Thermodynamic Equilibrium Modeling of Extractable Perfluoroalkyl Substances in Multi-Tiered Recycled Fiber Pulping Systems

Furnish

Incoming recovered paper bales arrive with uncharacterized loads of fluorochemical treatments applied during original paper and board conversion. Secondary fiber pulping operations disperse these inputs across hydrapulpers, coarse screening circuits, flotation deinking cells, and counter-current washing loops. Extractable perfluoroalkyl and polyfluoroalkyl substances partition between cellulose fibrils, mineral fillers, process water, and colloidal dissolved organic matter based on thermodynamic sorption equilibria.

Mills operating closed-loop water circuits concentrate short-chain homologues over time. Compliance with restricted substance limits requires rigorous phase-distribution modeling at each deinking and washing tier.

Analytical characterization under EN 17681-1 and EN 17681-2 targets extractable targets via liquid chromatography tandem mass spectrometry following methanol extraction. The total fluorine combustion assay provides a gross screening metric under DIN EN 15408, but thermodynamic equilibrium models predict how specific homologues migrate into finished recycled paper grades. Perfluorooctanoic acid and perfluorooctane sulfonate demonstrate elevated solid-water distribution coefficients compared to perfluorobutanoic acid.

Hydrophobic interactions drive long-chain perfluoroalkyl acids onto lignocellulose matrices, while short-chain replacements remain in the aqueous filtrate.

A solid-water distribution coefficient of 42 liters per kilogram holds for perfluorooctanoic acid on unbleached kraft furnish at four percent consistency.

Recovered fiber furnishes combine post-consumer corrugated containers, sorted graphic papers, and coated folding boxboard. Each raw furnish category carries distinct organofluorine profiles from grease-resistant barriers, sizing agents, and printing auxiliaries. The table below delineates the baseline thermodynamic input parameters and phase distribution characteristics observed across primary furnish categories entering high-consistency hydrapulpers.

Hydrapulper mechanical shear disintegrates fiber bundles at temperatures between forty-five and sixty-five degrees Celsius. Pulping releases non-covalently bound fluorosurfactants into the surrounding liquor. The thermodynamic drive toward equilibrium begins instantly as fibers suspend at consistencies between twelve and eighteen percent.

Mass transfer rates depend on boundary layer diffusion across swollen cell walls.

Process water pH shifts equilibrium distributions through headgroup ionization. Carboxylic and sulfonic acid moieties possess negative logarithmic acid dissociation constants, ensuring complete ionization across typical neutral and alkaline pulping ranges between pH 7.0 and 9.5. Deprotonated perfluoroalkyl anions interact with cationic retention aids, wet-strength resins, and multivalent cations in the furnish slurry.

Slurry

Fiber suspension mechanics govern the boundary conditions of the aqueous phase. Swollen secondary fibers present high specific surface areas ranging from two to six square meters per gram. Perfluoroalkyl chains exhibit hydrophobic exclusion from bulk water, driving sorption onto cellulose crystalline domains and residual lignin patches.

The sorption equilibrium follows non-linear isotherm behavior governed by both organic carbon partitioning and electrostatic surface complexation.

Mathematical modeling of this multiphase equilibrium requires coupling the Freundlich isotherm with electrostatic double-layer equations. At low aqueous equilibrium concentrations, sorption behaves linearly through the solid-water distribution coefficient. When surfactant concentrations rise near chemical barrier pulping batches, site-saturation effects emerge.

High dissolved organic carbon concentrations in recycled white water create a competing pseudo-phase that retains perfluoroalkyl acids in colloidal suspension.

Do Thermodynamic Models Predict Extractable PFAS Fractions?

Isotherm constants derived under laboratory bench conditions shift when applied to multi-component mill stocks. Dissolved calcium ions compress the electrical double layer around anionic cellulosic microfibrils. Divalent cation bridging accelerates the deposition of perfluorooctanoic acid onto negatively charged fiber surfaces.

The linear solid-water partition coefficient expands by a factor of three when calcium hardness rises from fifty to four hundred milligrams per liter.

Fiber consistency directly regulates the mass ratio between solid and aqueous compartments. In low-density screening circuits operating at one percent consistency, ninety-five percent of the total slurry mass consists of water. Short-chain compounds with distribution coefficients below ten liters per kilogram remain almost entirely dissolved in the free liquor.

The mathematical relationship governing the fraction of substance bound to the fiber phase derives from mass balance principles.

Cationic wet-strength polyamide-epichlorohydrin resins increase the solid-phase retention of perfluorinated carboxylates across neutral pulping stages.

