NBPhO 2026

Difficulty hard

Prerequisites

• Two-phase domes on $T$–$s$ diagrams; lever rule for phase mixtures
• Triple point and sublimation curve in the $p$–$T$ diagram
• Reversible adiabatic flow $\Leftrightarrow$ isentropic ($\Delta s = 0$)
• Clausius-Clapeyron equation $dp_\text{sat}/dT = L/(T\,\Delta v)$
• Maxwell relation $(\partial s/\partial p)_T = -(\partial v/\partial T)_p$
• Latent heats of vaporisation, fusion and sublimation

Learning objectives

• Read a $T$–$s$ diagram to extract specific entropies on saturation curves and inside two-phase domes
• Recognise reversible adiabatic flow as isentropic and use $\Delta s = 0$ as the master constraint for nozzle flow
• Apply the lever rule to compute mass fractions in two-phase end states
• Derive the slope of the saturated-vapour curve $ds_g/dT$ along the coexistence line by combining Maxwell relations with Clausius-Clapeyron
• Use the criterion $L > c_p T$ to predict spontaneous condensation upon adiabatic expansion of saturated vapour
• Cross-check $T$–$s$ diagram readings against $\Delta s = L/T$ across a saturation curve

Watch out for

• Below the triple-point pressure, liquid CO₂ cannot exist; the exit state must lie on the sublimation curve (solid + vapour), not on the vaporisation curve.
• The right boundary of a two-phase dome can have $ds_g/dT < 0$ — a counterintuitive feature near the critical point that is the entire reason a saturated-vapour feed still produces solid CO₂ in scenario (b).
• The ideal-gas approximation behind $L > c_p T$ silently fails for fluids near their critical point. CO₂ at $25\,°\text{C}$ is only $6\,\text{K}$ below $T_c \approx 31\,°\text{C}$, where $c_p$ diverges; the criterion's qualitative prediction survives but a numerical estimate of $c_p T$ for CO₂ at $25\,°\text{C}$ is meaningless.
• The dome at $25\,°\text{C}$ is narrow (close to the critical point), so small horizontal errors in reading $s_\ell$ and $s_g$ off the diagram translate into substantial errors in the lever-rule mass fraction. Use the consistency check $s_v - s_s = L_\text{sub}/T$ to validate the readings.
• The de Laval (converging-diverging) shape is necessary for supersonic exhaust but irrelevant to the phase-fraction calculation, which depends only on inlet entropy and exit pressure.