift to drag ratio and low thrust specific consumption become the keys to achieving long-range flight.

In the case of accelerated flight, where the vehicle continuously gains speed, and according to the definition of thrust specific consumption $S = \dot{m}/T$, the momentum equation can be conveniently rewritten from Equation (5) as follows:

$$dv = -\frac{1}{S}\frac{dm}{m} \qquad (57)$$

Then, by integrating over the powered flight and assuming *S* remains constant, the increase in vehicle velocity, $\Delta v$, can be calculated using the following expression:

$$S\Delta v = \ln\left(1 - \frac{\Delta m}{m}\right) \qquad (58)$$

For a given net propellant mass expelled, the velocity gained by the vehicle increases as the thrust specific consumption decreases. This equation, known as *the rocket equation*, was first formulated in this form by Konstantin E. Tsiolkovsky<fnref n="4" /> Russian scientist, widely recognized as the father of astronautics for his pioneering and fundamental contributions.

<footnote n="4">⁴ Konstantin Eduardovich Tsiolkovsky (1857-1935) established the fundamental principles of spaceflight and rocket propulsion that underpin modern space exploration. Largely self-taught due to hearing loss from childhood scarlet fever, this provincial teacher working in Kaluga became a founding father of theoretical astronautics. His landmark 1903 work "The Exploration of Cosmic Space by Means of Reaction Devices" introduced the foundational rocket equation and advocated for liquid propellant systems. His theoretical framework encompassed multi-stage rockets, space elevators, and rotating habitats for artificial gravity. The Tsiolkovsky rocket equation remains essential to spacecraft design today. His visionary contributions anticipated the Space Age by decades, earning him recognition alongside Hermann Oberth and Robert Goddard as one of the "fathers of modern rocketry".</footnote>
Both the Breguet equation and the rocket equation demonstrate that thrust specific consumption is a key parameter in evaluating aerospace vehicle performance. Moreover, both share the same dependence on the expelled propellant mass, expressed within the natural logarithm.

Figure 5 presents a classical plot where the inverse of the thrust specific consumption is presented against flight Mach number for various aerospace propulsion systems. The graph includes a theoretical curve given by the function $1/S = q_f/v$, assuming $q_f = 45$ MJ/kg, which represents the optimal case of unity thermal efficiency corresponding to the use a hydrocarbon fuel in air.

<figure><img src="image_7.png" /><figcaption>Figure 5: The figure shows the inverse of the thrust specific consumption, 1/S, for various aerospace propulsion systems as a function of flight Mach number.</figcaption></figure>

All propulsion systems displayed in Figure 5 exhibit efficiencies that rarely exceed 50% relative to this theoretical line. As the expression indicates, the inverse of specific thrust consumption increases with flight speed, which poses challenges to system performance at higher velocities. Notably, the graph orders different engine architectures along the flight speed axis according to their practical applicability. Flight speed thus constitutes a critical criterion when selecting an engine architecture. Only rocket engines remain unaffected by flight speed, although the specific thrust consumption of these autonomous systems is substantially higher than that of non-autonomous systems.

Having established the critical role of specific thrust consumption in aerospace vehicle performance, it becomes essential to develop an analytical framework that enables assessment of how engine type and configuration influence this fundamental parameter. This approach allows propulsion engineers to employ sound design and selection criteria, enabling them to deliver competitive and efficient products.

## Analysis and Discussion

To establish an appropriate reference framework for propulsion system analysis, a reference power is defined based on the propellant mass flow leaving the vehicle and the vehicle's velocity, that is,

$$ \dot{W}_{ref} = \frac{1}{2}(\dot{m} + \dot{m}_a)v^2 \qquad (59) $$
Using this reference, the dimensionless useful power is:

$$ \omega_u = \dot{W}_u / \dot{W}_{ref} = 2[x - 1/(1 + m)] \qquad (60) $$

where $m = \dot{m}/\dot{m}_a$ it is the dimensionless net mas flow, and $x = v'/v$ is the dimensionless jet velocity. Possible values of $m$ range from zero for a simple non-autonomous system, to infinity for a rocket engine. Meanwhile, the dimensionless jet velocity is valid for all $x \ge 0$, encompassing all possible flight conditions.

