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Liquid Rocket Motor
Design
This page outlines the simple design
requirements and equations for a simple rocket motor, including points to
ponder along the way. This section
is not a comprehensive outline of
rocket motor design, as such an outline becomes extremely complicated when
pursuing optimum efficiency. For
detailed information, refer to specialized books on the subject such as Rocket Propulsion Elements (G.
Sutton) and the AIAA publication Modern
Engineering for Design of Liquid-Propellant Rocket Engines (K Huzel and
H. Huang). In the case of rocket motors useful for ASA, optimal efficiency
is not useful - due to the prohibitive costs associated with such
efficiency. For ASA's purposes,
efficiency will be improved when possible, but more emphasis will be placed
on safety and reliability. Even this
level of efficiency and design considerations greatly exceeds the
instructive capabilities of this web page, though, so the design elements
outlined here are a broad overview of the issues faced by propulsion system
designers. Feel free to ask ASA
specific questions, though, as it is our purpose to improve the
capabilities of the amateur.
Motor design begins with
questions designed to identify as many of design parameters and limitations
as possible. There are virtually
countless different ways to design a rocket motor, these initial questions
help to limit the scope of a new motor project to into something easier to
handle. The following are some of
the typical questions one needs to answer to begin motor design:
·
What is the purpose of the rocket motor?
·
How much thrust does it need to produce?
·
What is the design envelope (how big can it
be)?
·
Are there financial limitations?
·
Are there environmental concerns (both for
the rocket vehicle and occupants and the location of operation)?
These factors help determine
what propellants to use, how to pressurize the propellants, the operational
environment of the motor, and the physical size of the motor. The eventual efficiency of the rocket
motor directly linked to the answers to the above questions, but increased
restrictions typically result in a less efficient rocket motor.
Another important factor to
consider when designing a rocket motor is the eventual operation of that
motor, including propellant delivery and loading, the person or computer
(preferable) actually controlling the rocket while operating, and
safety. One major category of issues
that some designers face are the complications associated with cryogenic
propellants. A very common cryogenic
oxidizer is liquid oxygen (LOX), but at -297 degrees F it requires massive
infrastructure and planning, not to mention the safety issues coupled with
such a potent oxidizer. LOX may be
the most efficient method of burning a given propellant, but are the
hassles worth it? Some designers
choose to use less efficient oxidizers due to these issues.
ASA has chosen to use LOX as
an oxidizer for the initial design series since LOX is a very good
oxidizer, LOX performance documentation is widely available, and (most
importantly) the ASA motor designers have past experience with this
propellant.
Rocket
Motor Design Issues
Once the required motor
thrust has been determined, the next steps determine how long the motor
will operate, what propellants are best suited for the application, the
best way to deliver those propellants to the rocket motor, and what method
will be used to cool the rocket during operation. These steps seem sequential, but one
quickly finds that the process can only be adequately completed through
iteration since each answer changes the answer to each other question.
Typical fuels for liquid
rocket motors are kerosene (aka "RP-1"or "JP-(4,5,6,7,8,
etc)", various alcohols, and liquid hydrogen. Typical oxidizers are liquid oxygen
(LOX), hydrogen peroxide, and nitrous oxide. The ASA rocket motors use LOX and either
alcohol or kerosene. They are
pressure-fed motors, meaning that propellants are delivered to the motor
via helium-pressurized fuel tanks.
Why not use pumps to pressurize the fuel? The inert weight and complexity of a pump
makes it impractical for the relatively small rocket motors designed by ASA. The efficiency crossover point between
pressurized tanks and the use of a pump is somewhere around 20-40,000
pounds of thrust, depending on the application. When the thrust requirement is greater
than ~10,000 pounds of thrust or the motor is expected to have a long burn
time, however, the designer should incorporate the potential of a pump into
the design iteration. Why use
helium? It is the lightest
pressurant gas (good for flight) and it does not dissolve readily into LOX
or fuels, as do other pressurant gases like nitrogen or compressed air
(compressed air should never be used
to pressurize LOX due to contamination concerns). Since the propellant tanks must be
pressurized, this limits the motor combustion pressure. High combustion pressures are more
efficient, but require increasingly heavy fuel tanks. The optimal trade between efficiency and
tank weight becomes another iterative calculation dependent on vehicle
flight performance.
