Model Forum / General / Rockets / January 2006

This is what I asked for in this NG.
I found my answer.
This is my last post to this NG, good bye, so long and thanks for all the
fish.
My thanks and highest regards to those who provided intelligent answers
here.
Tom the Canuck.
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-------------------------Read this and learn something

Isaac Newton stated in his third law of motion that "for every action there
is an equal and opposite reaction." It is upon this principle that a rocket
operates. Propellants are combined in a combustion chamber where they
chemically react to form hot gases which are then accelerated and ejected at
high velocity through a nozzle, thereby imparting momentum to the engine.
The thrust force of a rocket motor is the reaction experienced by the motor
structure due to ejection of the high velocity matter. This is the same
phenomenon which pushes a garden hose backward as water flows from the
nozzle, or makes a gun recoil when fired.

Thrust

Thrust is the force that propels a rocket or spacecraft and is measured in
pounds, kilograms or Newtons. Physically speaking, it is the result of
pressure which is exerted on the wall of the combustion chamber.

The figure to the right shows a combustion chamber with an opening, the
nozzle, through which gas can escape. The pressure distribution within the
chamber is asymmetric; that is, inside the chamber the pressure varies
little, but near the nozzle it decreases somewhat. The force due to gas
pressure on the bottom of the chamber is not compensated for from the
outside. The resultant force F due to the internal and external pressure
difference, the thrust, is opposite to the direction of the gas jet. It
pushes the chamber upwards.

To create high speed exhaust gases, the necessary high temperatures and
pressures of combustion are obtained by using a very energetic fuel and by
having the molecular weight of the exhaust gases as low as possible. It is
also necessary to reduce the pressure of the gas as much as possible inside
the nozzle by creating a large section ratio. The section ratio, or
expansion ratio, is defined as the area of the exit Ae divided by the area
of the throat At.

The thrust F is the resultant of the forces due to the pressures exerted on
the inner and outer walls by the combustion gases and the surrounding
atmosphere, taking the boundary between the inner and outer surfaces as the
cross section of the exit of the nozzle. As we shall see in the next
section, applying the principle of the conservation of momentum gives

where q is the rate of the ejected mass flow, Pa the pressure of the ambient
atmosphere, Pe the pressure of the exhaust gases and Ve their ejection
speed. Thrust is specified either at sea level or in a vacuum.

Conservation of Momentum

The linear momentum (p), or simply momentum, of a particle is the product of
its mass and its velocity. That is,

Newton expressed his second law of motion in terms of momentum, which can be
stated as "the resultant of the forces acting on a particle is equal to the
rate of change of the linear momentum of the particle". In symbolic form
this becomes

which is equivalent to the expression F=ma.

If we have a system of particles, the total momentum P of the system is the
sum of the momenta of the individual particles. When the resultant external
force acting on a system is zero, the total linear momentum of the system
remains constant. This is called the principle of conservation of linear
momentum. Let's now see how this principle is applied to rocket mechanics.

Consider a rocket drifting in gravity free space. The rocket's engine is
fired for time t and, during this period, ejects gases at a constant rate
and at a constant speed relative to the rocket (exhaust velocity). Assume
there are no external forces, such as gravity or air resistance.

The figure below-left (a) shows the situation at time t. The rocket and fuel
have a total mass M and the combination is moving with velocity v as seen
from a particular frame of reference. At a time t later the configuration
has changed to that shown below-right (b). A mass M has been ejected from
the rocket and is moving with velocity u as seen by the observer. The rocket
is reduced to mass M-M and the velocity v of the rocket is changed to v+v.

Because there are no external forces, dP/dt=0. We can write, for the time
interval t

where P2 is the final system momentum, figure (b), and P1 is the initial
system momentum, figure (a). We write

If we let t approach zero, v/t approaches dv/dt, the acceleration of the
body. The quantity M is the mass ejected in t; this leads to a decrease in
the mass M of the original body. Since dM/dt, the change in mass of the body
with time, is negative in this case, in the limit the quantity M/t is
replaced by -dM/dt. The quantity u-(v+v) is Vrel, the relative velocity of
the ejected mass with respect to the rocket. With these changes, equation
(2.4) can be written as

The right-hand term depends on the characteristics of the rocket and, like
the left-hand term, has the dimensions of a force. This force is called the
thrust, and is the reaction force exerted on the rocket by the mass that
leaves it. The rocket designer can make the thrust as large as possible by
designing the rocket to eject mass as rapidly as possible (dM/dt large) and
with the highest possible relative speed (Vrel large).

