Rocket Propulsion

Concept

Rocket Propulsion and Chemical Performance — Thermodynamics, Nozzle Flow, and Exergy Analysis

Scope: a rigorous first-principles treatment of non-airbreathing propulsion systems, including rocket thermodynamics, combustion chemistry, expansion flow, and performance metrics.

1. Fundamentals of Rocket Propulsion

1.1 Control Volume Formulation

For a steady control volume enclosing a rocket engine: F=m˙pVe+(pe−p0)Ae.F = \dot{m}_p V_e + (p_e - p_0)A_e.

Where:

  • FF: thrust (N)
  • m˙p\dot{m}_p: propellant mass flow rate (kg/s)
  • VeV_e: exhaust velocity (m/s)
  • pe,Aep_e, A_e: exit pressure and area
  • p0p_0: ambient pressure

1.2 Conservation of Momentum and Energy

From 1st law for steady adiabatic flow: hc+Vc22=he+Ve22.h_c + \frac{V_c^2}{2} = h_e + \frac{V_e^2}{2}. If inlet velocity Vc≈0V_c ≈ 0: Ve=2(hc−he).V_e = \sqrt{2(h_c - h_e)}.

For ideal gas and isentropic expansion: Ve=2γγ−1RTc(1−(pepc)(γ−1)/γ).V_e = \sqrt{\frac{2γ}{γ-1}R T_c\left(1 - \left(\frac{p_e}{p_c}\right)^{(γ-1)/γ}\right)}.

2. Performance Parameters

2.1 Specific Impulse

Isp=Fm˙pg0.I_{sp} = \frac{F}{\dot{m}_p g_0}.

Expressed in seconds, representing thrust per unit weight flow of propellant.

2.2 Characteristic Velocity (c*)

Defines combustion performance independent of nozzle: c∗=pcAtm˙p.c^* = \frac{p_c A_t}{\dot{m}_p}.

For ideal gas: c∗=RTcγ[γ+12]γ+12(γ−1).c^* = \sqrt{\frac{R T_c}{γ}} \left[\frac{γ+1}{2}\right]^{\frac{γ+1}{2(γ-1)}}.

2.3 Thrust Coefficient (C_f)

Cf=FpcAt=AeAtpe−p0pc+2γ2γ−1(2γ+1)(γ+1)/(γ−1)[1−(pepc)(γ−1)/γ].C_f = \frac{F}{p_c A_t} = \frac{A_e}{A_t}\frac{p_e - p_0}{p_c} + \sqrt{\frac{2γ^2}{γ-1}\left(\frac{2}{γ+1}\right)^{(γ+1)/(γ-1)}\left[1 - \left(\frac{p_e}{p_c}\right)^{(γ-1)/γ}\right]}.

Thrust equation then becomes: F=CfpcAt.F = C_f p_c A_t.

3. Nozzle Flow and Expansion Ratio

3.1 Isentropic Relations

At any nozzle section: TTc=(ppc)(γ−1)/γ,ρρc=(ppc)1/γ.\frac{T}{T_c} = \left(\frac{p}{p_c}\right)^{(γ-1)/γ}, \quad \frac{\rho}{\rho_c} = \left(\frac{p}{p_c}\right)^{1/γ}.

3.2 Area–Mach Relation

AA∗=1M[2γ+1(1+γ−12M2)](γ+1)/2(γ−1).\frac{A}{A^*} = \frac{1}{M}\left[\frac{2}{γ+1}(1 + \frac{γ-1}{2}M^2)\right]^{(γ+1)/2(γ-1)}.

Expansion ratio: ε=AeAt.ε = \frac{A_e}{A_t}.

3.3 Optimal Expansion

For maximum thrust, pe=p0p_e = p_0. Over-expanded nozzles cause flow separation; under-expanded nozzles waste potential thrust.

4. Combustion Thermodynamics

4.1 Adiabatic Flame Temperature

Determined by energy balance: ∑nrhfr+Qreaction=∑nphfp.\sum n_r h_f^r + Q_{reaction} = \sum n_p h_f^p.

For adiabatic combustion (no heat loss): ∑nrhfr=∑np(hfp+cppΔT).\sum n_r h_f^r = \sum n_p (h_f^p + c_p^p ΔT).

4.2 Chemical Equilibrium

Equilibrium constant for reaction aA+bB⇋cC+dDaA + bB \leftrightharpoons cC + dD: Kp=(pCcpDd)(pAapBb)=exp⁡(−ΔG0RT).K_p = \frac{(p_C^c p_D^d)}{(p_A^a p_B^b)} = \exp\left(-\frac{ΔG^0}{RT}\right).

Equilibrium composition found by minimizing G(T,P)G(T,P) subject to mass conservation.

4.3 Frozen vs. Shifting Flow

  • Frozen flow: composition fixed after throat (high-speed expansion).
  • Shifting equilibrium: chemical equilibrium maintained throughout nozzle.

Shifting flow yields slightly higher exhaust velocity and Isp due to continued energy release.

5. Propellant Types

5.1 Liquid Propellants

  • Two separate fluids: oxidizer and fuel.
  • Examples: LOX/RP-1, LOX/LH₂.
  • Advantages: controllable, restartable, high performance.
  • Disadvantages: complexity, cryogenic handling.

5.2 Solid Propellants

  • Homogeneous (single-base) or composite (fuel + oxidizer matrix).
  • Simpler but not throttleable.

