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[[!toc levels=3]]

#News and Updates#
[[!inline rootpage="news" pages="news/* and !news/*/* and tagged(propulsion)" archive="yes" show="5" sort="title" reverse="yes" template="titlepage"]]

#Program Justification#

It is possible to insert a payload into orbit with big dumb boosters alone; the Japanese proved that with the Lambda 4S. Benjamin's google-fu has proved insufficient to determine much about the L4S' control system, other than that the upper stages used chlorofluorocarbon injection into the exhaust stream for thrust vectoring. Injecting liquids and gases into the motor exhaust seems an *intuitively* inefficient method to vector thrust (an intuition validated by the fact that rear-stage impacts on the forward stages drove the control system outside of its authority domains). India's Polar Satellite Launch Vehicle uses a flex nozzle for thrust vectoring on the third stage. The Polar Satellite Launch Vehicle also has an entire stage fueled by hydrazine and nitrogen tetraoxide (!).

It would probably be worthwhile to elide the requirements for a long-term vehicle development plan:

 - Control of Pitch_a_, Pitch_b_ and roll for the entire flight duration.

This requirement is driven by fuel considerations. If the orbital vehicle can hit well-optimized trajectories it will save buckets of fuel. This will reduce gravity losses and deliver either equivalent payloads with smaller rockets or larger payloads with equivalent rockets.
 - Hit some optimized point on the Isp/complexity tradeoff curve.
 - Reduce stage numbers to reduce complexity penalties.

Several scenarios are immediately apparent (and others are the result of fun with combinations):

 - Solids for all stages with aerodynamic controls while in atmosphere and some form of vectored thrust when aerodynamic controls lose effectivity.
 - Liquids for all stages with vectored thrust in all stages.
 - Some numerically-optimized combination of liquids and solids with either vectored thrust on all stages or only some stages.

It would be nice to have some objective criteria ("that's crazy" not being good engineering criteria) to evaluate solids vectoring vs. liquids vectoring. It seems obvious (again, *intuitively* that vectoring solids is a crazy plan; the unknown unknowns being a perceived larger set than the unknown unknowns of liquid development.

#Plan of Attack#

 - Build Pump
 - Choose fuel profile
 - Determine motor operating parameters

#Current Issues List#

#Design Choices#

##Cooling Choice##
Obvious Options:
- Ablative
- Regenerative
- Open-cycle
- Film

##Fuel Choices##

A exercise in combinatorics!

 - H2/O2
 - Lox/Kerosene
 - Lox/Alcohol
 - H2O2/Kerosene
 - H2O2/Alcohol

#Motor Operating Parameters#
A nice place to start designing the motor is a desired thrust level. Choosing too small of a thrust will make manufacturing difficult, and too large of a thrust will entail mass flows of lethal scales. There's an optimum in there somewhere, right?

#Injector Element Design#
It has been shown elsewhere (and examined to the satisfaction of Propulsion Team members) that a liquid flow resulting from the impingement of two jets directly opposed will flow axially down the chamber if the following holds:

[[!teximg code="\overset{\centerdot}{m}_f * v_f * cos(\gamma_f)= \overset{\centerdot}{m}_o * v_o * cos (\gamma_o)"]]

where [[!teximg code="\gamma_f"]] and [[!teximg code="gamma_o"]] are the angles between the chamber axis and the fuel and oxygen flows respectively.

However, because we are working with a three-port injector element, the radial mass flows are not necessarily so simple. We can rewrite the above equation to account for the additional angle like so:

[[!teximg code="\overset{\centerdot}{m}_f * v_f * cos (\gamma_f) = 2 * \overset{\centerdot}{m}_o * v_o * cos(\gamma_z) * cos (\gamma_o)"]] 

A common injector parameter is the angle between the streams of oxygen and fuel flow in the plane of fuel flow. The value we have found and are designing from for this parameter is sixty degrees. We can describe the triangle formed by the fuel stream impingement angles and the angles at which the streams leave the injector plate like so:

[[!teximg code="180^o = 60^o + \gamma_f + \gamma_o"]]

Choosing [[!teximg code="\gamma_f = 45^o"]] arbitrarily, we can solve algebraically to show that[[!teximg code="\gamma_o = 75^o"]]

[[!teximg code="\gamma_z"]] can be derived from trigonometric and algebraic manipulations on the 3 port injector element maths above. The derivation is left as an exercise to the reader, but the final result is as shown:

[[!teximg code="\gamma_z = cos^{-1} \left[\frac{\overset{\centerdot}{m_f}v_fcos(\gamma_f)}{2\overset{\centerdot}{m_o}v_ocos(\gamma_o)} \right]"]]

There are not-immediately-obvious singularities to avoid, but the values of gamma<sub>f</sub> and gamma<sub>o</sub> set equal to 60<sup>o</sup> work well.

#Engine Parameters#

#Injector Plates#

Some random note on this:

Pressure drop across injectors - Typically 15-25% of chamber pressure. High pressure drops increase stability (RPE p.284). Seems logical. Some sources cite delta p across injector ~41% of chamber pressure to further reduce feed system induced combustion instability.

Angle of injection - Set to get axial flow after impingement, based on mass flow and angle. i.e. "Resultant momentum at the point of impingement between the fuel and oxidizer flow is axially directed"

What is a good injection velocity? One publication cited a study of 5 m/s up to 50 m/s, over a range of chamber pressures, vs. stability. Less than 18-20 m/s cited stability issues. This is likely due to fuel entering the chamber at speeds less than that of the flame front.

