Overview

Chapter 3

3.1.1 Classification

It is possible to categorize space propulsors following different criteria. The first type of classification is based on the purpose of the propulsor itself:

• Primary propulsion: it is designed and used in order to change the trajectory of the spacecraft.

• Auxiliary propulsion: it is designed and used in order to maintain the desired trajectory, withstanding and contrasting external disturbance actions.

On the other hand, it is possible to classify propulsor on the basis of the energy source used to accelerate the propellant. In particular, three different classes exist:

• Chemical propulsion: it exploits a chemical propellant or the reaction between two propellants — a fuel and an oxidizer — to generate thrust.

• Electrical propulsion: it exploits electromagnetic phenomena to accelerate the pro-pellant.

• Nuclear propulsion: it leverages nuclear power to generate thrust.

3.1.2 Relevant Entities

Since all the propulsors leverage the action-reaction principle, it is possible to describe in a general way their behaviour and to introduce some particularly relevant quantities, without specifying the class of the propulsor itself.

It is considered a body on which no external force is applied, thus its behaviour can be studied as a closed system. If this hypothesis is respected, the global momentum must be constant over time.

In a first moment, the body of mass m is moving with a velocity v. After an infinites-imal interval of time, the body has expelled a part of its mass: the propellant mass dm_P. Thus, the body loses a part of its mass but increases its velocity. The increment in ve-locity is related to the veve-locity with which the propellant infinitesimal mass is expelled:

c. This is the effective discharge velocity. It is worth highlighting that c is defined with respect to the spacecraft, thus the global velocity of the propellant is c − v.The scheme of the problem is shown in Fig. 3.1.

Figure 3.1: Momentum conservation scheme

This said, it is possible to impose the conservation of the total momentum of the system:

mv = (m − dm_P)(v + dv) − dm_P(c − v)

Carrying out the mathematical operations and neglecting the second degree infinites-imal terms, it is possible to obtain:

m dv = dm_Pc

Taking into account that the propellant is expelled in a continuous and not discrete way, it is possible to define the propellant flow:

m˙P = dmP

dt (3.1)

The preceding considerations yield to:

mdv

dt = ˙mPc

Since the term on the left is the product of mass and acceleration of the body, the term on the right must be the force applied to the body itself, according to Newton’s second law of motion. This force is nothing but the thrust:

T = ˙m_Pc (3.2)

with which is possible to express the thrust power : PT = 1

2T c = 1

2m˙Pc² (3.3)

which is the power necessary to accelerate the propellant at the velocity that generates a thrust of intensity T .

It is worth highlighting that the exit velocity is not, in general, equal to the effective discharge velocity. This is due to the fact that, when accelerated in the nozzle the propellant is subjected also to pressure forces. In particular, the portion of propellant that is still inside the nozzle — and therefore still belongs to m — exchanges a pressure force with the propellant that is already outside. The entity of this force, referred to as static thrust, depends on the difference between the exit pressure and the ambient pressure. Thus, the thrust is divided in two terms: a dynamic one — depending on the exhaust velocity — and a static one:

T = ˙mPue+ A_e(p_e− p₀)

where, u_eis the exhaust velocity, A_e is the exit section of the nozzle, p_eis the pressure in the exit section and p₀is the ambient pressure — equal to null in space —. Anyway, from the point of view of the mission analysis there is no interest in knowing how the thrust is divided between dynamic and static. This is the reason why the effective discharge

velocity is defined. As a matter of fact, it is c that is defined once that the thrust is known — and not vice versa —.

In general, for space propulsors the exit velocity is almost equal to the effective exhaust velocity:

c = T m˙P

(3.4) At this point it is worth introducing some other relevant quantities that describe the performances of the propulsor. The first one is the total impulse:

I_t= Z tf

t0

T dt (3.5)

which indicates the total propulsive power of the system. The bigger the total impulse, the higher is the propulsive cost of the mission that the spacecraft may carry out.

With the total impulse it is possible to define the specific impulse:

I_s= I_t

m_Pg₀ (3.6)

where m_P is the total propellant mass onboard and g₀ is the gravity acceleration on the Earth’s surface. If the thrust is constant:

It= T ∆t where ∆t is the functioning time of the propulsor.

Considering that if T and c are constants, also the propellant flow is:

mP = ˙mP∆t

Substituting these two expressions in equation (3.6), it is easy to obtain:

I_s = c

g₀ (3.7)

This equation means that, neglecting a constant, c and I_s have the same value. As a matter of fact, both the specific impulse and the effective discharge velocity are a measure of the efficiency with which the propellant is used to generate thrust. The higher the value, the more performant the thruster is.

Perhaps, it is easier to understand this crucial concept if the following example is introduced. If a propulsor which generates a thrust of an entity equal to the weight of its propellant mass on Earth is considered, its operative time is exactly the specific impulse. In fact:

T = m_Pg₀ −→ I_t= T ∆t = m_Pg₀∆t −→ I_s= I_t

m_Pg₀ = ∆t

This means that if two thrusters with the same propellant mass but different specific impulses are compared, the one with the highest specific impulse may:

• Function for the same time but with a higher thrust level

• Generate the same thrust but for a longer time

needless to say it is important to have a high specific impulse. This consideration is even more important if the concepts introduced in the following paragraph are considered.

3.1.3 Tsiolkovsky Equation

The Tsiolkovsky equation is perhaps the most important and surely the most iconic equation of space propulsion. As a matter of fact it is often referred to as the rocket equation. It relates the ideal propulsive cost of a manoeuvre with the propellant mass it needs to be carried out. The term ’ideal’ means that no losses or external disturbances are taken into account.

The starting point is the definition of the propulsive cost, or more precisely the characteristic velocity:

∆V = Z tf

t0

T

m dt (3.8)

which is the variation of the velocity of the body across the manoeuvre. Using equation (3.2), it is possible to express the characteristic velocity as:

∆V = Z tf

t0

c ˙m_P dt m

Considering that ˙m_P = − ˙m — the spacecraft’s mass decreases as the propellant is expelled — and taking into account equation (3.1):

∆V = − Z t_f

t0

c ˙m_P dm m

the integral is easy to determinate, if the assumption of constant c is adopted. In general, the effective discharge velocity is not constant, but an opportune average value is always possible to estimate. Anyway, solving the integral, the Tsiolkovsky equation is retrieved and expressed in its two forms:

∆V = c lnm₀ mf

⇐⇒ mf = m₀e^−∆Vc (3.9)

Since the propellant consumption and the propulsive cost are connected through an exponential relation, it is crucial to have an effective discharge velocity which is at least comparable with the characteristic velocity. If the specific impulse is too low the final mass is negligible if compared to the initial one, thus there is no possibility of carrying any payload. The exponential relation highlights a crucial concept. When accelerating a payload, the propulsor is not only accelerating the payload itself, but it is also accelerating the propellant that is needed to accelerate that payload. Thus, if the payload is increased, the propellant mass needed does not increase linearly but

exponentially, since the propulsor does not only have to accelerate extra payload, but to accelerate extra propellant.

Hence the need for a high specific impulse. The higher c, the less propellant is needed to accomplish the manoeuvre: this strongly hinders the vicious cycle of mass increasing.