Co-solvent Enhancement Calculation

Reference ID: MET-402A | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The Co-solvent Enhancement calculation is a fundamental procedure in Supercritical Fluid Extraction (SFE) used to estimate the solubility increase of a target solute when a polar co-solvent (entrainer) is added to a supercritical carbon dioxide (sc-CO₂) stream. In process engineering, pure sc-CO₂ is often limited by its non-polar nature, which restricts its ability to dissolve polar or high-molecular-weight compounds. By introducing a small fraction of a co-solvent, such as ethanol, the solvent polarity is modified, significantly increasing the extraction efficiency.

This calculation is typically employed during the process scoping and solvent formulation phase, including solubility parameter calculation for supercritical fluids to determine the optimal entrainer concentration required to achieve target extraction yields without necessitating complex, high‑cost Equation of State (EoS) modeling, provided the system remains within the dilute regime.

Methodology & Formulas

The methodology relies on the modification of the Hansen solubility parameter to account for the mixture composition. The process follows a sequential calculation of baseline solvent power, volumetric mixing, and the resulting enhancement factor.

1. Baseline Solvent Power

The solubility parameter of pure sc-CO₂ is determined by its density relative to a reference liquid state (often near the critical point):

\[ \delta_{\text{CO}_{2}} = 1.25 \cdot \sqrt{P_{c,\text{CO}_2}} \cdot \sqrt{\frac{\rho_{\text{sc}}}{\rho_{\text{liq,ref}}}} \]

Where \( \rho_{\text{liq,ref}} \) is typically taken as 850 kg/m³ for CO₂.

2. Volumetric Mixing

To determine the mixture properties, mass fractions are converted to volume fractions based on the density of the individual components. Given a total mass \( m_{\text{total}} \), the volume of each component is \( V_{i} = m_{i} / \rho_{i} \). The volume fraction (\(\phi\)) for the co-solvent and CO₂ is:

\[ \phi_{\text{cs}} = \frac{V_{\text{cs}}}{V_{\text{cs}} + V_{\text{CO}_{2}}} \quad ; \quad \phi_{\text{CO}_{2}} = 1 - \phi_{\text{cs}} \]

3. Mixed Solvent Solubility Parameter

For preliminary scoping with low co-solvent concentrations, a simplified linear mixing rule is often applied. Note: This is an approximation. For more accurate prediction of polar system behavior, the full Hansen combining rules should be used.

\[ \delta_{\text{mix}} = \phi_{\text{CO}_{2}} \cdot \delta_{\text{CO}_{2}} + \phi_{\text{cs}} \cdot \delta_{\text{cs}} \]

4. Solubility Enhancement Factor

The enhancement factor (\(E\)), representing the estimated ratio of solubility in the mixture to the solubility in pure sc-CO₂, is derived from an empirical correlation:

\[ E = 10^{\,m \cdot (\delta_{\text{mix}} - \delta_{\text{CO}_{2}})} \]

Where \( m \) is a solute-specific empirical constant determined from experimental data.

Validity and Operational Constraints

The following table outlines the empirical boundaries for the validity of this simplified model. Operating outside these ranges, or for accurate design, requires more rigorous thermodynamic modeling.

Parameter	Constraint/Range
CO₂ Density (\(\rho_{\text{sc}}\))	500 – 1200 kg/m³
Co-solvent Concentration	0 – 10 wt%
System Pressure	> \(P_{c,\text{CO}_2}\) & < 50 MPa
System Temperature	40 – 80 °C

Note: \(P_{c,\text{CO}_2} = 7.39\ \text{MPa}\).

A co-solvent (or entrainer) is an auxiliary solvent added to the primary solvent to modify its physicochemical properties.

Improves Solubility: Enhances the dissolution of target compounds that have poor solubility in the main solvent (e.g., polar molecules in non-polar sc-CO₂).
Modifies Polarity: Adjusts the overall solvent polarity, which can improve selectivity and mass transfer rates.
Process Efficiency: Can allow for operation at lower pressures or temperatures, reducing energy consumption and capital cost.

The theoretical enhancement factor (E) is estimated using the empirical correlation based on solubility parameters:

Determine the solubility parameter of pure sc-CO₂ (\( \delta_{\text{CO}_2} \)) using the operating density and critical properties.
Convert the co-solvent mass fraction to volume fractions (\( \phi_{\text{cs}}, \phi_{\text{CO}_2} \)) using component densities.
Calculate the mixed solvent solubility parameter (\( \delta_{\text{mix}} \)) using a volume-weighted mixing rule (simplified) or Hansen combining rules (more accurate).
Obtain the solute-specific constant (\( m \)) from literature or correlating experimental data.
Compute \( E = 10^{\,m \cdot (\delta_{\text{mix}} - \delta_{\text{CO}_2})} \).

Note: For final process design, experimental validation or rigorous thermodynamic modeling is required.

Key parameters to explore during optimization include:

Co-solvent Concentration: Primary variable, typically 0–10 wt% for sc-CO₂ processes.
System Pressure and Temperature: Directly affect sc-CO₂ density and solvent power.
Extraction Time/Flow Rate: Affects mass transfer and contact efficiency.
Co-solvent Type: Different co-solvents have varying polarity (\( \delta_{\text{cs}} \)) and selectivity.

A systematic approach like Design of Experiments (DoE) is recommended to evaluate interactions between these variables.