In rocketry, the basic thrust equation is written as

where q is the rate of the ejected mass flow, Ve is the exhaust gas ejection
speed, Pe is the pressure of the exhaust gases at the nozzle exit, Pa is the
pressure of the ambient atmosphere, and Ae is the area of the nozzle exit.
The product qVe, which we derived above (Vrel x dM/dt), is called the
momentum, or velocity, thrust. The product (Pe-Pa)Ae, called the pressure
thrust, is the result of unbalanced pressure forces at the nozzle exit. As
we shall see latter, maximum thrust occurs when Pe=Pa.

Click here for example problem #2.1
(use your browser's "back" function to return)

Equation (2.6) may be simplified by the definition of an effective exhaust
gas velocity, C, defined as

Equation (2.6) then reduces to

Impulse & Momentum

In the preceding section we saw that Newton's second law may be expressed in
the form

Multiplying both sides by dt and integrating from a time t1 to a time t2, we
write

The integral is a vector known as the linear impulse, or simply the impulse,
of the force F during the time interval considered. The equation expresses
that, when a particle is acted upon by a force F during a given time
interval, the final momentum p2 of the particle may be obtained by adding
its initial momentum p1 and the impulse of the force F during the interval
of time.

When several forces act on a particle, the impulse of each of the forces
must be considered. When a problem involves a system of particles, we may
add vectorially the momenta of all the particles and the impulses of all the
forces involved. When can then write

For a time interval t, we may write equation (2.10) in the form

Let us now see how we can apply the principle of impulse and momentum to
rocket mechanics.

Consider a rocket of initial mass M which it launched vertically at time
t=0. The fuel is consumed at a constant rate q and is expelled at a constant
speed Ve relative to the rocket. At time t, the mass of the rocket shell and
remaining fuel is M-qt, and the velocity is v. During the time interval t, a
mass of fuel qt is expelled. Denoting by u the absolute velocity of the
expelled fuel, we apply the principle of impulse and momentum between time t
and time t+t. Please note, this derivation neglects the effect of air
resistance.

We write

We divide through by t and replace u-(v+v) with Ve, the velocity of the
expelled mass relative to the rocket. As t approaches zero, we obtain

Separating variables and integrating from t=0, v=0 to t=t, v=v, we obtain

which equals

The term -gt in equation (2.15) is the result of Earth's gravity pulling on
the rocket. For a rocket drifting in space, -gt is not applicable and can be
omitted. Furthermore, it is more appropriate to express the resulting
velocity as a change in velocity, or V. Equation (2.15) thus becomes

Click here for example problem #2.2

Note that M represents the initial mass of the rocket and M-qt the final
mass. Therefore, equation (2.16) is often written as

where mo/mf is called the mass ratio. Equation (2.17) is also known as
Tsiolkovsky's rocket equation, named after Russian rocket pioneer Konstantin
E. Tsiolkovsky (1857-1935) who first derived it.

In practical application, the variable Ve is usually replaced by the
effective exhaust gas velocity, C. Equation (2.17) therefore becomes

Alternatively, we can write

where e is a mathematical constant approximately equal to 2.71828.

Click here for example problem #2.3

For many spacecraft maneuvers it is necessary to calculate the duration of
an engine burn required to achieve a specific change in velocity.
Rearranging variables, we have

Click here for example problem #2.4

Combustion & Exhaust Velocity

The combustion process involves the oxidation of constituents in the fuel
that are capable of being oxidized, and can therefore be represented by a
chemical equation. During a combustion process the mass of each element
remains the same. Consider the reaction of methane with oxygen

This equation states that one mole of methane reacts with two moles of
oxygen to form one mole of carbon dioxide and two moles of water. This also
means that 16 g of methane react with 64 g of oxygen to form 44 g of carbon
dioxide and 36 g of water. All the initial substances that undergo the
combustion process are called the reactants, and the substances that result
from the combustion process are called the products.

The above combustion reaction is an example of a stoichiometric mixture.
That is, there is just enough oxygen present to chemically react with all
the fuel. The highest flame temperature is achieved under these conditions.
However, it is often desirable to operate a rocket engine at a "fuel-rich"
mixture ratio. Mixture ratio is defined as the mass flow of oxidizer divided
by the mass flow of fuel.

