2. Balance of Mass¶

2.1 General Chamber Mass Balance¶

The conservation of mass for an open control volume (the rocket chamber) states that the rate of change of mass stored equals the difference between mass generated and mass out:

\[ \frac{dM_{stored}}{dt} = \dot{m}_{gen} - \dot{m}_{out} \]

Treating the chamber contents as an ideal gas at uniform state (pressure \(P_0\), temperature \(T_0\), volume \(V_0\)):

\[ M_{stored} = \rho_c V_0 = \frac{P_0 V_0}{R T_0} \]

Differentiating (constant \(T_0\), constant \(V_0\)):

\[ \frac{dM_{stored}}{dt} = \frac{V_0}{R T_0}\frac{dP_0}{dt} \]

Substituting into the mass balance:

\[ \boxed{\frac{dP_0}{dt} = \frac{R T_0}{V_0}\left(\dot{m}_{gen} - \dot{m}_{out}\right)} \]

This single ODE is the foundation of all internal-ballistics simulations in machwave. It is integrated numerically using the 4th-order Runge–Kutta solver in machwave.core.solvers.

2.2 Solid Rocket Motor (SRM)¶

Reference: Seidel, H. (1965). Transient Chamber Pressure and Thrust in Solid Rocket Motors. AFRPL.

2.2.1 Mass Generation Rate¶

Propellant regression exposes new surface at the burn rate \(r\) [m/s]. The mass generated per unit time is:

\[ \dot{m}_{gen} = \rho_p \cdot r \cdot A_b \]

where \(\rho_p\) is the solid propellant density and \(A_b\) is the instantaneous burn area. This feeds the \(\dot{m}_{in}\) argument of the unified compute_chamber_pressure_mass_balance.

2.2.2 Mass Exit Rate — Choked Flow¶

When the nozzle is choked (\(P_e / P_0 \leq P^*/P_0\), see §1.4), the throat is sonic and the choked mass flow from §1.7 applies:

\[ \dot{m}_{exit} = \frac{C_d P_0 A_t}{\sqrt{R T_0}}\,H_\text{choked}, \qquad H_\text{choked} = \sqrt{k}\left(\frac{2}{k+1}\right)^{(k+1)/[2(k-1)]} \]

2.2.3 Mass Exit Rate — Sub-critical Flow¶

When \(P_e / P_0 > P^*/P_0\) the throat is subsonic. Applying the isentropic energy equation (§1.1) and density relation (§1.3) between the stagnation state and the throat at back pressure \(P_e\), with \(P_r = P_e/P_0\):

\[ v_t = \sqrt{\frac{2k}{k-1}R T_0\left[1 - P_r^{(k-1)/k}\right]}, \qquad \rho_t = \frac{P_0}{R T_0}P_r^{1/k} \]

Mass flow through the throat area \(A_t\):

\[ \dot{m}_{exit} = \rho_t\, v_t\, A_t = \frac{C_d P_0 A_t}{\sqrt{R T_0}}\,H_\text{sub}, \qquad H_\text{sub} = \sqrt{\frac{2k}{k-1}}\,P_r^{1/k}\sqrt{1 - P_r^{(k-1)/k}} \]

(Seidel 1965, Eq. 35.)

2.2.4 SRM ODE¶

Substituting into §2.1:

\[ \boxed{\frac{dP_0}{dt} = \frac{R T_0}{V_0}\left(\rho_p r A_b - \frac{C_d P_0 A_t\, H}{\sqrt{R T_0}}\right)} \]

Evaluated by compute_chamber_pressure_mass_balance with \(\dot{m}_{in} = \rho_p r A_b\), as called from SolidMotorState.

2.3 Liquid Rocket Engine (LRE)¶

References: Sutton & Biblarz (2017), Ch. 6; Huzel & Huang (1992), Ch. 1, 4, 7.

For an LRE the mass-generation term is no longer set by surface regression but by the injector mass flow of two independent propellant streams. Mass exits through the same choked throat as in §2.2.2.

