Performance of a rocket engine

Introduction

A rocket engine is designed and built for a specific goal, namely to propel a launch vehicle or a spacecraft. The performance of a given rocket engine can be defined using different parameters.

This page explains (briefly) the theory of the performance of a rocket engine.

Characteristic velocity

The characteristic velocity represents the characteristics of the chosen propellant combination and is independent of the performance of the designed nozzle. So, the characteristic velocity provides the possibility to objectively compare different propellant combinations. The following equation can be used to determine the characteristic velocity:

Equation for the characteristic velocity [2]

In this equation the following parameters are defined:
- T₁: Combustion temperature in Kelvin
- R: gas constant per unit weight, which is equal to the gas constant devided by the molar mass.
- k: Specific heat ratio

The characteristic velocity reaches a maximum before the maximum combustion temperature is reached when varying the oxidizer-to-fuel ratio of a given propellant combination. [1]

As a rule of thumb, the characteristic velocity has range between 1600 m/s (red fuming nitric acid and RP-1) and 2430 m/s (Oxygen and hydrogen). This is based on a chamber pressure of 1000 psi and ideally expaned with an exit pressure of 1 bar. [2]

Thrust coeficient

The thrust coeficient describes the quality of the expansion of the gas and the quality of the nozzle. The thrust coeficient can be determined with the following equation:

Definition of thrust coeficient [1]

In this equation the following parameters are defined:
- C_f: Thrust coeficient [-]
- γ: Specific heat ratio [-]
- P_e: Exit pressure in Pa
- P_{c ns}: Chamber pressure Pa
- P_a: Ambiant pressure in Pa
- ε: Expansion ratio [-]

The thrust coeficient is also a function of the ambiant pressure, as can be seen in the above equation. The expansion ratio, which also influences the exit pressure of the nozzle, is a design parameter. Also the ratio of exit pressure over chamber pressure is a design parameter. The specific gas constant is a function of the choosen propellant combination. [1]

In general, the thrust coeficient has a value that is in a between 1.3 and 2.0. [1]

Thrust

Thrust is the reaction force that is generated by ejecting high velocity gas out of the nozzle [2].

The thrust can be determined with the following equation:

Definition of the thrust [1]

In this equation the following parameters are defined:
- c^*: characteristic velocity in m/s
- F: generated thrust in N
- m dot: mass flow in kg/s
- CF: Thrust coeficient

Specific impulse

Specific impulse describes the overall performance of the rocket engine. For the launch vehicles' engineers, the specific impulse is an important performance parameter. [2]

The specific impulse can be determined with the following equation:

Definition isp [1]

In this equation the following parameters are defined:
- c^*: characteristic velocity in m/s
- I_s: specific impulse in m/s
- g₀: gravitiational acceleration in m/s², which has a value of 9.81 m/s²
- CF: Thrust coeficient

As a general rule, the specific ipulse is in between 270 sec (red fuming nitric acid and RP-1) and 390 sec (Oxygen and hydrogen). This is based on a chamber pressure of 1000 psi and ideally expaned with an exit pressure of 1 bar. [2]

Sources

[1]: Design of liquid propellant rocket engines - Huzel and Huang
[2]: Rocket Propulsion Elements - George P. Sutton and Oscar Biblarz