PSAS/ 2011 liquid motor notes
  1. Renders (5/4/02011)
  2. Plan of Attack
  3. Current Issues List
  4. Injector Element Design
  5. Engine Parameters
    1. Engine
    2. Injector
  6. Injector Plates
      1. Preliminary design parameters
      2. Preliminary Flow Calculations
    1. Topics for discussion
  7. Useful Mathematical Relationships and Information:
    1. Notation Guide
    2. Formulae
  8. References

Renders (5/4/02011)

Plan of Attack

Current Issues List

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:

\overset{\centerdot}{m}_f * v_f * cos(\gamma_f)= \overset{\centerdot}{m}_o * v_o * cos (\gamma_o)

where \gamma_f and 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:

\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:

180^o = 60^o + \gamma_f + \gamma_o

Choosing \gamma_f = 45^o arbitrarily, we can solve algebraically to show that\gamma_o = 75^o

\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:

\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 gammaf and gammao set equal to 60o work well.

Engine Parameters



Characteristic Value
Thrust 72.779
Mass Flow 0.0917 kg/s
Chamber Pressure 6.895e6 Pa (1 ksi, 68.05 atm)
O/F Ratio 2.29
Ratio of Specific Heats 1.22
Adiabatic Flame Temperature 3525 K
Molar Weight of Gases 21.6
Exit Velocity 2831.7 m/s
Combustion Chamber Volume 1.7393 ( in3)
Throat Diameter 5.49 mm (0.2161 in)
Exit Diameter 16.1 mm (0.6323 in)


Characteristic Value
Elements 4
Fuel Hole Diameter .344 mm (0.01354 in)
Oxygen Hole Diameter .338 mm (0.0133 in)
Fuel Stream Impingement Angle 22.5o
Oxygen Stream Impingement Angle 13.19o

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.


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):

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

Useful Mathematical Relationships and Information:

Notation Guide

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


R = \frac{R'}{M} whereR' is the universal gas constant andM the average molecular weight of the exhaust gases which can be found here:

\overset{\centerdot}{w} = \frac{A_tv_t}{V_t} (RPE 3-24)

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

v_t = \sqrt{gkRT_t} = \sqrt{\frac{2gkRT_c}{k+1}} (RPE 3-23)

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}}

N_m^2 = \left(\frac{2}{k-1} \right) \left[\left(\frac{P_c}{P_e}\right)^\frac{k-1}{k}-1 \right] ( 1-29)

T_t = T_c \left(\frac{2}{k+1} \right) (RPE 3-22)

V_t = V_c \left(\frac{k+1}{2}\right)^\frac{1}{k-1} (RPE 3-21)

P_t = P_c * \left (1+\frac{k-1}{2} \right) ^\frac{-k}{k-1} (Isentropic compression? Sourced from

V_c = \frac{RT_c}{P_c} (Ideal Gas Law)

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

A_i = \frac{\overset{\centerdot}{w}V_i}{v_i} (RPE 3-24)

A_t = \frac{\overset{\centerdot}{w}}{P_t}*\sqrt{\frac{R'T_t}{Mk}} (Braeunig 1.26)

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)


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