2.3.1 Mass Inflow Rate — Injector¶

References: Sutton & Biblarz (2017) Ch. 8 (Thrust Chambers — Injectors); Huzel & Huang (1992) Ch. 4 §4.5 (Injector Design).

Each propellant stream is treated as an incompressible fluid flowing through an orifice from the upstream feed pressure \(P_\text{up}\) to the chamber pressure \(P_0\). Combining Bernoulli with continuity through an effective orifice area \(A_\text{eff}\) and applying a discharge coefficient \(C_d\) to lump together contraction and viscous losses gives (Huzel & Huang §4.5; Sutton & Biblarz §8.2):

\[ \boxed{\dot{m} = C_d\,A_\text{eff}\,\sqrt{2\rho\,(P_\text{up} - P_0)}} \]

The total inflow is the sum of the fuel and oxidiser streams:

\[ \dot{m}_{gen} = \dot{m}_{fuel} + \dot{m}_{ox}, \qquad \dot{m}_{i} = C_{d,i}\,A_{\text{eff},i}\,\sqrt{2\rho_i\,(P_{\text{up},i} - P_0)}, \quad i \in \{fuel, ox\}. \]

Implemented in get_mass_flow_orifice (module machwave.core.incompressible_flow) and called per stream by the feed system (see §2.3.2).

2.3.2 Upstream Pressure — Pressurised Tank¶

References: Sutton & Biblarz (2017) Ch. 6 §6.3 (Propellant Feed Systems); Huzel & Huang (1992) Ch. 5 (Gas-Pressurized Feed Systems) and Ch. 8 (Propellant Tanks).

machwave currently models a stacked-tank pressure-fed architecture: a single pressurant volume above the oxidiser also drives the fuel via a piston, with a constant pressure drop \(\Delta P_\text{piston}\) accounting for friction and piston weight (Huzel & Huang §5.2; Sutton & Biblarz §6.3). The injector upstream pressures are therefore:

\[ P_{\text{up},ox} = P_{\text{tank},ox}, \qquad P_{\text{up},fuel} = P_{\text{tank},ox} - \Delta P_\text{piston}. \]

Each tank is modelled as a two-phase isothermal vessel (constant \(T\), saturation pinning when liquid is present, ideal-gas vapour when only vapour remains; cf. Huzel & Huang Ch. 8 for tank thermodynamics). Properties come from CoolProp; details in Tank and the orchestrating StackedTankPressureFedFeedSystem.

2.3.3 Stoichiometric Limiting-Reagent Adjustment¶

References: Sutton & Biblarz (2017) Ch. 6 §6.1 (Mixture Ratio); Huzel & Huang (1992) Ch. 1 §1.3 (Performance Parameters).

The injectors deliver fuel and oxidiser independently, so over a finite step \(\Delta t\) one tank can run dry while the other still has propellant. machwave clamps the surplus to preserve the design oxidiser–fuel ratio \(\mathrm{O\!/\!F} = \dot{m}_{ox}/\dot{m}_{fuel}\) (Sutton & Biblarz §6.1):

\[ \text{if } \dot{m}_{fuel}\Delta t \geq m_{fuel}: \;\; \dot{m}_{fuel} \leftarrow m_{fuel}/\Delta t, \;\; \dot{m}_{ox} \leftarrow \mathrm{O\!/\!F}\cdot\dot{m}_{fuel} \]

(and symmetrically when oxidiser is the limiting reagent). This avoids unphysical post-burnout transients in which one stream continues for several steps after the other has been exhausted. Implemented in _adjust_flows_for_stoichiometry.

2.3.4 Mass Exit Rate — Choked Throat¶

References: Sutton & Biblarz (2017) Ch. 3 §3.3 (Isentropic Flow through Nozzles); Huzel & Huang (1992) Ch. 1 §1.4 (The Gas-Flow Processes).