The dimensionless mechanical power is expressed as:

$$ \omega_m = \dot{W}_m / \dot{W}_{ref} = x^2 - 1/(1 + m) \qquad (61) $$

The expression for dimensionless wasted power is:

$$ \omega_w = \dot{W}_w / \dot{W}_{ref} = (1 - x)^2 \qquad (62) $$

and the dimensionless released power is:

$$ \omega_r = \dot{W}_r / \dot{W}_{ref} = m/(m + 1) \qquad (63) $$

Note that, with this scaling, the dimensionless released power does not depend on $x$ and is bounded between zero and one.

On the other hand, using the variables $m$ and $x$, propulsion efficiency from Equation (46) is expressed as follows:

$$ \eta_p = 2 \frac{(1 + m)x - 1}{(1 + m)x^2 - 1 + m} \qquad (64) $$

Finally, from equation (52) the thrust specific consumption multiplied by the flight velocity, yielding a dimensionless quantity, is:

$$ Sv = \frac{m}{(1 + m)x - 1} \qquad (65) $$

In Figure 6a, the dimensionless powers, propulsion efficiency, and thrust specific consumption are plotted as functions of dimensionless jet speed for $m = 2$. The propulsion efficiency is zero at $x = 1/(1 + m)$, reaches its maximum value, $\eta_p = 1$, at $x = 1$ and then decreases monotonically as $x$ increases further. The dimensionless useful power grows linearly with $x$, starting from zero at $x = 1/(1 + m)$ and increases without bound for larger values of $x$. The dimensionless mechanical power follows a parabolic function, increasing over the interval of interest, beginning at zero when $x = 1/\sqrt{1 + m}$. As shown in the graph, there exists an interval, $1/(1 + m) < x < 1/\sqrt{1 + m}$, where the useful power is positive, but the mechanical power is negative. This unusual situation is supported by the released power. The wasted power forms a parabola independent of $m$, remaining always positive, and reaches zero at $x = 1$, which corresponds to an optimum from the standpoint of energy utilization.

In Figure 6b, thrust specific consumption and propulsion efficiency are represented for three values of dimensionless net propellant mass flow. The values of $m$ have been chosen as representative of an
autonomous system with $m = 1000$, an intermediate value $m = 2$, and a sufficiently small value $m = 0.1$, representative of a conventional non-autonomous system. In the graph, it is observed that propulsion efficiency grows with $m$ for each value of $x$, such that for $m \to \infty$ (rockets) the best propulsion efficiency is reached. The variation of propulsion efficiency with $m$ when $x > 1$ is not pronounced. However, when $x < 1$, the change in behavior is drastic. To be able to propel the vehicle, $m \ge (1-x)/x$ must be fulfilled. The restriction disappears if a rocket engine is used, that is, when $m \to \infty$ the system produces positive thrust for any value of the dimensionless jet speed.

<figure><img src="image_8.png" /><figcaption>Figure 6: On the left, graph a) shows the dimensionless powers, propulsion efficiency and thrust specific consumption as functions of the dimensionless speed for $m = 2$. On the right, graph b) presents the thrust specific consumption and propulsion efficiency as functions of the dimensionless jet velocity for three values of dimensionless mass flow: $m = 0.1, 2$ and $1000$.</figcaption></figure>

Thrust specific consumption behaves differently depending on whether $x$ is less than or greater than one. For $x < 1$, thrust specific consumption increases as $m$ decreases for each value of $x$. Conversely, for $x > 1$, the opposite occurs; thrust specific consumption decreases as $m$ decreases for each value of $x$. The graph in Figure 6b shows that vehicle propulsion in the range $x < 1$ is only possible with values of $m > 0$. Considering both thrust specific consumption and propulsion efficiency, the system becomes more effective as $m$ increases. If there are no constraints on this parameter, the design directive should be $m \to \infty$, corresponding to a rocket engine. For $x > 1$, the trend of the thrust specific consumption curves reverses, while the propulsion efficiency remains higher when increasing $m$, though with small variations. In this case, to decrease thrust specific consumption, the design action should be to reduce dimensionless net mass flow.

Finally, for very small values of $m$ approaching zero ($m \to 0$), the non-autonomous system behaves like a simple propulsion system. From an energy perspective, the most effective approach is to increase the mass flow through the system while minimizing the velocity difference between the jet velocity and the flight velocity (i.e., $v' \sim v$ or equivalent $x \to 1$).