Types:

- Coaxial injection
- Like On Like (LOL) Fuel impinges on Fuel, Ox on Ox.
- LOX on Fuel Impinging (armad?)
- Fuel Ox Fuel impinging (FOF) only seen with LH2 and LOX (Empirical Google presence)
- OFO Presumed higher efficiency, at expense of stability

###Preliminary design parameters###

delta P (injector) = 41% of Pc

Injection velocity = 30 m/s

Injection type: OFO

Impingement angle = Fuel: 45 deg.; Oxygen: 11.8586 deg.

Pre-Impingement distance = 5 mm

Orifice L/D ration = 18.4 or 80 (decide this)

###Preliminary Flow Calculations###

O/F ratio =~ 2.35

RPE (277):

[[!teximg code="Q = A * \sqrt{\frac{2\Delta p}{\rho}}"]]

Rearranging allows us to solve for total hole area in injector plate satisfying desired fuel flow and oxygen flow:

Ao = 1.158 E-6 m^2

Af = 6.123 E-7 m^2

The new script version computes these values and more, including from an input of number of impingement elements, the individual hole sizes.


##Topics for discussion##
 - Friction and surface tension effects along length of injector holes.


#Useful Mathematical Relationships and Information:

##Notation Guide
[[!table class="data" data="""
Symbol | Meaning 
[[!teximg code="R, R'"]] | Specific Gas Constant, Gas Constant.
[[!teximg code="\overset{\centerdot}{w}"]] | Mass flow (combined fuel and oxidizer).
[[!teximg code="A_i"]] | Cross-sectional area at a point in the engine.
[[!teximg code="v_i"]] | Gas velocity at a point in the engine.
[[!teximg code="V_i"]] | Gas specific volume.
[[!teximg code="P_i"]] | Gas pressure.
[[!teximg code="T_i"]] | Temperature.
[[!teximg code="_c"]] | Subscript denoting engine chamber.
[[!teximg code="_t"]] | ... engine throat.
[[!teximg code="_e"]] | ... exhaust exit.
[[!teximg code="_a"]] | ... ambient pressure.
[[!teximg code="g"]] | Gravitational acceleration.
[[!teximg code="k"]] | Ratio of specific heats at constant pressure and volume. Thermodynamic constant for specific gases.
[[!teximg code="{\eta}"]] | Thermal efficiency of the motor. Function of pressure and temperature ratios.
[[!teximg code="N_m"]] | Mach number (dimensionless ratio of speed to local speed of pressure wave propagation).
"""]]

##Formulae##
[[!teximg code="R = \frac{R'}{M}"]] where[[!teximg code="R'"]] is the universal gas constant and[[!teximg code="M"]] the average molecular weight of the exhaust gases which can be found here: <http://www.braeunig.us/space/comb.htm>.

[[!teximg code="\overset{\centerdot}{w} = \frac{A_tv_t}{V_t}"]] (RPE 3-24)

[[!teximg code="\overset{\centerdot}{w} = \frac{Fg}{c}"]] (RPE p. 52). This equation gives optimum fuel consumption as a function of thrust and exhaust speed c.

[[!teximg code="v_t = \sqrt{gkRT_t} = \sqrt{\frac{2gkRT_c}{k+1}}"]] (RPE 3-23)

[[!teximg code="v_e = \sqrt{\frac{2*k}{k-1}*\frac{R'T_c}{M}* \left(1-\frac{P_e}{P_c} \right)^\frac{k-1}{k}}"]]

[[!teximg code="N_m^2 = \left(\frac{2}{k-1} \right) \left[\left(\frac{P_c}{P_e}\right)^\frac{k-1}{k}-1 \right]"]] (<http://www.braeunig.us/space/propuls.htm> 1-29)

[[!teximg code="T_t = T_c \left(\frac{2}{k+1} \right)"]] (RPE 3-22)

[[!teximg code="V_t = V_c \left(\frac{k+1}{2}\right)^\frac{1}{k-1}"]] (RPE 3-21)

[[!teximg code="P_t = P_c * \left (1+\frac{k-1}{2} \right) ^\frac{-k}{k-1}"]] (Isentropic compression? Sourced from <http://www.braeunig.us/space/sup1.htm>)

[[!teximg code="V_c = \frac{RT_c}{P_c}"]] (Ideal Gas Law)

[[!teximg code="V_e = V_c * \left( \frac{P_c}{P_e} \right) ^\frac{1}{k}"]] (RPE 3-6 and p. 52)

[[!teximg code="A_i = \frac{\overset{\centerdot}{w}V_i}{v_i}"]] (RPE 3-24)

[[!teximg code="A_t = \frac{\overset{\centerdot}{w}}{P_t}*\sqrt{\frac{R'T_t}{Mk}}"]] (Braeunig 1.26)

[[!teximg code="A_e = \frac{A_t}{N_m} * \left[\frac{1+\frac{k-1}{2}*N_m^2}{\frac{k+1}{2}} \right]^\frac{k+1}{2*(k-1)}"]] (Braeunig 1.30)

#References#

*Liquid Rocket Engine Combustion Instability* by Vigor Yang, Yang, William E. Andersen (Editor) 

*Rocket Propulsion Elements* by George P. Sutton, Oscar Biblarz

*Spray Characteristics of Impinging Jet Injectors at High Back-Pressure*