Introduction & Context

Supercritical-fluid (SCF) extraction with carbon dioxide is a cornerstone separation technique in the chemical, food, and pharmaceutical industries. The economic viability of a plant is governed by three cost pillars: annualised capital charge, operating expenditure, and solvent make-up. This reference sheet provides the algebraic framework used to estimate the cost per unit mass of extract for a continuous-batch process, enabling rapid screening of design alternatives and feed specifications without disclosing proprietary data.

Methodology & Formulas

1. Operating Regime Check

Ensure the fluid is supercritical before applying density-based correlations.

Parameter	Regime Criterion
Temperature	$ T_{\text{op}} \ge T_{\text{c,CO}_2} $
Pressure	$ P_{\text{op}} \ge P_{\text{c,CO}_2} $

2. Solvent Density Ratio

Define the reduced density for correlation range validation:

\[ \phi = \frac{\rho_{\text{g}}}{\rho_{\text{l}}} \]

3. Solubility Parameter (Hildebrand)

Estimate the solubility parameter of supercritical CO₂:

\[ \delta = 1.25\,\sqrt{\phi}\,\sqrt{P_{\text{c,CO}_2}} \]

Validity interval: 0.2 ≤ φ ≤ 0.9. Outside this window the correlation accuracy degrades.

4. Mass Balances

Annual feed throughput:

\[ m_{\text{feed}} = m_{\text{batch}} \cdot N_{\text{batches}} \cdot t_{\text{days}} \]

Annual product (extract) mass:

\[ m_{\text{product}} = m_{\text{feed}} \cdot Y \]

Annual solvent circulation:

\[ m_{\text{solvent}} = m_{\text{feed}} \cdot R \]

Annual solvent make-up (loss):

\[ m_{\text{make-up}} = m_{\text{solvent}} \cdot f_{\text{loss}} \]

5. Cost Contributions

Capital charge (annualised):

\[ C_{\text{cap}} = I \cdot r \]

Operating cost:

\[ C_{\text{op}} = m_{\text{feed}} \cdot c_{\text{op}} \]

Solvent replacement cost:

\[ C_{\text{solv}} = m_{\text{make-up}} \cdot p_{\text{CO}_2} \]

Total cost per unit mass of extract:

\[ \text{Unit Cost} = \frac{C_{\text{cap}} + C_{\text{op}} + C_{\text{solv}}}{m_{\text{product}}} \]

6. Nomenclature

Symbol	Description	Units
$ T_{\text{op}} $	Operating temperature	°C
$ P_{\text{op}} $	Operating pressure	bar
$ T_{\text{c,CO}_2} $	CO₂ critical temperature	°C
$ P_{\text{c,CO}_2} $	CO₂ critical pressure	MPa
$ \rho_{\text{g}} $	SC-CO₂ density at operating conditions	kg m^-3
$ \rho_{\text{l}} $	Liquid reference density (water)	kg m^-3
$ \phi $	Density ratio $ \rho_{\text{g}}/\rho_{\text{l}} $	—
$ \delta $	Estimated solubility parameter	(MPa)^0.5
$ m_{\text{batch}} $	Mass per batch	kg
$ N_{\text{batches}} $	Batches per day	d^-1
$ t_{\text{days}} $	Operating days per year	d yr^-1
$ Y $	Target mass yield (extract/feed)	—
$ R $	Solvent ratio (CO₂/feed)	kg kg^-1
$ f_{\text{loss}} $	Fractional solvent loss	—
$ I $	Installed capital cost	USD
$ r $	Annualised capital charge rate	yr^-1
$ c_{\text{op}} $	Unit operating cost (per kg feed)	USD kg^-1
$ p_{\text{CO}_2} $	CO₂ purchase price	USD kg^-1

Unit production cost ($ kg^-1 product) versus market price to maintain positive gross margin
Specific energy consumption (kWh kg^-1) because supercritical fluids require high-pressure compression and heating
Solvent loss rate (% w w^-1 feed) which dictates make-up chemical expense
Annualized capital charge or CAPEX per kg installed capacity to benchmark against alternative separation technologies
Overall plant availability (%) since high-pressure equipment downtime can erode NPV quickly

Start with mass balance: determine total CO₂ circulation rate (kg h^-1) from extraction column and stripper design
Calculate ideal compression work using isentropic relations across the pressure ratio (P₂/P₁) and correct with vendor-supplied adiabatic efficiency (typically 75-80%)
Add inter-stage cooling duty using CO₂ heat capacity at average pressure; include 10% fouling factor
Convert kWh to local electricity tariff ($ kWh^-1) and cooling water cost ($ m^-3) to obtain $ h^-1 utility cost
Scale to annual basis using plant operating hours and apply 5% contingency for control valve and seal losses

Feedstock price volatility—every 10% increase can extend payback by 6-9 months for high-volume botanical extractions
Product purity specification—tighter residual solvent limits raise both capital (extra separators) and operating costs (longer batch times)
Energy price forecasts; high-pressure CO₂ recycle compressors can represent 35-45% of OPEX
Plant utilization factor; campaigns below 70% capacity utilization shift fixed costs onto fewer kilograms, stretching payback beyond 5 years

