Liquid Rocket Engines - Rocket Propulsion notes

This modules covers the design and operation of liquid rocket engines, including propellant types, feed systems, engine cycles, propellant tanks, and combustion chamber sizing.

Overview¶

Figure 1:Schematic of a pressure-fed liquid bipropellant rocket engine showing: high-pressure gas (He or N₂) pressurant system, separate fuel and oxidizer tanks, injector assembly, combustion chamber with regenerative cooling (coolant flows through channels in chamber/nozzle walls), and converging-diverging nozzle forming the thrust chamber.

Advantages of liquid engines:

Higher $I_{sp}$ compared to solid motors
Throttleable: can vary thrust during flight
Restartable: can shut down and reignite

Disadvantages:

Complex: many components, valves, plumbing, turbomachinery
Cryogenic propellants require cooling and careful tank filling

Propellant Classifications¶

Monopropellant Systems¶

A single liquid that decomposes exothermically, typically over a catalyst. $I_{sp}$ values range 200-230 s.

Common monopropellants:

H₂O₂: Hydrogen peroxide
N₂H₄: Hydrazine

Characteristics:

Simple system design
Lower impulse (lower $\Delta V$ capability)
Stable and storable

Bipropellant Systems¶

Two separate propellants (fuel + oxidizer) that combust together. Used for larger rockets requiring higher total impulse.

Storable propellants, which can be stored at ambient conditions for extended periods:

RP-1: Refined kerosene (fuel)
N₂O₄ (NTO): Nitrogen tetroxide (oxidizer)
MMH: Monomethylhydrazine (fuel)

Cryogenic propellants, which must be stored at very low temperatures; boil-off is a concern:

LH₂ — Liquid hydrogen (fuel)
LOX — Liquid oxygen (LO₂) (oxidizer)
CH₄ — Liquid methane (fuel); when used with LOX, called methalox

Performance comparison:

Propellant Combination	$I_{\text{vac}}$ (s)
Hydrocarbons (RP-1, CH₄) with LOX	250–350
LOX/LH₂	425–500

Hypergolic Propellants¶

Hypergolic propellants ignite spontaneously upon contact, no ignition system required. A common example is MMH/NTO (monomethylhydrazine with nitrogen tetroxide).

Advantages:

Reliable ignition
Restartable without complex igniter
Storable

Primary applications include spacecraft attitude control, orbital maneuvering systems.

Feed Systems¶

The feed system delivers propellants from tanks to the combustion chamber at the required pressure and flow rate.

Pressure-Fed (Blowdown) Systems¶

High-pressure gas (typically helium) pressurizes the propellant tanks directly, forcing propellants into the combustion chamber.

Figure 2:Three configurations for pressure-fed systems: (a) separate pressurant tank with helium, (b) pressurant stored in bladder/diaphragm within propellant tank, (c) pressurant ullage above propellant with helium and fuel separated.

Advantages:

Simple, reliable
No turbomachinery

Disadvantages:

Heavy tanks (must withstand high pressure)
Limited to moderate chamber pressures

Pump-Fed Systems¶

Turbopumps pressurize the propellants. A turbine (driven by hot gas) powers the pumps.

\text{Hot gas} \longrightarrow \text{Turbine} \longrightarrow \text{Pumps}

(1)

Advantages:

No additional tanks needed, and relatively lower pressure tanks
High chamber pressures achievable

Disadvantages:

Complex turbomachinery
Development cost and risk

Thrust Chamber Design¶

Chamber pressure $P_c$ is a critical design choice that affects multiple system parameters. Given thrust $F$ and mass flow rate $\dot{m}$ , the throat area is:

\boxed{A_t = \frac{\dot{m} c^*}{g_0 P_c}}

(2)

Trade-off:

A_t \downarrow \quad \text{as} \quad P_c \uparrow

(3)

Higher $P_c$ → smaller throat/chamber → lower thrust chamber weight
But higher $P_c$ → heavier feed system (stronger pumps, tanks)

