Air-Breathing Propulsion in Spaceplanes

Propulsion 1: Requirements and Characteristics

Course Overview

  • Instructor: Dr. Neil Taylor

  • Institution: Swansea University (Prifysgol Abertawe)

  • Academic Year: 2025-26

  • Course Focus: Air-breathing propulsion suitable for spaceplanes.

    • Lectures: 3

    • Objectives:

    • Examine performance requirements and characteristics of air-breathing engines.

    • Provide an overview of turbine and ramjet based cycles.

    • Provide an overview of rocket and pre-cooled cycles.

    • Introduce general engine characteristics and examine various engine options.

    • Conduct basic analysis related to propulsion systems.

Rationale for Air-Breathing Engines

  • Context: Before delving into performance analysis, it's critical to understand why air-breathing engines are considered.

  • Significance of Rockets:

    • Rockets have been effective from sea level to vacuum for over 60 years.

    • The question arises: Why develop air-breathing engines?

  • Definition: A rocket is characterized by carrying all its reaction mass, enabling operation in a vacuum, albeit with a relatively fuel-inefficient process at lower speeds compared to other propulsion types.

Characteristics of an Ideal Air-Breathing Engine

  • Baseline System: The reference system is a rocket. Air-breathing engines must demonstrate superiority over equivalent rockets.

  • Operational Constraints:

    • The terminal phase of ascent will be performed using rockets, either as part of the same stage or via a dedicated rocket-powered second stage (the only type functioning in a vacuum).

    • Air-breathing (a/b) propulsion must complement the rocket phase and provide substantial benefits during the active flight phase to justify the costs.

  • Key Characteristics of Ideal Systems:

    • High specific impulse (Isp)

    • High transition Mach number

    • Minimal additional mass attributable to the system

    • Minimal complexity to reduce technical challenges

    • Target: Reduce the lifecycle cost of the entire propulsion system, a challenging feat due to the competing properties.

Desirable Properties of a/b Engines

  • Specific Impulse (Isp):

    • Utilizing air as both the reaction mass and oxidizer enhances Isp compared to simple rockets.

    • Note: Provided performance data is generalized; installed performance of specific engines can vary significantly. For instance, rockets are quite effective at speeds exceeding Mach 10.

    • A graph compares Isp versus Mach number for various engine types (clarification of types indicated as uninstalled performance).

  • Mach Range:

    • The operational capacity at high Mach numbers affects the final rocket stage performance.

    • For Two-Stage-To-Orbit (TSTO) configurations, the second stage is generally an independent rocket.

    • The best mass ratio for hydrocarbon propellants in expendable systems is approximately 6-8 and 5-6 for reusable configurations.

    • For liquid hydrogen (LH), mass ratios are about 5 for expendable and 4 for reusable designs.

  • Mass Considerations:

    • Minimize additional mass, particularly important in Single-Stage-To-Orbit (SSTO) designs where the engine must reach orbit.

    • The importance extends to TSTO configurations; a heavier engine increases empty weight, necessitating more fuel for similar missions.

    • A single engine design from sea level to rocket start is favored, emphasizing active compression.

  • Complexity:

    • Low cost typically correlates with low complexity, which poses challenges in high-speed engines due to complex aerodynamics, particularly in intake and high-temperature environments.

    • Incorporating multiple propulsion systems can complicate transitions, with the rule of thumb being: more transitions result in more challenges.

    • Bottom Line: Achieve good specific impulse and thrust-to-weight (T/W) ratios. The trade-off differs from cruise engines, and the air-breathing system must demonstrate advantages over equivalent rocket systems performing the same function.

Performance Metrics

  • The previous discussions regarding desirable characteristics now lead to the necessity for precise comparisons of different propulsion concepts.

  • Mass, while equally important, becomes more dependent on specific engine data.

  • Estimation of cost and complexity remains difficult to quantify and will require future analysis.

Generating Thrust

  • Newton's Law: Generation of thrust fundamentally relies on changing momentum in the air.

    • For engines adding minimal fuel mass, the exhaust velocity must exceed flight speed.

    • Requirement for High-Speed Flight: For flights above approximately Mach 2, exhaust velocities need to surpass 600 ext{ m/s}.

    • Typical speeds for turbojet exhaust range around 550 ext{ to } 600 ext{ m/s}, leading to supersonic exhaust conditions.

    • Key characteristic: Nearly all high-speed engines utilize choked nozzles to optimize performance.

Thrust Calculations

  • Modeling the thrust chamber as a rocket:

    • Fg = ext{m}˙c Ve + (pe - p∞) Ae

    • Parameters defined as follows:

    • C = Ve + (pe - p∞) Ae / ext{m}˙_c

    • For choked nozzles: ext{m}˙c C^∗ = pc A_t, leading to a thrust equation:

    • Fg = ext{m}˙c C = pc At C_F

    • By definition of thrust coefficient: CF = Fg / pc At, which implies C = C^∗ C_F

    • Effective net thrust (F_N) can be calculated:

    • FN = Fg - ext{m}˙a V∞

    • Where:

    • P_c = Total pressure at the throat of the combustion chamber

    • p_e = Static pressure at the nozzle exit

    • A_t = Nozzle throat area

    • A_e = Nozzle exit area

    • ext{m}˙_c = Mass flow rate in the combustion chamber

    • ext{m}˙_a = Mass flow rate into the intake

    • V_e = Actual (1D) exhaust velocity

    • C = Effective (1D) exhaust velocity

    • C^∗ = Characteristic velocity of propellants

    • C_F = Thrust coefficient.