Integrating co-solvent use requires analysis of downstream unit operations:

Solvent Recovery: The co-solvent's boiling point and vapor pressure affect the energy demand for separation from the extract and from CO₂.
Phase Behavior: Co-solvent may alter the phase separation in flash tanks or require additional distillation/absorption steps.
Product Purity: Residual co-solvent in the final product must meet specification; additional washing or purification may be needed.
Material Balances: Flowsheets and equipment sizing must be updated to include co-solvent recycle and make-up streams.

Worked Example: Co-solvent Enhancement in Supercritical CO₂ Extraction

A process engineer is scoping a supercritical CO₂ extraction process for caffeine from coffee beans. To improve efficiency, they are evaluating the addition of a small amount of ethanol as a co-solvent. This example estimates the solubility enhancement factor for caffeine under typical operating conditions using the simplified linear mixing rule.

Knowns (Input Parameters):

Temperature, \( T = 50.0 \, ^\circ\text{C} \)
Pressure, \( P = 200.0 \, \text{bar} = 20.0 \, \text{MPa} \)
Supercritical CO₂ density, \( \rho_{\text{sc}} = 800.0 \, \text{kg/m}^3 \)
Co-solvent concentration, \( w_{\text{cs}} = 5.0 \, \text{wt\%} \) ethanol
Co-solvent solubility parameter (ethanol), \( \delta_{\text{cs}} = 26.5 \, \text{MPa}^{0.5} \)
Solute-specific constant for caffeine, \( m = 0.18 \)
Critical pressure of CO₂, \( P_{c,\text{CO}_2} = 7.39 \, \text{MPa} \)
Reference liquid CO₂ density, \( \rho_{\text{liq,ref}} = 850.0 \, \text{kg/m}^3 \)
Ethanol density, \( \rho_{\text{cs}} = 789.0 \, \text{kg/m}^3 \)

Calculation Steps:

Calculate the baseline solubility parameter for pure supercritical CO₂, \( \delta_{\text{CO}_2} \):
\[ \delta_{\text{CO}_2} = 1.25 \cdot \sqrt{P_{c,\text{CO}_2}} \cdot \sqrt{\frac{\rho_{\text{sc}}}{\rho_{\text{liq,ref}}}} = 1.25 \cdot \sqrt{7.39} \cdot \sqrt{\frac{800.0}{850.0}} \]

\[ \delta_{\text{CO}_2} = 1.25 \cdot 2.7185 \cdot 0.9701 \approx 3.296 \ \text{MPa}^{0.5} \]
Calculate the volume fractions of CO₂ and ethanol. Basis: 100 g total solvent mixture.
- Mass of ethanol, \( m_{\text{cs}} = 5.0 \, \text{g} = 0.005 \, \text{kg} \)
- Mass of CO₂, \( m_{\text{CO}_2} = 95.0 \, \text{g} = 0.095 \, \text{kg} \)
- Volume of ethanol, \( V_{\text{cs}} = \dfrac{m_{\text{cs}}}{\rho_{\text{cs}}} = \dfrac{0.005}{789.0} = 6.337 \times 10^{-6} \ \text{m}^3 \)
- Volume of CO₂, \( V_{\text{CO}_2} = \dfrac{m_{\text{CO}_2}}{\rho_{\text{sc}}} = \dfrac{0.095}{800.0} = 1.1875 \times 10^{-4} \ \text{m}^3 \)
- Total volume, \( V_{\text{total}} = V_{\text{cs}} + V_{\text{CO}_2} = 1.2509 \times 10^{-4} \ \text{m}^3 \)
- Volume fraction of ethanol, \( \phi_{\text{cs}} = \dfrac{V_{\text{cs}}}{V_{\text{total}}} = \dfrac{6.337 \times 10^{-6}}{1.2509 \times 10^{-4}} \approx 0.0507 \)
- Volume fraction of CO₂, \( \phi_{\text{CO}_2} = 1 - \phi_{\text{cs}} \approx 0.9493 \)
Calculate the mixed solvent solubility parameter, \( \delta_{\text{mix}} \):
\[ \delta_{\text{mix}} = \phi_{\text{CO}_2} \cdot \delta_{\text{CO}_2} + \phi_{\text{cs}} \cdot \delta_{\text{cs}} \]

\[ \delta_{\text{mix}} = (0.9493 \cdot 3.296) + (0.0507 \cdot 26.5) \approx 3.130 + 1.343 = 4.473 \ \text{MPa}^{0.5} \]
Calculate the solubility enhancement factor, \( E \):
\[ E = 10^{\,m \cdot (\delta_{\text{mix}} - \delta_{\text{CO}_2})} \]

\[ \delta_{\text{mix}} - \delta_{\text{CO}_2} = 4.473 - 3.296 = 1.177 \ \text{MPa}^{0.5} \]

Exponent: \( m \cdot (\delta_{\text{mix}} - \delta_{\text{CO}_2}) = 0.18 \cdot 1.177 = 0.2119 \)

\[ E = 10^{0.2119} \approx 1.629 \]

Final Answer:

The estimated solubility enhancement factor for caffeine using 5 wt% ethanol in supercritical CO₂ at 50°C and 200 bar is \( E \approx 1.63 \). This indicates the solubility of caffeine is estimated to increase by approximately 63% compared to pure supercritical CO₂ under these conditions, based on the simplified linear mixing model.

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