The exit term is identical to §2.2.2 — the choked-flow expression derived in §1.7 (Sutton & Biblarz §3.3). For LRE the implementation drops the throat discharge coefficient (\(C_d \equiv 1\)) and uses the chamber-state isentropic exponent \(k = k_\text{chamber}\):

\[ \dot{m}_{out} = \frac{P_0\, A_t}{\sqrt{R T_0}}\,\sqrt{k}\left(\frac{2}{k+1}\right)^{(k+1)/[2(k-1)]}. \]

2.3.5 LRE Chamber-Pressure ODE¶

References: Huzel & Huang (1992) Ch. 1 §1.4 and Ch. 4 §4.1 (Combustion-Chamber Processes); Sutton & Biblarz (2017) Ch. 8 §8.1 (Combustion Chamber Basic Configurations).

Substituting §2.3.1 and §2.3.4 into §2.1 yields the LRE form of the well-stirred reactor balance (Huzel & Huang §1.4; Sutton & Biblarz §8.1):

\[ \boxed{\frac{dP_0}{dt} = \frac{R T_0}{V_0}\left[ \dot{m}_{fuel} + \dot{m}_{ox} - \frac{P_0\,A_t}{\sqrt{R T_0}}\sqrt{k}\!\left(\frac{2}{k+1}\right)^{\!(k+1)/[2(k-1)]} \right]} \]

Evaluated by the same compute_chamber_pressure_mass_balance used in §2.2 — with \(\dot{m}_{in} = \dot{m}_{fuel} + \dot{m}_{ox}\), \(C_d = 1\), and chamber-state \(\{T_0, R, k\}\) — and integrated with the same RK4 solver as §2.2.

The thermochemical state \(\{T_0, R, k\}\) is evaluated by NASA-CEA at each step from the current \(P_0\) and the design expansion ratio, then held constant within the RK4 sub-stages — an explicit lag that is acceptable because chamber properties are only weakly pressure-dependent (cf. Sutton & Biblarz §5.4 on equilibrium thermochemistry).

2.3.6 Modelling Assumptions and Limitations¶

References: Sutton & Biblarz (2017) Ch. 8-9; Huzel & Huang (1992) Ch. 1, 4, 8.

The LRE mass balance is built on a number of simplifying assumptions; users should keep these in mind when interpreting transient results:

Well-stirred reactor / instantaneous combustion. Cold liquid propellant is assumed to burn to equilibrium products immediately upon entering the chamber. Atomisation, vaporisation, and finite reaction times are not resolved (Huzel & Huang Ch. 4 §4.1; Sutton & Biblarz Ch. 9 covers the real combustion process and its instabilities).
Uniform chamber state. Pressure, temperature, and composition are spatially uniform — there is no L* effect, no residence-time penalty, and no chamber-cooling energy loss (cf. Huzel & Huang §4.1 on \(L^*\) sizing).
Constant \(T_0\), constant \(V_0\). Flame temperature is treated as the CEA equilibrium value at the current \(P_0\) and design \(\varepsilon\) (Sutton & Biblarz Ch. 5); free volume is fixed (no regenerative cooling jacket displacement, no throat erosion — see Huzel & Huang Ch. 4 for cooling-jacket geometry).
Isothermal tank. The two-phase model assumes constant tank temperature; the energy of vaporisation that would normally cool a self-pressurised tank during blowdown is not modelled (Huzel & Huang Ch. 8 covers tank thermodynamics in more depth).
Bulk fluid density at the injector. When the tank is two-phase, the orifice flow uses the bulk mixture density rather than the liquid saturation density — acceptable while the tank is mostly liquid, less accurate as it empties (Huzel & Huang §4.5 on injector hydraulics).
Stoichiometric clamping. When one tank empties first, the simulation drops the surplus reagent rather than tracking the fuel-rich (or ox-rich) tail of a real engine (Sutton & Biblarz §6.1).

References¶

Seidel, H. (1965). Transient Chamber Pressure and Thrust in Solid Rocket Motors. Air Force Rocket Propulsion Laboratory (AFRPL).
Sutton, G. P., & Biblarz, O. (2017). Rocket Propulsion Elements (9th ed.). Wiley. Ch. 6, 12.
Huzel, D. K., & Huang, D. H. (1992). Modern Engineering for Design of Liquid-Propellant Rocket Engines. AIAA Progress in Astronautics and Aeronautics, Vol. 147. Ch. 1, 4, 7.