Until now, a generic situation has been assumed in which $x$ and $m$ can be chosen independently; however, this is only feasible if an external energy source is used. In aerospace propulsion systems, fuels or, more generally, energetic propellants are commonly employed. The amount of energy consumed is proportional to the value of $m$. Consequently, $x$ and $m$ are not independent variables but are linked by the system architecture and specific design parameters. This interrelation is reflected in
the value of the thermal efficiency of the system, which, using Equation (44), can be expressed as follows:

$$ \eta_T = \frac{(1+m)x^2 - 1}{m(2q_f/v^2)} \qquad (66) $$

where $2q_f/v^2$ is the ratio between the energy per unit mass of the propellant and the specific kinetic energy corresponding to the vehicle speed. Equation (66) imposes a constraint that the values of $x$ and $m$ must satisfy, since thermal efficiency is determined by the engine design and its operation. Specifically, for air-breathing jet engines, the thermal efficiency depends on system design parameters and the dimensionless speed itself, expressed as $\eta_T = \eta_T(p,x)$, where $p$ represents design variables such as the maximum temperature achievable by the system. For simplicity, it will be assumed that the engine efficiency is constant over a specific range of $x$, deteriorating sharply outside this range and preventing engine operation.

Solving Equation (66) for $m$ yields:

$$ m = \frac{x^2 - 1}{x^{*2} - x^2} \qquad (67) $$

where the following definition has been used,

$$ x^* = \sqrt{(\eta_T q_f)/(\frac{1}{2}v^2)} \qquad (68) $$

In other words, $x^*$ is the square root of the ratio between the energy per unit mass supplied by the power system from the source and the specific kinetic energy of the vehicle. As shown below, this dimensionless parameter plays a key role in determining the propulsive capabilities of each system in each assigned mission.

<figure><img src="image_9.png" /><figcaption>Figure 7: Graphical representation of Equation (67) for two values of $x^*$. The asymptotes are shown as dashed lines, and the regions with physically meaningful values ($m > 0$ and $x > 0$) are highlighted with thicker lines.</figcaption></figure>

Expression (67) is represented in Figure 7 for $x^* = 0.5$ in graph (a) and for $x^* = 2$ in graph (b). As can be easily observed, $m$ is symmetric with respect to $x = 0$, vanishes at $x = \pm 1$, has two vertical asymptotes at $x = \pm x^*$, and a horizontal asymptote where $m \to -1$ as $x \to \pm\infty$. Since negative values
of the net mass flow are not admissible, the physically meaningful range of dimensionless velocities extends between $x = \pm 1$ and $x = \pm x^*$. Therefore, as shown in graphs (a) and (b):

* If $x^* > 1$, $m$ is positive for $1 < |x| < x^*$.

* If $x^* < 1$, $m$ is positive for $x^* < |x| < 1$.

Only positive values of dimensionless jet velocity ($x > 0$) are considered, and the valid regions are indicated with arrows in the graphs (a) and (b) of Figure 7. In other words, the type of propellant and the flight condition determine the value of $x^{*2} = 2\eta_T(q_f/v^2)$, which establishes the valid intervals of $x = v'/v$.

Additionally, the energy parameter $x^*$ can reach extreme values. In general, the thermal efficiency $\eta_T$ takes good values only within a specific range of the dimensionless velocity $x$, while the parameter ($q_f/v^2$) can vary widely depending on the propellant and flight conditions. A typical aviation fuel (Jet-A or JP-4) has an energy content of about 45 MJ/kg, whereas propellants used in thermochemical rocket engines range from 5 to 20 MJ/kg. Nuclear rocket engines can reach up to 90 MJ/kg, and modern electrical thrusters can achieve as much as 500 MJ/kg. Flight speeds range from 300 to 3000 m/s for atmospheric flight, and from 7 to 15 km/s for launch and deep-space missions. Accordingly, the energy parameter $q_f/v^2$ can vary from values as high as about 500 for a civil turbofan to as low as 0.03 for propulsion of a deep-space probe powered by a thermochemical rocket.

