Problem Set

NBPhO 2026

9. CO₂ Fire Extinguisher Jaan Kalda 8 pts

Thermodynamics · Phase transitions, isentropic flow, Clausius-Clapeyron

Use a TT–ss diagram of CO₂ to find the dry-ice fraction in the exhaust of a fire extinguisher for two cylinder orientations, then derive the condition under which adiabatic expansion of saturated vapour spontaneously condenses.

Problem by Jaan Kalda.

A fire extinguisher contains liquid CO₂ in equilibrium with its saturated vapour at room temperature T0=298 KT_0 = 298\,\text{K}. Consider two scenarios: (a) the container is held upside-down so that the liquid phase flows to the nozzle; (b) it is held upright so that the saturated vapour flows to the nozzle. The nozzle has the shape of a converging–diverging channel (see figure), and the flow through it can be modelled as reversible and adiabatic. Atmospheric pressure is patm=1.0×105 Pap_\text{atm} = 1.0 \times 10^5\,\text{Pa}; the CO₂ triple point is at (Tt,pt)=(216.6 K,0.518 MPa)(T_t, p_t) = (216.6\,\text{K}, 0.518\,\text{MPa}).

Graph: Schematic of a converging–diverging nozzle (de Laval nozzle); on the left side is the CO₂ reservoir labelled "liquid or vapour", in the middle is the bottleneck (throat), and on the right the channel opens into the atmosphere.

The temperature–entropy diagram of CO₂ with isobars is provided below.

Graph: T–s diagram of CO₂ from s = 0 to 4000 J/(kg·K) and T from −100 °C to +150 °C. A two-phase dome covers the central region, with isobars labelled 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20 MPa crossing the diagram; isobars are nearly horizontal inside the dome and curve upward outside.

i) (1.5 points) Identify the phase or phases present and their temperature in the stream emerging from the nozzle.

ii) (3.5 points) Find the mass fraction xx of solid CO₂ in the stream for both scenarios.

iii) (3 points) Now consider an arbitrary pure substance in equilibrium with its saturated vapour at temperature TT, and suppose that only the vapour (not the liquid) escapes through a nozzle, undergoing reversible adiabatic expansion. Under what condition on the vaporisation latent heat LL (per unit mass), the isobaric specific heat of the vapour cpc_p, and the temperature TT does a fraction of the escaping vapour condense into droplets, even for a vanishingly small pressure drop? Does condensation occur for water vapour at T=373 KT = 373\,\text{K} (L≈2260 kJ⋅kg−1L \approx 2260\,\text{kJ}\cdot\text{kg}^{-1}, cp≈2.0 kJ⋅kg−1⋅K−1c_p \approx 2.0\,\text{kJ}\cdot\text{kg}^{-1}\cdot\text{K}^{-1})?