Another driving factor for
propellant and combustion pressure decisions is the expected combustion
temperature inside the thrust chamber.
Typical combustion temperatures are hot enough to melt the
combustion chamber, so the chamber must be protected. Two methods are commonly used to protect
the walls of the thrust chamber; active cooling using propellants which are
about to be burned or passive cooling using a ablative liner which is
designed to burn away during motor operation. Active cooling is usually better at
protecting the chamber walls, but is much more expensive and complex than
an ablative liner. ASA uses ablative
liners made of composite materials like fiberglass, carbon fiber, and high
temperature epoxies. There are three
significant limitations associated with ablative liners - motor operation
time, combustion temperature extremes, and thrust chamber geometry
changes. During operation, the
ablative is worn away, resulting in a gradual increase in the inner
diameter of the thrust chamber. This
geometry change can cause combustion instabilities when the chamber
diameter gets too large, hence ablative liners are only useful for short-duration
motors. If long burn times are
needed, some sort of active cooling is necessary. Additionally, ablative liners cannot
withstand the highest combustion temperatures expected of some
fuel/combustion chamber pressure combinations. If the expected combustion temperature is
too high, then active cooling (or very short motor operation) is the only
option. Since the ASA rocket burn
times are relatively short (under 60 seconds), and as a long as the
combustion temperature is carefully predicted, an ablative liner is the
logical choice for ASA's thrust chamber protection.
All of the parameters
mentioned to this point relate to each other inextricably. From a global perspective, every device,
bolt, camera, fin, and article of payload affects the design or mass
allowance of every other component on board the rocket. This interrelation
wreaks havoc on the design of a rocket motor, since the change of each item
can change the initial conditions of design for all items. The designer must determine how to relate
these parameters in order to iterate the design effectively. ASA,
therefore, has developed a spreadsheet application called REPSOP in an
attempt to relate these parameters.
Every component on board the rocket that can be expressed
mathematically (even if just by weight) is related in REPSOP, and this
provides the user with instant information of how a given design change
affects the rocket as a whole. This
kind of tool is extremely useful to the propulsion systems designer. As mentioned on the ISP calculation page,
ASA also uses third party chemical analysis documents and tools to obtain
expected combustion properties of various fuels. This data is incorporated into REPSOP to
further improve the iterative capabilities of the tool.
The Math:
Propellant Mass Flowrate
Starting with the thrust and
a range of combustion pressures/temperatures, a potential fuel/oxidizer
combination emerges. From this, one
can estimate the motor's specific impulse, as outlined on the ISP page. Since thrust is known at this point, the
required propellant flowrate (mdot) can be calculated by dividing thrust by
ISP: mdot=thrust/ISP
Nozzle
The nozzle design is
influenced by the propellant combustion properties and attempts to improve
efficiency (since the effectiveness of a nozzle in accelerating gas to
supersonic speeds is directly related to motor thrust). The nozzle inlet area, throat area, and
exit area are defined by combustion parameters and the optimum expansion
altitude (expansion is briefly outlined on the ISP page). The curvy shape of the nozzle exit
(sometimes called a "Bell Nozzle" is best calculated by CFD
analysis. These shapes, and the
equations used to create them are carefully guarded secrets of most rocket
motor designers. A simple nozzle
which is easy to construct and of relatively high efficiency is a
"45/15" nozzle, meaning that the nozzle inlet is a 45 degree
cone, and the nozzle exit is an opposite 15 degree cone.