Specific Thrust and Impulse Definitions

  • Specific Thrust (T_{sp}): Defined as thrust per unit air flow,

    • T{sp} = rac{Fg - ext{m}˙a V∞}{ ext{m}˙a} = rac{ ext{m}˙c}{ ext{m}˙a} (C^∗ CF - V_∞)

    • Units: ext{N/s kg}.

  • Specific Impulse (I_{sp}): Thrust per unit fuel flow,

    • I{sp} = rac{Fg - ext{m}˙a V∞}{ ext{m}˙f} = rac{ ext{m}˙c}{ ext{m}˙f} (C^∗ CF - rac{ ext{m}˙a}{ ext{m}˙f} V_∞)

    • Units: ext{N/s kg}, occasionally presented in seconds, requiring multiplication by g_0 ext{ (approx. } 9.807~ ext{m/s}^2) for consistent unit analysis.

Stoichiometry in Combustion

  • Definition: During combustion, propellants chemically react to produce products, illustrated by:

    • Example Reaction: 2H2 + O2 → 2H_2O

    • This shows no propellant remaining on the product side, indicating a stoichiometric reaction.

    • Stoichiometric mass ratio derived:

    • rac{32}{2 imes 2} = 8

    • Counter Example: LOxLH rockets typically operate with mass ratios of approximately rac{5}{3}. An example of hydrogen and oxygen combustion is presented:

    • 3H2 + O2 = 2H2O + H2 (here propellant remains on the right).

Equivalence Ratio Definitions

  • Fuel-Air Ratio:

    • fr = rac{m˙f}{m˙_a}

    • f_{stoi} = σ (the fuel-air ratio at stoichiometric conditions).

    • Equivalence Ratio (ER) is defined as:

    • ER = rac{fr}{f{stoi}}

    • Ratios for common fuels with air:

    • Kerosene: 0.068

    • Hydrogen: 0.0291

    • Methane: 0.0559

    • Note: fr will always be < 1 for air-breathing engines (with air-fuel ratios equal to mass ratios: = f^{-1}r).

Recap of Performance Equations Tsp and Isp

  • Equations reformulated from earlier definitions:

    • Specific Thrust:

    • T{sp} = (1 + σER) C^∗CF - V_∞

    • Specific Impulse:

    • I{sp} = C^∗CF + rac{1}{σER}(C^∗CF - V∞)

    • Importance of ER: Notably, at stoichiometric conditions, ER is consistently defined as 1 across all fuels, which simplifies comparisons between different fuel types.

Performance Analysis at System Level

Specific Thrust

  • Increasing specific thrust leads to lower air mass flows for a given thrust F_N, thus:

    • Smaller engine size can lead to lighter components

    • Reduced installation size minimizes impacts on airframe, promoting smaller intake and nozzle configurations.

    • Increases in specific thrust can be achieved by raising ER and/or C^∗C_F values.

    • Base condition demonstrates that with adequate performance, (1 + σER) C^∗CF > V∞ ext{ and } C^∗CF > V∞ for any thrust requirement.

Specific Impulse

  • Amplifying specific impulse results in lower fuel mass flow rates needed for equivalent thrust F_N, thereby:

    • High specific impulse signifies reduced fuel requirements.

    • Increased by optimizing C^∗C_F while also maintaining lower ER levels.

    • Although not immediately apparent, specific impulse is maximized when C^∗CF ightarrow V∞, signifying a delicate trade-off balancing specific thrust against specific impulse.

    • The equivalence ratio functions as a throttle setting under defined thermal constraints.

Characteristic Exhaust Velocity (C^∗)

  • Conceptual Understanding: Treat the captured stream tube as a black box.

  • Assumptions: Air enters with specific enthalpy, Ho = cp T∞ + rac{V∞^2}{2}, while fuel is delivered from a tank with enthalpy denoted as H_f.

    • Total enthalpy must be conserved, allowing recovery through relations dependent on air's handling.

  • Turbojet Example:

    • Air initially possesses enthalpy of cp T∞ + rac{V_∞^2}{2} before undergoing processes that increase enthalpy during compression, combustion, and passage through turbines.

    • Outcomes in a re-heat or afterburner follow thermal behaviors consistent with initial air and all fuel combustion.

    • Caution: Theoretical calculations are corrective with specific efficiency factors ( ext{η}_{C^∗}) depending on engine specifics, which may fluctuate under different throttle conditions.

Thrust Coefficient (C_F)

  • Choked Conditions: For a perfectly flowing, choked nozzle, with ideal gas considerations:

    • CF = C{0F} + rac{Ae}{At} rac{(pe - pa)}{p_c}

  • Performance at ideal expansion results in the thrust coefficient defined as:

    • C{0F} = igg( rac{γ}{2} igg)^{ rac{2}{γ - 1}} igg( rac{pc}{p_e} igg)^{ rac{(γ - 1)}{γ}}

  • Dependencies are explicitly linked to fuel characteristics, total chamber pressure, exhaust gas properties at varying altitudes.

  • Observations: Generally applies circumstances might correct thrust coefficient through loss factors that are affected by engine design intricacies.

Summary Impacts and Performance Evaluation

  • Characteristic exhaust velocity C^∗ is predominantly influenced by flight conditions and chosen fuels, with minimal influence from the specifics of engine cycles.

  • Thrust coefficients C_F adjust based on fuel properties and temperature, plus overall chamber pressures impacting performance at any given altitude.

  • Outcome: If (ER, pc, ext{ and } ϵ) are established, performance can be estimated across varying V∞ and p_a with little else required about the engine. However, calculations could deviate significantly under less-than-ideal conditions due to potential losses.

  • These concepts will be further explored in subsequent sessions.