Consider the following reaction of kerosene(1) with oxygen,

Given the molecular weight of C12H26 is 170 and that of O2 is 32, we have a
mixture ratio of

which is typical of many rocket engines using kerosene, or RP-1, fuel.

As we have seen previously, impulse thrust is equal to the product of the
propellant mass flow rate and the exhaust gas ejection speed. The ideal
exhaust velocity is given by

where k is the specific heat ratio, R' is the universal gas constant
(8,314.51 N-m/kg-oK in SI units, or 49,720 ft-lb/slug-oR in U.S. units), Tc
is the combustion temperature, M is the average molecular weight of the
exhaust gases, Pc is the combustion chamber pressure, and Pe is the pressure
at the nozzle exit.

Specific heat ratio(2) varies depending on the composition and temperature
of the exhaust gases, but it is usually about 1.2. The thermodynamics
involved in calculating combustion temperatures are quite complicated,
however, flame temperatures generally range from about 2,500o to 3,600o C
(4,500o-6,500o F). Chamber pressures can range from about about 7 to 250
atmospheres. Pe should be equal to the ambient pressure at which the engine
will operate, more on this later. Click Here for charts providing optimum
mixture ratio, adiabatic flame temperature, gas molecular weight, and
specific heat ratio for some common rocket propellants.

From equation (2.22) we see that high chamber temperature and pressure, and
low exhaust gas molecular weight results in high ejection velocity, thus
high thrust. Based on this criterion, we can see why liquid hydrogen is very
desirable as a rocket fuel.

Click here for example problem #2.5

It should be pointed out that in the combustion process there will be a
dissociation of molecules among the products. That is, the high heat of
combustion causes the separation of molecules into simpler constituents
which are then capable of recombining. Consider the reaction of kerosene
with oxygen. The true products of combustion will be an equilibrium mixture
of atoms and molecules consisting of C, CO, CO2, H, H2, H2O, HO, O, and O2.
Dissociation has a significant effect on flame temperature.

     (1) In dealing with combustion of liquid hydrocarbon fuels it is
convenient to express the composition in terms of a single hydrocarbon, even
though it is a mixture of many hydrocarbons. Thus gasoline is usually
considered to be octane, C8H18, and kerosene is considered to be dodecane,
C12H26.

     (2) Specific heat, or heat capacity, represents the amount of heat
necessary to raise the temperature of one gram of a substance one degree C.
Specific heat is measured at constant-pressure, Cp, or at constant-volume,
Cv. The ratio Cp/Cv is called the specific heat ratio, represented by k.

Specific Impulse

The specific impulse of a rocket, Isp, is the ratio of the thrust to the
flow rate of the weight ejected, that is

where F is thrust, q is the rate of mass flow, and g is the acceleration of
gravity at ground level.

Specific impulse is expressed in seconds. When the thrust and the flow rate
remain constant throughout the burning of the propellant, the specific
impulse is the time for which the rocket engine provides a thrust equal to
the weight of the propellant consumed.

For a given engine, the specific impulse has different values on the ground
and in the vacuum of space because the ambient pressure is involved in the
expression for the thrust. It is therefore important to state whether
specific impulse is the value at sea level or in a vacuum.

There are a number of losses within a rocket engine, the main ones being
related to the inefficiency of the chemical reaction (combustion) process,
losses due to the nozzle, and losses due to the pumps. Overall, the losses
affect the efficiency of the specific impulse. This is the ratio of the real
specific impulse (at sea level, or in a vacuum) and the theoretical specific
impulse obtained with an ideal nozzle from gases coming from a complete
chemical reaction. Calculated values of specific impulse are several percent
higher than those attained in practice.

Click here for example problem #2.6

From Equation (2.8) we can substitute qC for F in Equation (2.23), thus
obtaining

Equation (2.24) is very useful when solving Equations (2.18) through (2.21).
It is rare we are given the value of C directly, however rocket engine
specific impulse is a commonly given parameter.

Engines & Nozzles

A typical rocket motor consists of the combustion chamber, the nozzle, and
the injector, as shown in the figure below. The combustion chamber is where
the burning of propellants takes place at high pressure. The chamber must be
strong enough to contain the high pressure generated by, and the high
temperature resulting from, the combustion process. Because of the high
temperature and heat transfer, the chamber and nozzle are usually cooled.
The chamber must also be of sufficient length to ensure complete combustion
before the gases enter the nozzle.