Combustion rate law: r˙=apn.\dot{r} = a p^n.

Where a,na, n are empirical constants.

5.3 Hybrid Propellants

  • Solid fuel with liquid oxidizer.
  • Combines simplicity with throttling capability.

Regression rate: r˙=aGoxn,Gox=m˙oxAport.\dot{r} = a G_{ox}^n, \quad G_{ox} = \frac{\dot{m}_{ox}}{A_{port}}.

6. Chamber Processes and Heat Transfer

6.1 Mass and Energy Balance

m˙p=ρcAtVt.\dot{m}_p = ρ_c A_t V_t. Combustion chamber must provide complete reaction and mixing prior to nozzle throat.

6.2 Cooling Methods

MethodDescription
RegenerativePropellant circulates through walls before injection
FilmThin layer of coolant or unreacted fuel protects wall
AblativeMaterial designed to sublimate/erode carrying heat away

Heat flux at wall: q′′=hg(Tg−Tw).q'' = h_g (T_g - T_w). Typical gas-side heat transfer coefficients: 1000–3000 W/m²·K.

7. Multiphase and Two-Phase Effects

In solid or hybrid rockets, condensed particles (Al₂O₃, etc.) influence momentum and heat transfer: Ftwo−phase=Fgas+m˙p(Vp,e−Vp,0).F_{two-phase} = F_{gas} + \dot{m}_p (V_{p,e} - V_{p,0}).

Two-phase losses:

  • Particle slip velocity reduces effective momentum.
  • Condensed phase increases throat erosion.

8. Performance Optimization

8.1 Mixture Ratio (O/F)

Optimum mixture ratio maximizes specific impulse: (OF)opt=argmax(Isp(O/F)).\left(\frac{O}{F}\right)_{opt} = \text{argmax}(I_{sp}(O/F)).

Typical ratios:

PropellantO/F
LOX/LH₂5.5–6.0
LOX/RP-12.5–2.8
N₂O₄/MMH1.6–2.0

8.2 Expansion Optimization

For altitude compensation, variable geometry (aerospike, plug, or expansion-deflection nozzles) maintains near-optimal pe≈p0p_e ≈ p_0.

9. Example Performance Metrics

PropellantTypeIsp (s, vacuum)Chamber T (K)γNotes
LOX/LH₂Liquid450–46036001.22Highest chemical Isp
LOX/RP-1Liquid330–34037001.25High density, lower cost
N₂O₄/MMHLiquid32034001.25Hypergolic, storable
APCPSolid260–29033001.20Composite propellant

10. Exergy and Irreversibility

10.1 Exergy of Chemical Reaction

ψchem=∑i(hi−T0si)products−∑j(hj−T0sj)reactants.ψ_{chem} = \sum_i (h_i - T_0 s_i)_{products} - \sum_j (h_j - T_0 s_j)_{reactants}.

10.2 Sources of Irreversibility

MechanismEffect
Finite-rate chemistryNon-equilibrium losses
Viscous dissipationConversion of kinetic → thermal energy
Heat transferTemperature gradients in chamber and wall
Flow separationShock formation and entropy increase

10.3 Exergy Efficiency

ηex=Useful Work RateChemical Exergy Input=m˙p(Ve2/2)m˙pψchem.η_{ex} = \frac{Useful\,Work\,Rate}{Chemical\,Exergy\,Input} = \frac{\dot{m}_p(V_e^2/2)}{\dot{m}_p ψ_{chem}}.

Typical chemical-to-kinetic conversion efficiencies: 60–70% for high-performance engines.

11. Advanced Concepts

11.1 Staged Combustion

Sequential preburners drive turbopumps using partial combustion gases; used in high-efficiency engines (e.g., Space Shuttle SSME).

11.2 Expander Cycle

Fuel absorbs chamber heat → drives turbines → injected into chamber (e.g., RL10 engine).

11.3 Hybrid and Electric Augmentation

Combining electric pumps or electrothermal assistance with classical cycles improves throttling and reusability.

12. Summary of Key Relations

ConceptEquationNotes
ThrustF=m˙pVe+(pe−p0)AeF = \dot{m}_p V_e + (p_e - p_0)A_eMomentum balance
Specific impulseIsp=F/(m˙pg0)I_{sp} = F/(\dot{m}_p g_0)Efficiency metric
Characteristic velocityc∗=pcAt/m˙pc^* = p_c A_t / \dot{m}_pChamber performance
Thrust coefficientCf=F/(pcAt)C_f = F/(p_c A_t)Nozzle efficiency
Exhaust velocityVe=2γRTc/(γ−1)[1−(pe/pc)(γ−1)/γ]V_e = \sqrt{2γRT_c/(γ-1)[1 - (p_e/p_c)^{(γ-1)/γ}]}Ideal gas model
Exergy efficiencyηex=(Ve2/2)/ψchemη_{ex} = (V_e^2/2)/ψ_{chem}Chemical → kinetic conversion
  • Fluid_Dynamics/12_Propulsion_and_Jet_Engines.md — airbreathing propulsion.
  • Thermodynamics/09_Phase_Transitions_and_Reactive_Mixtures.md — combustion equilibrium.
  • Heat_Transfer/HighTemperature_Flows.md — nozzle wall heat transfer.
  • Thermodynamics/10_NonEquilibrium_Thermodynamics.md — irreversibility and exergy foundations.