Page History

...

Specific impulse is a quantity effectively measuring the efficiency of an engine through the relation of thrust and mass flow rate; from the equation, this efficiency may be viewed as the impulse obtained per unit mass of propellant. 

Image Modified

F=thrust

ṁ=mass flow rate

g₀=9.8 m/s²

ṁ_cc=combustion chamber mass flow rate

V_e=exit velocity

A_e=exit area

P_e=exit pressure

P_a=ambient pressure

Assuming a higher efficiency is more desirable, our goal is to maximize our I_sp, thus maximizing efficiency. In the case of our engine, we may make a few simplifications in our model of I_sp. Because our nozzle is matched, we may eliminate the 2nd term of the thrust equation thus allowing us to also eliminate the mass flow term under the key assumption that all propellants flow through the combustion chamber. This leaves us with the following model:

Image Modified

V_e=exit velocity

g₀=9.8 m/s²

It is important to note the discrepancy between calculated I_sp values and those achieved in practice. While our model assumes ideal combustion and lossless flow, these factors and others account for the lower efficiency observed in practice compared to calculations. 

...

C* is related to the chemical properties of the fuel and oxidizer within the combustion chamber (different combinations of fuel and oxidizer will affect its value). The most typical way to increase C* with a given fuel and oxidizer is to increase the combustion temperature, which is done by altering the ratio of fuel and oxidizer used. However, other considerations taken into account such as max temperature for the combustion chamber wall material force non-ideal ratios to be used in practice. C* can be found both theoretically and experimentally with the equations below.

Image Modified(theoretical)

C*=characteristic velocity

R=gas constant

g_c=acceleration due to gravity

T_c=combustion chamber temperature

𝛾=specific heat ratio of propellant mixture

Image Modified(experimental)

C*=characteristic velocity

A_t=throat area

P_c=chamber pressure

g_c=acceleration due to gravity

ṁ=mass flow rate (of gas)

C_f - thrust coefficient

C_f is a measure of the thrust amplification by the engine’s nozzle, essentially being a measure of how efficient the nozzle is. The nozzle expands the exhaust gas to lower pressures and higher velocities, increasing thrust, and C_f is a measure of how well it does that.

C_f depends on the chemical characteristics of the fuel and oxidizer, the expansion ratio of the nozzle exit area and throat area, as well as the different pressures within and outside of the engine. The expansion ratio alters the ratio of exit to chamber pressure, thus changing C_f itself. To have the best possible C_f, an engine should have a very high chamber pressure which the nozzle turns into a low exit pressure matching the ambient pressure around the nozzle.

Image Modified

C_f=thrust coefficient

𝛾=specific heat ratio of propellant mixture

P_e=exit pressure

P_c=chamber pressure

𝜺=expansion ratio (exit area/throat area)

P_a=ambient pressure

Combustion Chamber Geometry

...

The combustion chamber functions most notably as a point for propellant mixture and combustion, ideally at a high efficiency. Propellant stay time is the time required of propellants within the combustion chamber for complete mixing and combustion and is reliant on many factors. These factors include propellant combination, the state of injected propellants (liquid, gas), chamber geometry, initial conditions from injector design, and rate-limiting factors. Propellant stay time is related to chamber volume by the following equation:

Image Modified

 V_c=chamber volume

 W_tc=propellant mass flow rate

 V=average specific volume

 t_s=propellant stay time

L* - Characteristic Length

...

After choosing an L* within the desired range through previously listed considerations, we may use the following equation to derive a chamber volume:

Image Modified

V_c=chamber volume

A_t=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.

...

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

Image Modified

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:

Image Modified

R_t=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:

Image Modified

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:

Image Modified

ε_c=contraction ratio

A_c=chamber cross-sectional area

A_t=throat cross-sectional area

Step 3 Extract Cylindrical Lengths

...

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:

Image Modified

L_c=chamber length

ε_c=contraction ratio

A_t=throat cross-sectional area

θ=contraction angle

a=1 (approximation)

This concludes the calculation of major parameters within the combustion chamber.

...

“The divergent cone half-angle 𝛼 varies from approximately 12 to 18 deg.” (Huzel and Huang 1992)

Image Modified

R_t=throat radius

ε=contraction ratio

R=arc radius=1.5*Rt

a=𝛼=divergent angle