<figure><img src="image_10.png" /><figcaption>Figure 8: Thrust specific consumption multiplied by flight speed as a function of dimensionless jet velocity for different values of the dimensionless energy capacity. Graphs (a) and (b) show the same values given by Equation (69), but graph (b) highlights the regions where the solution is physically admissible, and marks the limits given by the extreme values of $m$.</figcaption></figure>

Using the value of the dimensionless net mass flow from Equation (67) in Equation (65), the thrust specific consumption becomes:

$$Sv = \frac{x + 1}{x + x^*^2} \qquad (69)$$

This simple function is plotted for different values of $x^*$ in Figure 8a. A logarithmic scale is used to capture the extreme values of $x^*$. For $x > 0$, if $x^* > 1$, the function is monotonically increasing; if
$x^* < 1$, it is monotonically decreasing. When $x^* = 1$, the thrust specific consumption satisfies $Sv = 1$. The effect of $x^* \gg 1$ on thrust specific consumption is significant because it scales as $Sv \sim 1/x^{*2}$. Conversely, for very small values, $x^* \ll 1$, the thrust specific consumption remains of the order of unity, or if $x$ is sufficiently small, the curves tend to toward $Sv \to 1/x$. Note that in this case, $S \sim 1/v'$ and only the autonomous system solution becomes possible. Not all points in the graph in Figure 7 are physically possible because the dimensionless net mass flow takes positive values only in certain regions. Figure 8b highlights only the regions where $m > 0$, bounded by $m = 0$ and $m = \infty$. The first condition corresponds to $x = 1$, while the second correspond to the limit $Sv = 1/x$, representing the thrust specific consumption of an autonomous system ($m \to \infty$ in Equation (63)) which is the value reached as $x \to x^*$.

In Figure 8b, the two valid quadrants have been highlighted depending on the value of $x$. In the $x < 1$ quadrant, which correspond to vehicle velocity greater than jet velocity, the thrust specific consumption of non-autonomous propulsion systems ($m > 0$ and finite) is higher than that of a rocket engine (the $m = \infty$ line, which represents the lower bound in the quadrant). However, in the $x > 1$ quadrant, the thrust specific consumption of non-autonomous systems worsens as the dimensionless jet velocity increases, whereas the rocket engine maintains the same trend, decreasing as $x$ increases. In this quadrant, air-breathing propulsion is always more efficient than rocket engines.

Figure 8b provides a comprehensive synthesis of the parameters governing thrust specific consumption evaluation in generic propulsion systems. This analysis provides a unified treatment of both autonomous and non-autonomous systems, demonstrating that autonomous systems represent limiting cases of non-autonomous system definitions space. The operational ranges of the dimensionless parameters $x^*$ and $x$ are determined by the properties of the energized propellant, the characteristics of the engine architecture, and flight conditions. Inspection of the diagram shows that it is not possible to fly faster than the jet velocity while maintaining low specific fuel consumption. Under these conditions, the best option is to use a rocket engine, since any system with a finite $m$ exhibits a higher specific consumption. In the opposite quadrant, where flight velocities are lower than the jet velocity, the situation is quite different: the most efficient system is the non-autonomous one, with the highest possible value of $x^*$ and the lowest possible value of $m$, which corresponds to air-breathing jet engines as the most efficient propulsion systems.

The previous analysis makes it possible to identify the most efficient propulsion systems, distinguishing between autonomous and non-autonomous configurations. This framework also allows a direct comparison between rocket engines and air-breathing jet engines in terms of thrust specific consumption. To illustrate this aspect, a diagram is drawn below that improves the information provided by the diagram in Figure 5.

The best specific fuel consumption achievable with an air-breathing jet engine is

$$S_A v \sim 1/[\eta_{T,A}(q_f/v^2)] \qquad (70)$$

where $\eta_{T,A}$ is the thermal efficiency of the airbreathing jet engine.

For a rocket engine, thrust specific consumption is $S_R v \sim 1/x$ and, since the value of the dimensionless velocity can be expressed in terms of thermal efficiency, it follows that:

$$S_R v \sim 1/\sqrt{2\eta_{T,R}(q_f/v^2)} \qquad (71)$$

where $\eta_{T,R}$ is the thermal efficiency of the rocket engine.
Equating both expressions, $S_R = S_A$, the dimensionless energy capacity is:

$$ (q_f/v^2)_{S_R=S_A} = \frac{2\eta_{T,R}}{\eta_{T,A}^2} \qquad (72) $$

This relation between $q_f$ y $v$ delineates the regions in which one system has a lower thrust specific consumption than the other. Figure 9 shows only the isolines of best thrust specific consumption. The plot uses $q_f/v^2 = 4$ as the separation line, which corresponds to $\eta_{T,R} = \eta_{T,A} = 0.5$ or $x^* = 2$. To the left of the line, airbreathing jet engines have a lower thrust specific consumption, whereas to the right of the line, rocket engines become more efficient.