The oxidizer-to-fuel ratio determines combustion properties:

r = \frac{\dot{m}_{ox}}{\dot{m}_F}

(4)

Given geometry (which sets $C_F$ ):

r \longrightarrow c^* \longrightarrow I_{sp}

(5)

For a required thrust $F$ and burn time $t_b$ :

\dot{m} = \frac{F}{I_{sp} \, g_0} = \frac{m_p}{t_b}

(6)

Individual flow rates:

\boxed{\dot{m}_F = \frac{\dot{m}}{1+r}} \qquad \boxed{\dot{m}_{ox} = \frac{r\dot{m}}{1+r}}

(7)

The bulk density $\rho_b$ characterizes the combined propellant density accounting for the mixture ratio:

\boxed{\rho_b = \frac{1+r}{\dfrac{1}{\rho_F} + \dfrac{r}{\rho_{ox}}} = \frac{m_p}{V_p}}

(8)

Example: LOX/LH₂ with $r = 6$ :

Property	Value
$\rho_{ox}$ (LOX)	1140 kg/m³
$\rho_F$ (LH₂)	64 kg/m³
$\rho_b$	335 kg/m³

Monopropellant Thruster Design¶

Figure 3:Hydrazine monopropellant thruster showing: propellant inlet, catalyst bed (often called “cat bed”) with upper and lower sections containing catalyst granules, and nozzle. Hydrazine decomposes exothermically as it passes through the heated catalyst.

Hydrazine decomposition:

\text{N}_2\text{H}_4 \xrightarrow{\text{catalyst}} \text{NH}_3 \longrightarrow \text{N}_2 + \text{H}_2 + \text{energy} + \text{N} + \text{H} + \cdots

(9)

Advantages of monopropellant systems:

Single tank and simplified feed system
Simplified injector design
Less sensitive to temperature variations** $\delta T$ ; bipropellant systems have “outage” issues (residual propellant due to mixture ratio drift with temperature): $\delta\rho = f(\delta T)$

Catalyst Bed Loading¶

The bed loading $G$ is the mass flux through the catalyst bed:

G = \frac{\dot{m}}{A}

(10)

Design considerations:

Higher $G$ is better for compact design
Given bed area $A$ : increasing $\dot{m}$ increases $G$
But: $G \uparrow \Rightarrow u \uparrow \Rightarrow t_{\text{res}} \downarrow$
If $t_{\text{res}}$ too short → flooding (incomplete decomposition)
Also: higher flow rates increase pressure drop $\Delta p$

Bipropellant engine cycles¶

Most bipropellant engines are pump-fed and use various thermodynamic cycles to drive the turbopumps. These can be categorized as open cycles or closed cycles, depending on whether the flow driving the turbine(s) is reincorporated into the combustion chamber or effectively lost (by not being expanded in the nozzle).

Gas generator cycle (“GG”)¶

Figure 4:Gas generator cycle schematic: A small portion of propellants is burned in a separate gas generator (fuel-rich to limit temperature) to produce hot gas that drives the turbine(s). Fuel pump and oxidizer pump are driven by separate or shared turbines. The turbine exhaust (low pressure) is dumped overboard.

Operation:

Small fraction of propellants combusted in gas generator
Hot gas drives turbine(s)
Turbine powers fuel and oxidizer pumps
Turbine exhaust dumped at low pressure

Variants:

Single turbine vs. separate turbines for fuel and oxidizer
Turbine exhaust directed to main nozzle in low-pressure diverging section, or separate nozzle (for roll control or thrust vectoring)

Example engines using the gas-generator cycle include F-1 (Saturn V), RS-68 (Delta IV), and Merlin (Falcon 9).

Combustion Tap-Off Cycle¶

Figure 5:Tap-off cycle schematic: Hot combustion gases are bled from the main combustion chamber (near the injector face) to drive the turbine. No separate gas generator is needed. Shows fuel pump, oxidizer pump, fuel turbine, oxidizer turbine, injector, and low-pressure exhaust path.