The following calculations
show how the nozzle inlet, throat, and exit areas can be obtained:
User-defined values:
T: Combustion Temperature
P: Combustion Pressure
Pe: Nozzle Exhaust
Pressure
R: Gas Constant of Exhaust Products
k: Specific Heat Ratio
M: Chamber mach number
ISP: Motor Specific Impulse
First, the nozzle inlet
conditions must be calculated:
Inlet Stagnation Pressure =
P*(1+0.5*(k-1)*M^2)^(k/(k-1))
Inlet Stagnation Temp =
T*(1+(0.5*(k-1)*M^2))
Inlet Velocity = M*(k* R *
Inlet Stagnation Temp)^0.5
Inlet Specific Volume =
(R*T)/(144*P*32.174)
Inlet Flow Area (or,
Combustion Chamber Diameter) = (144*mdot* Inlet Specific Volume)/ Inlet
Velocity
Next, the nozzle throat area
can be calculated:
Throat Pressure = Inlet Stagnation
Pressure *((2/(k+1))^(k/(k-1)))
Throat Temperature = Inlet
Stagnation Temp *( Throat Pressure / Inlet Stagnation Pressure)^((k-1)/k)
Throat Velocity (Mach 1) =
(k*R* Throat Temperature)^0.5
Throat Specific Volume = (R*
Throat Temperature)/(144* Throat Pressure*32.174)
Throat Flow Area = (144*mdot*
Throat Specific Volume)/ Throat Velocity
Finally, the nozzle exit area
can be calculated
Exit Area Ratio =
1/(((k+1)/2)^(1/(k-1))*( Pe /P)^(1/k)*(((k+1)/(k-1))*(1-( Pe
/P)^((k-1)/k)))^0.5)
Exit Flow Area = Throat Flow
Area * Exit Area Ratio
A simple method of
manufacturing a nozzle is to carve the inside of a cylinder of graphite
into a nozzle shape, and then use a metal thrust chamber to contain the
cylinder. ASA uses this method for
simple rocket nozzles, since the added weight is worth the ease in
manufacturing. For large nozzles,
this method is not efficient. When
using the above calculations, one finds that the throat temperature is
still above the melting point of most metals, indicating that some sort of
cooling is needed in the throat of the nozzle, in addition to the
combustion chamber. An ablative
liner is not practical for this location, due to the erosion issues
discussed earlier. If the nozzle
throat area grows during motor operation, the efficiency of the motor would
be greatly compromised. Since the
melting point of graphite is extremely high (at ~6700 degrees F, carbon
sublimes), a graphite nozzle is a good alternative to active cooling.
Propellant Injector
There are many different types
of injectors, each suited for a different type of rocket motor. To keep this discussion focused, this
page will only review "shower head" type liquid injectors. Like other intricate parts of rocket
motors, injector design is a highly guarded secret of motor
manufacturers. Designing a manifold
to inject propellants is not complex, but designing one which is
lightweight, has minimal pressure losses, and is robust is much more
complicated. Modern CFD analysis
makes injector design easier, but this is still complicated. There is no "standard' way of
designing an injector, since each application will be different. This page,
therefore, will review the basic design ideas behind showerhead injectors. As the name implies, this type of injector
looks like a shower head, and injects small streams of each propellant into
the combustion chamber. These
streams must atomize as fast as possible for efficient combustion, so
liquid injectors typically force the streams of propellant to hit each
other, encouraging mechanical atomization.
The positioning of these streams across the face of the injector,
the diameters of the streams, and the impingement angles are all up to the
designer, but follow "good ideas" such as - the injected mass
should be evenly distributed across the face to reduce hot spots; and the
streams should be small enough, fast enough, and at an impingement angle
best suited to force atomization.
Since we know the total motor
mdot and the mixture ratio of propellants (MR=oxidizer/fuel), we know how
many pounds per second of each propellant must be injected into the
combustion chamber.
Fuel mdot = (motor
mdot*MR)/(MR+1)
Oxidizer mdot = motor
mdot/(MR+1)
Once these values are known,
they are be converted into volume flowrates using known propellant density,
and then divided by the user-defined number of injector elements to find
the injector velocity of each small jet.