The function of the nozzle is to convert the chemical-thermal energy
generated in the combustion chamber into kinetic energy. The nozzle converts
the slow moving, high pressure, high temperature gas in the combustion
chamber into high velocity gas of lower pressure and temperature. Since
thrust is the product of mass and velocity, a very high gas velocity is
desirable. Nozzles consist of a convergent and divergent section. The
minimum flow area between the convergent and divergent section is called the
nozzle throat. The flow area at the end of the divergent section is called
the nozzle exit area. The nozzle is usually made long enough (or the exit
area is great enough) such that the pressure in the combustion chamber is
reduced at the nozzle exit to the pressure existing outside the nozzle. It i
s under this condition, Pe=Pa where Pe is the pressure at the nozzle exit
and Pa is the outside ambient pressure, that thrust is maximum and the
nozzle is said to be adapted, also called optimum or correct expansion. When
Pe is greater than Pa, the nozzle is under-extended. When the opposite is
true, it is over-extended.

We see therefore, a nozzle is designed for the altitude at which it has to
operate. At the Earth's surface, at the atmospheric pressure of sea level
(0.1 MPa or 14.7 psi), the discharge of the exhaust gases is limited by the
separation of the jet from the nozzle wall. In the cosmic vacuum, this
physical limitation does not exist. Therefore, there have to be two
different types of engines and nozzles, those which propel the first stage
of the launch vehicle through the atmosphere, and those which propel
subsequent stages or control the orientation of the spacecraft in the vacuum
of space.

The figure above-right shows three different exhaust nozzles. The most
efficient nozzle (1) is contoured to the exhaust stream, allowing the
escaping gas to expand just enough to fill the nozzle. A nozzle that lets
the gas expand too much (2), or too little (3), wastes the energy and thrust
potential of the exhaust system.

The nozzle throat area, At, can be found if the total propellant flow rate
is known and the propellants and operating conditions have been selected.
Assuming perfect gas law theory, we have

where q is the propellant mass flow rate, Pt is the gas pressure at the
nozzle throat, Tt is the gas temperature at the nozzle throat, R' is the
universal gas constant, and k is the specific heat ratio. Pt and Tt are
given by

where Pc is the combustion chamber pressure and Tc is the combustion chamber
flame temperature.

Click here for example problem #2.7

The hot gases must be expanded in the diverging section of the nozzle to
obtain maximum thrust. The pressure of these gases will decrease as energy
is used to accelerate the gas. We must find that area of the nozzle where
the gas pressure is equal to the outside atmospheric pressure. This area
will then be the nozzle exit area.

Mach number Nm is the ratio of the gas velocity to the local speed of sound.
The Mach number at the nozzle exit is given by the perfect gas expansion
expression

where Pa is the pressure of the ambient atmosphere.

The nozzle exit area, Ae, corresponding to the exit Mach number is given by

The section ratio, or expansion ratio, is defined as the area of the exit Ae
divided by the area of the throat At.

Click here for example problem #2.8

For additional information, please see Supplement #1: Optimizing Expansion
for Maximum Thrust.

Power Cycles

Liquid bipropellant rocket engines can be categorized according to their
power cycles, that is, how power is derived to feed propellants to the main
combustion chamber. Described below are some of the more common types.

Gas-generator cycle: The gas-generator cycle, also called open cycle, taps
off a small amount of fuel and oxidizer from the main flow (typically 3 to 7
percent) to feed a burner called a gas generator. The hot gas from this
generator passes through a turbine to generate power for the pumps that send
propellants to the combustion chamber. The hot gas is then either dumped
overboard or sent into the main nozzle downstream. Increasing the flow of
propellants into the gas generator increases the speed of the turbine, which
increases the flow of propellants into the main combustion chamber, and
hence, the amount of thrust produced. The gas generator must burn
propellants at a less-than-optimal mixture ratio to keep the temperature low
for the turbine blades. Thus, the cycle is appropriate for moderate power
requirements but not high-power systems, which would have to divert a large
portion of the main flow to the less efficient gas-generator flow.

As in most rocket engines, some of the propellant in a gas generator cycle
is used to cool the nozzle and combustion chamber, increasing efficiency and
allowing higher engine temperature and efficiency.