Operation:

No gas generator: combustion chamber gases bled off near injector face
Hot gas drives turbine
Simpler than gas generator cycle

Example engines: J-2S, Blue Origin BE-3 (New Glenn rocket)

Expander bleed cycle¶

This cycle uses a small amount of evaporated coolant (fuel) from regenerative cooling to drive the turbines. Application is limited to cryogenic fuels (H₂, CH₄) that can absorb enough heat and easily reach their boiling point. Examples include the BE-3U engine (New Glenn upper stage).

Open vs. Closed Cycles¶

All open cycles (GG, tap-off):

Turbine exhaust is dumped into nozzle diverging section or separate nozzle
Lost propellant reduces overall efficiency

Closed cycles:

Turbine exhaust gases kept at high pressure and reintroduced into combustion chamber
All propellant contributes to thrust → higher exit velocity

\boxed{I_{sp,\text{closed}} > I_{sp,\text{open}}}

(11)

Expander Cycle¶

Figure 6:Expander cycle: Cryogenic fuel (typically LH₂) is heated in regenerative cooling passages, vaporizes, drives the turbine, then enters the combustion chamber. All propellant goes through the main chamber—a closed cycle.

Operation:

Vaporized coolant drives turbine
Turbine exhaust → combustion chamber

Limitations:

Limited to moderate $P_c$ (limited heat transfer area)
Great for upper-stage engines

Applications include the RL-10 (Centaur upper stage of Atlas V) and Vinci engine (Ariane 6).

Staged Combustion Cycles¶

Preburner combusts a portion of one propellant with all of the other propellant, producing gas to drive turbines. The preburner exhaust (still fuel-rich or ox-rich) enters the main chamber to be combusted with the rest of the other propellant (oxidizer or fuel).

Figure 7:Staged combustion cycle schematic showing preburner, fuel pump, oxidizer pump, fuel turbine, oxidizer turbine, and injector. The preburner operates either fuel-rich or oxidizer-rich depending on cycle variant.

Fuel-Rich Staged Combustion (FRSC)¶

Preburner: all fuel with some oxidizer
Fuel-rich gas drives turbine, then enters main chamber
Compatible with hydrogen (less coking)

Example engines: SSME/RS-25 (LH₂/LOX)

Oxidizer-Rich Staged Combustion (ORSC)¶

Preburner: all oxidizer with some fuel
Requires oxidizer-compatible turbine materials

Example engines: RD-180 (Russian design)

Full-Flow Staged Combustion (FFSC)¶

The most complex and highest-performing cycle:

Two preburners: one fuel-rich, one oxidizer-rich
All fuel goes through fuel-rich preburner
All oxidizer goes through oxidizer-rich preburner
Both streams enter main chamber as hot gas

Advantages:

No inter-propellant seal needed (each turbine sees only one propellant)
“Gas-gas” mixture enters combustion chamber → better mixing and faster combustion
Highest potential $I_{\text{sp}}$ and thrust density

Example engine: SpaceX Raptor

Propellant Tanks¶

Figure 8:Common tank arrangements: Spacecraft typically use spherical tanks in various arrangements. Launch vehicles typically use cylindrical tanks for structural efficiency, in configurations including: (a) Tandem — fuel and oxidizer tanks stacked vertically, (b) Concentric — one tank inside the other (ox surrounding fuel, or vice versa), (c) Twin/Multi-tank — parallel tanks side by side.

Total tank volume consists of several components:

V_t = V_p + V_u + V_{tr} + V_b

(12)

Component	Description
$V_p$	Propellant volume = $m_p/\rho$
$V_u$	Ullage volume (gas space above liquid); few % of total; $\delta\rho = f(\delta T)$
$V_{tr}$	Trapped propellant (feed lines, etc.)
$V_b$	Boil-off allowance (cryogenics only); depends on propellant

Figure 9:Cylindrical tank under flight loads showing: internal pressure $P_n$ , external atmospheric pressure $P_a(t)$ , tank radius $R$ , initial propellant head height $h_0$ , current propellant height $h(t)$ , axial thrust load $F_{ax}$ , and vehicle acceleration $a(t)$ .