An "element" is defined as a small grouping of oxidizer
and fuel jets. Common injector
elements are FO, FOF, OFFO, etc (F for fuel and O for oxidizer) and
empirical data of different element types is available in various rocket
design books. Element testing shows
that some elements are better for one type of propellant or another, but
the most important design criteria is that the element must maximize
propellant atomization. If the
injected propellants are not atomized fast enough, they may pass out of the
motor without burning, wasting the fuel and reducing efficiency. Books should be consulted to properly
pick an injector and element type for a given application. The orifice size for each propellant
should be small enough to ensure atomization but large enough to prevent
unnecessary pressure drop. ASA uses
FOOF, FOF, and FO elements, and uses orifice sizes smaller than
0.1”. Lastly, the velocity of
the injected propellant should fast enough so that the injector is not
sensitive to chamber pressure oscillations, which could travel upstream
into the injector (if the pressure drop and injection velocity is not high enough). ASA targets injection velocities of 75
ft/sec and overall injection pressure drops of ~15% of the combustion
chamber pressure.
Combustion Chamber
The size of the combustion
chamber must be adequate to provide enough time o fully burn all injected
propellant, but not so big as to be susceptible to pressure oscillations
and instability. Instead of
calculating the length of time it takes each droplet of fuel to atomize and
fully burn, motor design typically uses empirical data from previously successful
rocket motors to determine the proper size of the combustion chamber. This parameter is called L*
("ell-star") or "Combustion Chamber Characteristic
Length", one targets values of L* that are appropriate for a give
fuel/oxidizer combination. LOX/Kerosene
L*'s for example, should be in the 40-50 range. L* can be expressed as:
L* = Chamber Volume/Throat
Area
Since the throat area is
already known, and L* is available from known data, the chamber volume is
the only variable. For simplicity,
ASA uses cylindrical thrust chambers with one end being the injector and
the other being the nozzle. Since
the diameter of the chamber was already calculated (the nozzle inlet area),
the only remaining variable in the size of the combustion chamber is the
length, which can be estimated using the L* equation, above.
Propellant Tanks
Since burn time and mixture
ratio are known, the total mass requirement for each propellant can be
determined by:
Total fuel mass = Burn time*
Fuel mdot
Total oxidizer mass = Burn
time* Oxidizer mdot
The tank volumes are
calculated using known propellant densities. Again, ASA is using pressurized tanks to
deliver the propellants to the motor at the required pressures. The propellant tank pressure setting is defined
by the target combustion pressure and the line losses associated in
delivering the propellants to the combustion chamber:
Propellant Tank Pressure =
Injection pressure + Injector pressure loss + Plumbing line loss -
Acceleration induced head pressure (if applicable)
The injector pressure losses
can be estimated using standard fluid dynamics equations and assuming
average flowpath lengths, turns, and restrictions inside the injector. The overall feedline pressure losses can
be calculated using the same fluid dynamics equations for the various
components of the plumbing system (valves (ball, check, solenoid), tubing,
fittings, etc). Once the system is
assembled, the accuracy of the initial estimates can be measured by running
pressurized water through the completed plumbing system and then using
conversion equations to relate the water pressure losses to that of actual
propellants. A notable advantage of
a pressurized tank system is the ability to change the propellant injection
pressure by simply opening or closing the flow setting on the helium
regulator delivering helium to the pressurized tank. If the measured pressure drop through the
completed plumbing/injector system has a lower or higher pressure loss than
was calculated, this can be accounted for by adjusting the regulator
(within reason).
Overview
The proceeding sections
reviewed basic design for the components of a rocket motor. Detailed design requires a deeper
understanding of the various components of the rocket motor, and can be obtained
through the two books recommended at the beginning of this page or a
variety of other resources. More
research is required to properly design a liquid fueled rocket motor -
rocket motors are highly dangerous, and one should take their design very seriously.
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