<figure><img src="image_11.png" /><figcaption>Figure 9: Minimum achievable thrust-specific consumption for air-breathing jet engines and rocket engines, showing performance regions for conventional aerospace propulsion systems. Air-breathing engines include both conventional fuel (kerosene) and non-conventional (hydrogen).</figcaption></figure>

In Figure 9 conventional non-autonomous systems are represented by the shaded region labeled *kerosene*, corresponding to flight speeds between 300 m/s and 2500 m/s, from high-bypass-ratio turbofans to sub- and supersonic ramjets. The use of hydrogen as a fuel reduces thrust specific consumption in inverse proportion to its energy capacity, as shown in the region labelled *hydrogen*, and extends the maximum velocity achievable by the vehicle. Finally, rocket engines have been positioned from orbital speeds and beyond.

Following the lines of constant thrust specific consumption in Figure 9, it is possible to compare the efficiency of all propulsion types. Air-breathing jet engines remain highly competitive at subsonic speeds and are unmatched by any other known propulsion system. As design flight speed increases, engine architectures must evolve, from high-bypass-ratio turbofans to single-flow engines, and eventually to ramjets and scramjets without turbomachinery. However, this drastic architectural shift alone is insufficient, as increasing flight speeds inevitably result in higher thrust-specific fuel consumption. By contrast, rocket engines exhibit thrust-specific fuel consumption that is independent
of flight speed and depends solely on the energy available from the propulsion system. Nevertheless, the achievable values are modest, and conventional systems using thermal energy (chemical or nuclear) exhibit thrust-specific consumption at least an order of magnitude higher than that of jet engines. To approach the excellent thrust-specific efficiency of jet engines, it becomes necessary to employ independent propellant acceleration systems based on electromagnetic energy, such as space electric propulsion.

The diagram in Figure 9 also highlights several key aspects absent from Figure 5. It plots propulsion systems against their achievable thrust specific consumptions and shows the operational ranges of different engines. It illustrates the limitations of rocket propulsion in atmospheric flight and the advantages of different propellant types. Representations such as Figure 9 should replace diagrams like Figure 5, as they more clearly depict engine characteristics, explicitly indicate the influence of propellant type and flight speed, and provide accurate information on thrust specific consumption.

## Summary and Conclusions

This work has reviewed the basic principles governing aerospace propulsion, using a simple yet effective model that accurately represents thrust generation and enables the establishment of energy balance. This framework allows the identification of key aspects in aerospace propulsion, beginning with the fundamental classification of systems into autonomous and non-autonomous according to the vehicle-propellant relationship. The energy-balance approach shows how the energy supplied to the system is ultimately distributed, and how propulsion efficiency can be rigorously defined. The role of thrust specific consumption in determining overall vehicle performance has been illustrated through simplified forms of the Breguet and rocket equations.

A general analysis of propulsion systems led to the identification of the dimensionless parameter $x^*$, defined as

$$x^* = \sqrt{\frac{\eta_T q_f}{\frac{1}{2} v^2}} \qquad (73)$$

This parameter —the square root of the ratio between the energy per unit mass available from the power source and the specific kinetic energy of the vehicle— emerges as a key variable for comparing propulsion concepts. It links the thermal efficiency of the engine, the energy capacity of the propellant, and the flight speed, enabling direct estimation of the lowest thrust specific consumption achievable under given flight conditions.

The $x^*$-based framework enables a unified representation of autonomous and non-autonomous propulsion systems, showing how their performance domains complement each other. Airbreathing engines dominate in low to moderate flight speed regimes due to their superior specific consumption, while rocket engines become indispensable at very high speeds and outside the atmosphere, where airbreathing propulsion is inoperable. The diagrams developed also make explicit the influence of propellant energy content on performance, highlighting the gains achievable with high-energy propellants such as hydrogen.

This approach provides a concise but comprehensive tool for identifying the most efficient propulsion system for a given mission profile, clarifying the inherent trade-offs between architecture, propellant type, and flight regime. The results confirm that no single technology is optimal for all conditions, and that effective aerospace propulsion relies on the complementary strengths of both autonomous and non-autonomous systems.