Introduction & Context

Super-critical fluids (SCFs) are widely used as green solvents in extraction, reaction, and particle-formation processes. Determining whether the operating temperature and pressure place a pure fluid above its critical point is the first step in every process design: only then can the fluid exhibit the liquid-like densities and gas-like diffusivities that make SCF technology attractive. This reference sheet provides the algebraic criteria and dimensionless ratios used to verify that carbon dioxide (or any other pure solvent) is in the super-critical regime.

Methodology & Formulas

Convert Celsius to absolute temperature \[T_{\text{K}}=T_{^{\circ}\text{C}}+273.15\]
Form dimensionless ratios \[ \frac{T}{T_{\text{c}}}=\frac{T_{\text{K}}}{T_{\text{c,K}}} \quad\text{and}\quad \frac{P}{P_{\text{c}}}=\frac{P_{\text{abs}}}{P_{\text{c}}} \]

Criticality test

Region	Condition
Super-critical	\(T/T_{\text{c}}>1\) and \(P/P_{\text{c}}>1\)
Sub-critical	either ratio ≤ 1

Ideal-gas sanity check (optional) \[ v_{\text{ideal}}=\frac{R\,T_{\text{K}}}{P_{\text{abs}}} \]

Validity range	Condition
Approx. ideal gas	\(P/P_{\text{c}}\le 2\) and \(T/T_{\text{c}}\ge 1.2\)

Start with a variable-volume view-cell and a high-precision syringe pump.

Load the blend at a sub-critical density and ramp temperature in 2 °C steps while recording pressure.
Plot P vs. T; the inflection where (∂P/∂T)_ρ becomes discontinuous flags the vicinity of the critical point.
Fine-tune by approaching at constant T in 0.1 MPa increments until meniscus disappearance/reappearance is observed; the average of the two values gives T_c within ±0.3 °C.
Confirm by measuring the critical opalescence intensity with a fiber-optic probe; the intensity peak should coincide with the inflection point.

Always verify with a second sample to rule out impurities that can shift T_c by several degrees.

For quick screening use Peng-Robinson with binary interaction parameters (k_ij) fitted to VLE data at 0.8–1.2 T_c.

If the entrainer is polar (ethanol, water), switch to PSRK or VTPR; both handle hydrogen bonding more reliably.
Above 15 MPa, PC-SAFT with 2B association scheme reduces deviation in ρ to <1 % compared to 4 % for PR.
Always validate at two compositions bracketing your plant range; a single k_ij tuned at 5 mol % can over-predict P_c by 0.4 MPa at 10 mol %.

Base margin on instrument uncertainty plus feed variability.

If your plant thermocouples are calibrated to ±1 °C, set the interlock 3 °C below T_c to cover 3σ drift.
Add another 2 °C if the solvent is recovered and recycled; trace accumulation of light hydrocarbons can depress T_c by up to 1.5 °C per 0.1 wt %.
Keep the reactor pressure ≥1.15 P_c at the chosen margin temperature to avoid two-phase flow that can foul the compressor.

CO₂ acts as a light impurity, lowering both T_c and P_c.

At 2 mol % CO₂ in ethanol, T_c drops ~1.2 °C and P_c drops ~0.25 MPa.
The shift is nonlinear; from 2 to 5 mol % the additional T_c depression is only 0.7 °C.
Vent the receiver headspace or run a 5 min N₂ purge between batches to keep CO₂ below 0.5 mol % and avoid re-optimizing the entire temperature profile.

Worked Example: Verifying Super-critical CO₂ for a High-pressure Extraction

A specialty-chemicals plant plans to use carbon-dioxide as a solvent for continuous extraction of plant-based antioxidants. Before the detailed column design, the process engineer must confirm that the operating temperature and pressure place CO₂ well into the super-critical region so that solvent power remains high while surface tension is negligible.

Knowns

Critical temperature of CO₂: 31.1 °C
Critical pressure of CO₂: 7.4 MPa
Specific gas constant for CO₂: 188.9 J kg^-1 K^-1
Proposed operating temperature: 45.0 °C
Proposed operating pressure: 10.0 MPa

Step-by-Step Calculation

Convert the critical temperature to kelvin: \(T_c = 31.1 + 273.15 = 304.25\ \text{K}\).
Convert the operating temperature to kelvin: \(T_{op} = 45.0 + 273.15 = 318.15\ \text{K}\).
Compute the reduced temperature: \(T_r = \frac{T_{op}}{T_c} = \frac{318.15}{304.25} = 1.046\).
Compute the reduced pressure: \(P_r = \frac{P_{op}}{P_c} = \frac{10.0}{7.4} = 1.351\).
Because both \(T_r > 1\) and \(P_r > 1\), the state is super-critical.
Estimate the ideal specific volume with the ideal-gas relation: \[v_{ideal} = \frac{R\ T_{op}}{P_{op}} = \frac{188.9 \times 318.15}{10.0 \times 10^6} = 0.006\ \text{m}^3\ \text{kg}^{-1}\]

Final Answer

At 45.0 °C and 10.0 MPa, CO₂ is in the super-critical state (reduced temperature 1.046, reduced pressure 1.351) with an ideal specific volume of 0.006 m³ kg^-1. The chosen conditions are therefore suitable for the intended super-critical extraction process.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle

Critical Point Determination for SCF Solvents

Introduction & Context

🚀 Skip the Manual Math!

Methodology & Formulas

Worked Example: Verifying Super-critical CO2 for a High-pressure Extraction

Worked Example: Verifying Super-critical CO₂ for a High-pressure Extraction