...
The contraction ratio is the ratio of the cylindrical cross-sectional area of the chamber to the cross-sectional area of the throat of the nozzle. Larger engines typically have a low contraction ratio with a longer chamber length, and smaller engines typically have a larger contraction ratio with a smaller chamber length to have a large enough L* for complete fuel combustion. Along with L*, the contraction ratio governs how wide vs. how long the chamber will be. If the ratio is too small, the chamber might be needlessly long while increasing weight and cost, and if the ratio is too big the chamber might be either impractically wide or not allow for adequate gas flow.
Engine Design Process
After considering the parameters outlined above, the engine design process may begin. This process focuses heavily on engine geometry and dimensions including the comparison of our engine to others of similar size and performance and weighing benefits of certain design choices over others guided by established empirical data.
The design process must begin by making initial design choices as to derive other parameters of the engine. While this may appear to require experimentation to establish such parameters, empirical data from other engines is quite useful in doing so in a much more practical fashion. In our design these initial parameters include L*, contraction ratio, contraction angle, and divergent angle all of which will be chosen from empirical data.
Step 1 Derive Chamber Volume from L*
L* is a function of many propellant-specific factors. As our propellant choices are LOX and RP-1, we begin by choosing an L* value from the table below:
After choosing an L* within the desired range through previously listed considerations, we may use the following equation to derive a chamber volume:
Vc=chamber volume
At=throat area
We have now defined a chamber volume as a product of our chosen L* and throat area, allowing us to further define our chamber geometry through the following steps.
Step 2 Calculate Converging Section Parameters
The calculation of converging section parameters requires two design choices, the contraction ratio and the contraction angle, both of which are chosen from empirical data:
Contraction Ratio
Contraction Angle
“The half-angle of the nozzle convergent cone section can range from 20 to 45 deg.” (Huzel and Huang 1992)
With a contraction ratio and contraction angle defined, we may use the follow equation to solve for the converging chamber section length:
Rt=throat radius
ε=contraction ratio
R=arc radius=1.5*Rt
a=𝛼=contraction angle
Using the converging section length and the formula for the volume of a truncated cone, we may now derive the volume of the converging section of the chamber:
V=converging section volume
r1=chamber radius
r2=throat radius
h=converging section length
The chamber radius is derived from the contraction ratio equation shown below:
εc=contraction ratio
Ac=chamber cross-sectional area
At=throat cross-sectional area
Step 3 Extract Cylindrical Lengths
We would now like to extract both the cylindrical length and the complete length of the chamber. As we have obtained all necessary parameters for such derivations, the following calculations are possible through simple algebra and geometry.
Using the total chamber volume and the converging section volume, calculating the cylindrical volume is simply the difference between the two. Subsequently the cylindrical length is also easily calculated geometrically using the cylindrical volume and chamber radius. This leaves the complete chamber length as the sum of the cylindrical length and the converging section length.
Step 4 Define Chamber Surface Area
As outlined prior, the surface area of the chamber is an important consideration in our design regarding thermal and structural parameters. Using the equation below will yield a surface area:
Lc=chamber length
εc=contraction ratio
At=throat cross-sectional area
θ=contraction angle
a=1 (approximation)
This concludes the calculation of major parameters within the combustion chamber.
Step 5 Calculate Diverging Section Length
The final aspect of engine design is the nozzle (diverging section), specifically the length. This may be calculated using the same equation as converging length. As before this calculation requires a design choice, in this case, the divergent angle.
“The divergent cone half-angle 𝛼 varies from approximately 12 to 18 deg.” (Huzel and Huang 1992)
Rt=throat radius
ε=contraction ratio
R=arc radius=1.5*Rt
a=𝛼=divergent angle