Expander Cycle

Rayleigh Flow

• Heat addition in constant-area, frictionless ducts
• Let $(\cdot {)}_1\to (\cdot {)}_2$ be upstream/downstream states of a perfect gas
• Heat addition drives the flow toward a Rayleigh choking point at $M=1$ (maximum entropy on the Rayleigh line)
• Stagnation pressure $p_0$ decreases with heat addition
• Stagnation temperature $T_0$ increases when net heat is added

Qualitative regime trends

• Subsonic ($M<1$):
  • adding heat $\Rightarrow$ $M\uparrow$ toward 1
  • $p$ generally $\downarrow$, $T$ $\uparrow$
• Supersonic ($M>1$):
  • adding heat $\Rightarrow$ $M\downarrow$ toward 1
  • $p$ generally $\uparrow$, $T$ $\uparrow$
• Removing heat produces the opposite trends in each regime.

Design notes

• Expander lines are typically low-Mach and sized to avoid choking
• If choking is desired (flow metering), ensure margin to the Rayleigh limit and thermal stress limits

Subsonic Case

• Typical expander coolant passages
• Keep $M$ low to minimize compressibility losses and maintain stable $q^{\prime \prime }$ distribution
• Heat addition raises $h_f$ so turbine work $W_t={\dot{m}}_f(h_{f,\text{t,in}}-h_{f,\text{t,out}})$ can meet pump loads

Supersonic Case

• Generally avoided in coolant circuits
• Supersonic sections with heat addition are highly sensitive; small $\dot{Q}$ can drive large $M$ changes toward choking
• Use diffusers/area control to remain in a benign subsonic envelope

Heat Transfer Through Walls

• Let
  • hot-gas side $T_g$
  • wall inner/outer temperatures $T_{w,g},T_{w,c}$
  • coolant bulk $T_c$
• Local balance:
  • $$q^{\prime \prime }=h_g(T_g-T_{w,g})=\frac{k_w}{{\delta }_w}(T_{w,g}-T_{w,c})=h_c(T_{w,c}-T_c).$$
• Axial energy pickup:
  • $$\dot{Q}={\int }_0^L{q}^{\prime \prime }(x)P_{\text{wet}}(x)\mathrm{d}x={\dot{m}}_f(h_{f,\text{out}}-h_{f,\text{in}}).$$
• Coupling to turbine:
  • $$P_{\text{turb,out}}={\dot{m}}_f\Delta h_t{\eta }_{\text{turb}}$$
  • $$P_{\text{turb,out}}{\eta }_{\text{mech}}\ge \sum P_{\text{pump}}.$$

Practical notes

• Limit wall heat flux hot-spots; manage $T_w$ versus allowable stress/oxidation.
• If fuel partially boils, include latent heat and two-phase pressure-drop models in the coolant path.