For CO₂, pressure-swing is favored when electricity is <0.08 $ kWh^-1 and high-pressure (80-120 bar) storage vessels are already required for extraction
Temperature-swing becomes attractive if low-pressure steam (<3 barg) is available at <5 $ t^-1 because heater duty replaces expensive compression
Hybrid schemes (warm pressure reduction) often minimize total cost when extract has moderate volatility and heat-sensitive components
Always compare net present utility cost over 10-year plant life; crossover point typically occurs around 6 kt y^-1 CO₂ circulation rate

Worked Example – SCF CO₂ Extraction of High-Value Antioxidants

A specialty-chemicals plant is evaluating supercritical CO₂ extraction to recover 2% (mass) of antioxidants from dried rosemary leaves. The process runs 300 days per year in 4 batches per day; each batch treats 500 kg of feed. A solvent ratio of 40 kg CO₂ per kg feed is required, and 1% of the circulating CO₂ is lost and must be replaced at 0.20 USD kg^-1. The installed extractor cost is 3 M USD and is depreciated at 10% yr^-1; other operating costs (utilities, labour, maintenance) are 0.12 USD per kg of feed. Determine the total annual cost and the cost per kilogram of purified product.

Knowns

Annual operating days: 300 d
Batches per day: 4
Batch size: 500 kg
Target yield (mass): 0.02 kg product kg^-1 feed
Solvent ratio: 40 kg CO₂ kg^-1 feed
CO₂ make-up fraction: 0.01
CO₂ price: 0.20 USD kg^-1
Capital charge rate: 0.10 yr^-1
Installed cost: 3,000,000 USD
Operating cost: 0.12 USD kg^-1 feed

Step-by-step calculation

Annual feed throughput
\[ \text{Annual feed} = 300\ \text{d} \times 4\ \text{batch d}^{-1} \times 500\ \text{kg batch}^{-1} = 600,000\ \text{kg} \]
Annual product mass
\[ M_{\text{product}} = 600,000\ \text{kg} \times 0.02 = 12,000\ \text{kg} \]
Annual solvent circulation
\[ \text{Annual solvent} = 600,000\ \text{kg} \times 40 = 24,000,000\ \text{kg} \]
CO₂ make-up requirement
\[ \text{Make-up} = 24,000,000\ \text{kg} \times 0.01 = 240,000\ \text{kg} \]
Annual solvent cost
\[ C_{\text{solv}} = 240,000\ \text{kg} \times 0.20\ \text{USD kg}^{-1} = 48,000\ \text{USD} \]
Annual capital charge
\[ C_{\text{cap}} = 3,000,000\ \text{USD} \times 0.10 = 300,000\ \text{USD} \]
Annual operating cost
\[ C_{\text{op}} = 600,000\ \text{kg} \times 0.12\ \text{USD kg}^{-1} = 72,000\ \text{USD} \]
Total annual cost
\[ \text{Total cost} = 48,000 + 300,000 + 72,000 = 420,000\ \text{USD} \]
Cost per kilogram of product
\[ \text{Unit cost} = \frac{420,000\ \text{USD}}{12,000\ \text{kg}} = 35.0\ \text{USD kg}^{-1} \]

Final Answer

Total annual cost: 420,000 USD
Cost per kilogram of purified antioxidant: 35 USD kg^-1

"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

Parameter	Regime Criterion
Temperature	\( T_{\text{op}} \ge T_{\text{c,CO}_2} \)
Pressure	\( P_{\text{op}} \ge P_{\text{c,CO}_2} \)

Symbol	Description	Units
\( T_{\text{op}} \)	Operating temperature	°C
\( P_{\text{op}} \)	Operating pressure	bar
\( T_{\text{c,CO}_2} \)	CO₂ critical temperature	°C
\( P_{\text{c,CO}_2} \)	CO₂ critical pressure	MPa
\( \rho_{\text{g}} \)	SC-CO₂ density at operating conditions	kg m^-3
\( \rho_{\text{l}} \)	Liquid reference density (water)	kg m^-3
\( \phi \)	Density ratio \( \rho_{\text{g}}/\rho_{\text{l}} \)	—
\( \delta \)	Estimated solubility parameter	(MPa)^0.5
\( m_{\text{batch}} \)	Mass per batch	kg
\( N_{\text{batches}} \)	Batches per day	d^-1
\( t_{\text{days}} \)	Operating days per year	d yr^-1
\( Y \)	Target mass yield (extract/feed)	—
\( R \)	Solvent ratio (CO₂/feed)	kg kg^-1
\( f_{\text{loss}} \)	Fractional solvent loss	—
\( I \)	Installed capital cost	USD
\( r \)	Annualised capital charge rate	yr^-1
\( c_{\text{op}} \)	Unit operating cost (per kg feed)	USD kg^-1
\( p_{\text{CO}_2} \)	CO₂ purchase price	USD kg^-1