The pressure at the tank bottom is:

P_t = P_n(t) + \rho \, g \, a(t) \, h(t) - P_a(t)

(13)

Peak pressure occurs when acceleration $a$ and propellant head $h$ combine to give maximum. Wall thickness design for cylindrical tanks:

\boxed{t_c = \frac{P_{t,\text{max}} \cdot R}{\sigma_w}}

(14)

where:

$R$ = tank radius
$\sigma_w$ = maximum allowable working stress

The allowable stress is the minimum of yield and ultimate limits:

\sigma_w = \min\left(\frac{F_y}{1.1}, \frac{F_{tu}}{f_s}\right)

(15)

where:

$F_y$ = yield strength
$F_{tu}$ = ultimate tensile strength
$f_s$ = factor of safety

Factor of Safety	Application
$f_s = 1.25$	Uncrewed vehicles
$f_s = 1.4$	Crewed vehicles

In addition to designing for a burst pressure, we also need to consider buckling under an axial load. The axial stress in the tank wall is

\sigma_{ax} = \max\left[\frac{(P_n - P_a)\pi R^2 - F_{ax}}{2\pi R \, t_c}\right] \;.

(16)

If $\sigma_{ax} < 0$ , the tanks walls are under compression and the tank could buckle. A useful empirical buckling criterion for large thin-walled tanks:

\sigma_{cr} = -E\left[9\left(\frac{t_c}{R}\right)^{1.6} + 0.16\left(\frac{t_c}{L}\right)^{1.3}\right] \;,

(17)

where $E$ is the elastic modulus. There will be no buckling if $\sigma_{ax} > \sigma_{cr}$ .

An alernative strategy is to increase the tank pressure so $\sigma_{ax} > 0$ , putting the tank walls into tension: pressure stabilization.

Combustion Chamber Sizing¶

Figure 10:Combustion chamber and nozzle showing chamber length $L_c$ , chamber cross-sectional area $A_c$ , throat area $A_t$ , and the characteristic length $L^*$ .

Propellants must spend sufficient time in the chamber to complete combustion, represented by a residence time $\tau_{\text{res}}$ . From mass conservation: $\dot{m} = \rho A u$ → $u = \dot{m}/(\rho A)$

\tau_{res} = \frac{L_c}{u_c} = \frac{\rho_c A_c L_c}{\dot{m}}

(18)

Using $\dot{m} = P_c A_t / c^*$ and $\rho_c = P_c/(R T_c)$ :

t_{res} = \frac{c^*}{R T_c}\left(\frac{A_c L_c}{A_t}\right)

(19)

The characteristic length ( $L^*$ ) combines chamber geometry:

\boxed{L^* = \frac{A_c}{A_t} L_c = \frac{V_c}{A_t}}

(20)

where $V_c$ is the chamber volume.

\tau_{res} \propto L^*

(21)

Minimum $L^*$ values by propellant (from empirical results):

Propellant Combination	Minimum $L^*$ (m)
LOX/RP-1	1.0 – 1.3
LOX/LH₂	0.7 – 1.0
LOX/GH₂	0.5 – 0.7
N₂H₄/NTO	0.7 – 0.9
H₂O₂/RP-1	1.5 – 1.8

Then to calculate the minimum necessary combustion chamber length, given $L^*$ and the throat/chamber areas:

L_c = \frac{L^* A_t}{A_c}

(22)

The contraction ratio is

CR = \frac{A_c}{A_t} \quad \text{(typical range: 1.3–7.0)}

(23)

Trade-offs:

$CR \uparrow$ → Rayleigh heating losses increase
$CR \uparrow$ → Weight increases, $D_c$ increases
$CR \downarrow$ → May not achieve complete combustion (insufficient $\tau_{res}$ )