The system partition equation calculates the equilibrium fiber-bound fraction across varying stock consistencies and partition coefficients:

F_solid = (Kd C_fiber) / (1 + Kd C_fiber)

Here, Kd represents the distribution coefficient in liters per kilogram, and C_fiber represents the consistency expressed in kilograms of dry fiber per liter of total slurry. At high pulping consistencies, even weakly sorbing short-chain homologues show noticeable fiber retention due to limited solvent volume.

Temperature elevation inside the pulping drum alters standard free energy values. Desorption processes are generally endothermic for fluorotelomer-based sizing agents attached via ester linkages, whereas physical sorption of ionic perfluoroalkyl substances demonstrates exothermic characteristics. Higher temperatures promote migration into the aqueous phase during initial pulping, increasing the load carried into downstream screening filtrate.

Sorption kinetics reach ninety percent equilibrium within twelve minutes of mechanical agitation. Secondary fiber processing cycles maintain residence times sufficient to establish thermodynamic equilibrium prior to coarse screening. Mill engineers utilize these equilibrium values to calculate extractable chemical carried forward into subsequent process loops.

Foam

Flotation deinking units exploit surface tension gradients to remove hydrophobic ink particles, creating a distinct interfacial phase. Perfluoroalkyl substances act as exceptionally efficient surface-active agents, concentrating preferentially at the air-water interface of rising bubbles. Thermodynamic enrichment factors in deinking froth exceed solid-phase distribution coefficients by several orders of magnitude.

The gas-liquid equilibrium constant dictates fluorochemical accumulation in the reject froth stream.

Froth removal rates depend on bubble surface area flux and surfactant surface excess. The Gibbs adsorption isotherm relates equilibrium surface excess to the derivative of surface tension with respect to aqueous concentration. Long-chain homologues like perfluorooctane sulfonate exhibit high surface excess values, stripping rapidly into the mechanical reject stream.

Short-chain perfluorobutanoic acid displays low interfacial affinity, remaining behind in the deinked accept stock.

Does Closed Loop Washing Concentrate Short Chains?

Flotation froth extraction creates an unexpected divergence in homologue distribution across the mill. While air flotation efficiently purges perfluorooctyl moieties from the accept stream, it leaves hydrophilic short-chain carboxylic acids unaffected. Counter-current clarification circuits recycle flotation clarify water back to the hydrapulper, progressively elevating steady-state concentrations of persistent short-chain compounds.

The interfacial mass transfer rate governs foam fractionation performance in mechanical deinking cells. Air flow rates, bubble diameter distributions, and impeller speeds dictate the total interfacial area generated per cubic meter of pulp stock. The following operational parameters define the interfacial partition dynamics observed in typical four-cell flotation banks:

• Interfacial Surface Excess defines the molar concentration of fluorosurfactant molecules packed at the gas-liquid boundary layer per square meter of bubble area.
• Reject Froth Volume determines the hydraulic drag carrying entrained liquid and suspended solids into the mechanical skimmer trough.
• Aerosol Stripping Coefficient quantifies the volatile loss of neutral fluorotelomer alcohols into the exhaust air handling ductwork.
• Bubble Coalescence Rate limits the drainage velocity of liquid lamellae, establishing the final chemical enrichment in the deinking reject cake.

Counter-current deinking configurations maximize contaminant purge efficiency by routing cleaner water toward the finished stock deckers. Clarified water from downstream thickening stages flows upstream to feed pulping and flotation stages. This hydraulic arrangement pushes extractable perfluorinated contaminants into high-solids waste streams, but incomplete foam separation allows recalcitrant anions to bypass primary purges.

Equilibrium partitioning models must incorporate air-water partition coefficients alongside solid-liquid parameters. Multiphase mass balances establish that twenty to forty percent of input perfluorooctane sulfonate mass exits via flotation froth. Conversely, less than three percent of input perfluorobutanoic acid partitions into the froth phase under identical hydrodynamic conditions.

OEKO-TEX STANDARD 100 Class I limits total extractable perfluorooctanoic acid to twenty-five parts per billion on dry fiber weight.

Surfactant competition at the bubble interface alters extraction efficiency during mixed-office waste pulping. Fatty acid collectors and synthetic non-ionic surfactants added to aid ink detachment compete directly with fluorosurfactants for interface sites. High collector addition rates suppress fluorochemical foam enrichment, forcing extractable residues to remain within the aqueous pulp slurry.