Staged combustion cycle: In a staged combustion cycle, also called closed
cycle, the propellants are burned in stages. Like the gas-generator cycle,
this cycle also has a burner, called a preburner, to generate gas for a
turbine. The preburner taps off and burns a small amount of one propellant
and a large amount of the other, producing an oxidizer-rich or fuel-rich hot
gas mixture that is mostly unburned vaporized propellant. This hot gas is
then passed through the turbine, injected into the main chamber, and burned
again with the remaining propellants. The advantage over the gas-generator
cycle is that all of the propellants are burned at the optimal mixture ratio
in the main chamber and no flow is dumped overboard. The staged combustion
cycle is often used for high-power applications. The higher the chamber
pressure, the smaller and lighter the engine can be to produce the same
thrust. Development cost for this cycle is higher because the high pressures
complicate the development process. Further disadvantages are harsh turbine
conditions, high temperature piping required to carry hot gases, and a very
complicated feedback and control design.

Staged combustion was invented by Soviet engineers and first appeared in
1960. In the West, the first laboratory staged combustion test engine was
built in Germany in 1963.

Expander cycle: The expander cycle is similar to the staged combustion cycle
but has no preburner. Heat in the cooling jacket of the main combustion
chamber serves to vaporize the fuel. The fuel vapor is then passed through
the turbine and injected into the main chamber to burn with the oxidizer.
This cycle works with fuels such as hydrogen or methane, which have a low
boiling point and can be vaporized easily. As with the staged combustion
cycle, all of the propellants are burned at the optimal mixture ratio in the
main chamber, and typically no flow is dumped overboard; however, the heat
transfer to the fuel limits the power available to the turbine, making this
cycle appropriate for small to midsize engines. A variation of the system is
the open, or bleed, expander cycle, which uses only a portion of the fuel to
drive the turbine. In this variation, the turbine exhaust is dumped
overboard to ambient pressure to increase the turbine pressure ratio and
power output. This can achieve higher chamber pressures than the closed
expander cycle although at lower efficiency because of the overboard flow.

Pressure-fed cycle: The simplest system, the pressure-fed cycle, does not
have pumps or turbines but instead relies on tank pressure to feed the
propellants into the main chamber. In practice, the cycle is limited to
relatively low chamber pressures because higher pressures make the vehicle
tanks too heavy. The cycle can be reliable, given its reduced part count and
complexity compared with other systems.

Engine Cooling

The heat created during combustion in a rocket engine is contained within
the exhaust gases. Most of this heat is expelled along with the gas that
contains it; however, heat is transferred to the thrust chamber walls in
quantities sufficient to require attention.

Thrust chamber designs are generally categorized or identified by the hot
gas wall cooling method or the configuration of the coolant passages, where
the coolant pressure inside may be as high as 500 atmospheres. The high
combustion temperatures (2,500 to 3,600o K) and the high heat transfer rates
(up to 16 kJ/cm2-s) encountered in a combustion chamber present a formidable
challenge to the designer. To meet this challenge, several chamber cooling
techniques have been utilized successfully. Selection of the optimum cooling
method for a thrust chamber depends on many considerations, such as type of
propellant, chamber pressure, available coolant pressure, combustion chamber
configuration, and combustion chamber material.

Regenerative cooling is the most widely used method of cooling a thrust
chamber and is accomplished by flowing high-velocity coolant over the back
side of the chamber hot gas wall to convectively cool the hot gas liner. The
coolant with the heat input from cooling the liner is then discharged into
the injector and utilized as a propellant.

Earlier thrust chamber designs, such as the V-2 and Redstone, had low
chamber pressure, low heat flux and low coolant pressure requirements, which
could be satisfied by a simplified "double wall chamber" design with
regenerative and film cooling. For subsequent rocket engine applications,
however, chamber pressures were increased and the cooling requirements
became more difficult to satisfy. It became necessary to design new coolant
configurations that were more efficient structurally and had improved heat
transfer characteristics.

This led to the design of "tubular wall" thrust chambers, by far the most
widely used design approach for the vast majority of large rocket engine
applications. These chamber designs have been successfully used for the
Thor, Jupiter, Atlas, H-1, J-2, F-1, RS-27 and several other Air Force and
NASA rocket engine applications. The primary advantage of the design is its
light weight and the large experience base that has accrued. But as chamber
pressures and hot gas wall heat fluxes have continued to increase (>100
atm), still more effective methods have been needed.

