Mechanical-Drivetrain Efficiency

Flow Pathing

Oxidizer

• Tank $\to$ pre-valves $\to$ LOX pump (boost + main) $\to$ manifold $\to$ injector $\to$ chamber
• Losses: valve $\Delta p$, line friction, injector $\Delta p_{\text{inj}}$
• Design aims for stable atomization and mixture stratification control

Fuel

• Tank $\to$ fuel pump $\to$ cooling channels (regen) around chamber/nozzle $\to$ (cycle dependent) turbine or main injector
• Thermal pickup in walls raises $h_f$: ${\dot{Q}}_{\text{wall}}={\dot{m}}_f(h_{f,\text{out}}-h_{f,\text{in}})$.

Gas Generator (GG)

• A small combustor drives the turbine with hot gas, exhausting to nozzle (open cycle)
• Balance: $P_{\text{turb,out}}{\eta }_{\text{mech}}\ge P_{\text{pump},f}+P_{\text{pump},o}+P_{\text{aux}}$.

$\mathrm{LH}{}_2/\mathrm{LOX}$

Given chamber pressure $p_c$, mixture ratio $O/F$, and injector drop $\Delta p_{\text{inj}}$:

• Mass flows:
  • $${\dot{m}}_o=\frac{O/F}{1+O/F}{\dot{m}}_{\text{tot}}$$
  • $${\dot{m}}_f=\frac{1}{1+O/F}{\dot{m}}_{\text{tot}}$$
• Pump discharge pressures:
  • $$p_{o,\text{out}}\approx p_c+\Delta p_{\text{inj}}+\Delta p_{\text{lines}}$$
  • $$p_{f,\text{out}}\approx \max \{p_c+\Delta p_{\text{inj}},p_{\text{coolreq}}\}.$$
• Hydraulic power:
  • $$P_{\text{pump}}=\dot{V}\Delta p=\frac{\dot{m}}{\rho }\Delta p$$
  • $$\Rightarrow P_{\text{shaft}}=\frac{P_{\text{pump}}}{{\eta }_{\text{pump}}}$$
• Turbine power (GG or heated fuel):
  • $$P_{\text{turb,out}}={\dot{m}}_t\Delta h_t{\eta }_{\text{turb}}$$
  • $${\eta }_{\text{mech}}=\frac{P_{\text{shaft,delivered}}}{P_{\text{turb,out}}}$$
• Overall drivetrain efficiency:
  • $${\eta }_{\text{drive}}={\eta }_{\text{turb}}{\eta }_{\text{mech}}{\eta }_{\text{pump}}^{(o)}{\eta }_{\text{pump}}^{(f)}.$$

Drivetrain

• Shaft groupings (common shaft vs split-shaft) and gear sets map turbine work to both pumps
• High‐speed turbines with reduction gearing trade speed for torque
• ${\eta }_{\text{mech}}$ captures bearing/gear losses

Qualitatively

• Injector $\Delta p$ stabilizes mixing but drives pump power.
• Coolant heating (regen) boosts turbine potential (expander or GG assist) but adds channel pressure drop
• Cycle choice (GG, expander, staged) reshapes where the heat/work is taken and thus the achievable ${\eta }_{\text{drive}}$ for a given $p_c$