Filtrate

Counter-current fiber washing and thickening stages shift chemical equilibrium through successive solid-liquid separations. Screw presses, disc filters, and gravity deckers squeeze process water from the pulp web, raising consistency from one percent to thirty percent. The squeezed filtrate carries dissolved perfluoroalkyl homologues into water clarification loops.

Chemical retention on finished sheet structures depends entirely on the equilibrium established in the final dilution zone before the paper machine headbox.

Dissolved organic carbon in recycled white water binds hydrophobic perfluorocarboxylic acids through micellar and humic-like interactions. Standard solid-liquid modeling fails if it neglects this colloidal phase. A three-phase equilibrium framework accounts for fiber-bound, colloid-bound, and truly dissolved aqueous fractions.

The apparent distribution coefficient decreases as dissolved organic carbon levels rise, mobilizing perfluorooctane sulfonate into the press filtrate.

Dissolved organic carbon complexation parameters dictate the mobility of persistent organofluorines throughout multi-tiered pulping circuits. The table below outlines the three-phase distribution modeling coefficients calculated across typical mechanical washing operations.

Mathematical simulation of a three-stage counter-current washing line demonstrates how chemical build-up occurs. Let a recycled pulp line operate with fresh water makeup restricted to eight cubic meters per metric ton of air-dry pulp. Intermediate filtrates circulate backward from stage three to stage one.

The model assumes instant phase equilibrium at each dilution stage followed by complete mechanical separation at the press nip.

1. Primary Hydrapulper Dilution mixes raw furnish with stage two press filtrate, establishing the initial dissolved and fiber-bound mass ratio.
2. Interstage Thickening Press raises stock consistency from three to twenty-eight percent, expelling eighty-nine percent of the total water volume and its associated dissolved fluorochemical load.
3. Secondary Counter-Current Displacement injects clarified white water from the third washing tier, displacing residual interstitial water containing concentrated short-chain anions.
4. Final Mat Formation Press consolidates fibers to forty percent consistency, locking remaining equilibrium-bound chemicals into the finished recycled base paper.

Calculations show that short-chain homologues achieve steady-state aqueous concentrations four to seven times higher than incoming furnish feed concentrations. Long-chain homologues do not build up in the water loop to the same extent because they continually purge with the outgoing fiber mat and deinking sludge cakes. High-consistency washing stages favor the retention of long-chain compounds on the fiber web.

Polymer coagulants and bentonite microparticle systems applied during dissolved air flotation clarification destabilize colloidal organic matter. When dissolved organic carbon precipitates into clarify sludge, it co-extracts bound perfluorooctane sulfonate and perfluorodecanoic acid. Clarifier underflow sludge dewatering represents a major environmental exit pathway for extractable organofluorines in closed-loop mills.

Settlement

Commercial contracts for recycled packaging and molded fiber tableware increasingly stipulate strict limit values for extractable fluorochemicals. Brand specifications invoke international eco-labels and food-contact legislation, referencing limits under the European Union Persistent Organic Pollutants Regulation and California Assembly Bill 1200. Mill laboratories verify compliance on composite finished product samples using targeted liquid chromatography tandem mass spectrometry techniques.

Thermodynamic models allow mill operators to establish incoming furnish thresholds based on allowable final sheet concentrations. If a food-grade recycled linerboard specification limits perfluorooctanoic acid to twenty-five micrograms per kilogram, the equilibrium model defines the maximum permissible loading in incoming recovered paper bales. Sourcing managers utilize these mathematical back-calculations to accept or reject secondary fiber lots before unloading at the warehouse dock.

A compliance failure on total fluorine testing triggers mandatory batch quarantine and costly third-party isomer quantification under EN 17681-1.

Disputes arise when laboratory solvent extraction methods overestimate bioavailable or migrating fractions during standard migration cell testing. Methanol extraction under elevated temperature and sonication strips both surface-adsorbed and intra-crystalline fluorochemical residues from cellulose matrices. Aqueous food simulant extractions recover only the fraction governed by the liquid-solid distribution coefficient at ambient temperatures.

Procurement agreements must define test methods and sample preparation protocols with explicit precision. General references to restricted substance lists fail to account for the thermodynamic behavior of recycled fibers across varying moisture, temperature, and contact matrices. The final commercial settlement between fiber suppliers and paper converters depends on verifiable, batch-specific chemical partition modeling grounded in thermodynamic equilibrium reality.

Mill technical teams manage compliance exposure by balancing furnish recipes. Blending virgin mechanical pulp with sorted secondary fiber dilutes incoming fluorochemical burdens below critical equilibrium thresholds. Chemical compliance remains an operational reality dictated by phase partition thermodynamics across every stage of the recycled fiber pulping circuit.