One solution has been "channel wall" thrust chambers, so named because the
hot gas wall cooling is accomplished by flowing coolant through rectangular
channels, which are machined or formed into a hot gas liner fabricated from
a high-conductivity material, such as copper or a copper alloy. A prime
example of a channel wall combustion chamber is the SSME, which operates at
204 atmospheres nominal chamber pressure at 3,600o K for a duration of 520
seconds. Heat transfer and structural characteristics are excellent.

In addition to the regeneratively cooled designs mentioned above, other
thrust chamber designs have been fabricated for rocket engines using dump
cooling, film cooling, transpiration cooling, ablative liners and radiation
cooling. Although regeneratively cooled combustion chambers have proven to
be the best approach for cooling large liquid rocket engines, other methods
of cooling have also been successfully used for cooling thrust chamber
assemblies. Examples include:

Dump cooling, which is similar to regenerative cooling because the coolant
flows through small passages over the back side of the thrust chamber wall.
The difference, however, is that after cooling the thrust chamber, the
coolant is discharged overboard through openings at the aft end of the
divergent nozzle. This method has limited application because of the
performance loss resulting from dumping the coolant overboard. To date, dump
cooling has not been used in an actual application.

Film cooling provides protection from excessive heat by introducing a thin
film of coolant or propellant through orifices around the injector periphery
or through manifolded orifices in the chamber wall near the injector or
chamber throat region. This method is typically used in high heat flux
regions and in combination with regenerative cooling.

Transpiration cooling provides coolant (either gaseous or liquid propellant)
through a porous chamber wall at a rate sufficient to maintain the chamber
hot gas wall to the desired temperature. The technique is really a special
case of film cooling.

With ablative cooling, combustion gas-side wall material is sacrificed by
melting, vaporization and chemical changes to dissipate heat. As a result,
relatively cool gases flow over the wall surface, thus lowering the
boundary-layer temperature and assisting the cooling process.

With radiation cooling, heat is radiated from the outer surface of the
combustion chamber or nozzle extension wall. Radiation cooling is typically
used for small thrust chambers with a high-temperature wall material
(refractory) and in low-heat flux regions, such as a nozzle extension.

Solid Fuel Geometry

A solid fuel's geometry determines the area and contours of its exposed
surfaces, and thus its burn pattern. There are two main types of solid fuel
blocks used in the space industry. These are cylindrical blocks, with
combustion at a front, or surface, and cylindrical blocks with internal
combustion. In the first case, the front of the flame travels in layers from
the nozzle end of the block towards the top of the casing. This so-called
end burner produces constant thrust throughout the burn. In the second, more
usual case, the combustion surface develops along the length of a central
channel. Sometimes the channel has a star shaped, or other, geometry to
moderate the growth of this surface.

The shape of the fuel block for a rocket is chosen for the particular type
of mission it will perform. Since the combustion of the block progresses
from its free surface, as this surface grows, geometrical considerations
determine whether the thrust increases, decreases or stays constant.

Fuel blocks with a cylindrical channel (1) develop their thrust
progressively. Those with a channel and also a central cylinder of fuel (2)
produce a relatively constant thrust, which reduces to zero very quickly
when the fuel is used up. The five pointed star profile (3) develops a
relatively constant thrust which decreases slowly to zero as the last of the
fuel is consumed. The 'cruciform' profile (4) produces progressively less
thrust. Fuel in a block with a 'double anchor' profile (5) produces a
decreasing thrust which drops off quickly near the end of the burn. The
'cog' profile (6) produces a strong inital thrust, followed by an almost
constant lower thrust.

Staging

Multistage rockets allow improved payload capability for vehicles with a
high V requirement such as launch vehicles or interplanetary spacecraft. In
a multistage rocket, propellant is stored in smaller, separate tanks rather
than a larger single tank as in a single-stage rocket. Since each tank is
discarded when empty, energy is not expended to accelerate the empty tanks,
so a higher total V is obtained. Alternatively, a larger payload mass can be
accelerated to the same total V. For convenience, the separate tanks are
usually bundled with their own engines, with each discardable unit called a
stage.

Multistage rocket performance is described by the same rocket equation as
single-stage rockets, but must be determined on a stage-by-stage basis. The
velocity increment, Vi, for each stage is calculated as before,

where moi represents the total vehicle mass when stage i is ignited, and mfi
is the total vehicle mass when stage i is burned out but not yet discarded.
It is important to realize that the payload mass for any stage consists of
the mass of all subsequent stages plus the ultimate payload itself. The
velocity increment for the vehicle is then the sum of those for the
individual stages where n is the total number of stages.