Catalytic threephase reactors l Gas liquid and solid
- Slides: 182
Catalytic three-phase reactors l Gas, liquid and solid catalyst
Function principle l Some reactants and products in gas phase l Diffusion to gas-liquid surface l Gas dissolves in liquid l Gas diffuses through the liquid film to the liquid bulk l Gas diffuses through the liquid film around the catalyst particle to the catalyst, where the reaction takes place l Simultaneous reaction and diffusion in porous particle
Three-phase reactors – catalyst l Small particles (micrometer scale < 100 micrometer) l Large particles (< 1 cm)
Catalyst design
Reactors
Bubble column
Flow pattern in bubble column
Tank reactor l Often called slurry reactor
Packed bed – trickle bed l Trickle bed ¡Liquid downflow – trickling flow l Packed bed, if liquid upflow
Packed bed- fixed bed – trickle bed
Flow chart: trickle bed
Trickle flow
Packed bed
Three-phase fluidized bed
Fluidized bed – flow chart
Monolith catalysts
Flow in monoliths
Monolith channel
Three-phase monolith reactor
Three-phase reactors Mass balances l Plug flow and axial dispersion ¡Columnr eactor ¡Tube reactor ¡Trickle bed ¡Monolith reactor l Backmixing ¡Bubble column ¡Tank reactor
Three-phase reactor Mass balances l Mass transfer from gas to liquid, from liquid to catalyst surface l Reaction on the catalyst surface l In gas- and liquid films only diffusion transport l Diffusion flow from gas to liquid
Three-phase reactor Mass transfer
Three-phase reactor Mass balances l For physical absorption the fluxes through the gas- and liquid films are equal l Flux from liquid to catalyst particle = component generation rate at steady state
Three-phase reactor Mass balances l Flux through the liquid film defined with concentration difference and liquid-film coefficient l Catalyst bulk density defined by
Three-phase reactor Mass balances l ap = total particle surface/reactor volume
Three-phase reactor Mass balances l If diffusion inside the particle affects the rate, the concept of effectiviness factor is used as for two-phase reactor (only liquid in the pores of the particles) l The same equations as for two-phase systems can be used for porous particles
Three-phase reactor – plug flow
Three-phase reactor - plug flow, liquid phase l For volume element in liquid phase l Liquid phase l
Three-phase reactor Plug flow - gas phase l For volume element in gas phase l Gas phase l - concurrent l + countercurrent
Three-phase reactor Plug flow l Initial conditions ¡ Liquid phase ¡ Gas phase, concurrent ¡ Gas phase, countercurrent
Three-phase reactor - plug flow model l Good for trickle bed l Rather good for a packed bed , in which liguid flows upwards l For bubble column plug flow is good for gas phase but not for liquid phase which has a higher degree of backmixing
Three-phase reactor - complete backmixing l Liquid phase l Gas phase
Three-phase reactor - semibatch operation l Liquid phase in batch l Gas phase continuous l Initial condition
Parameters in three-phase reactors l Gas-liquid equilibrium ratio (Ki) from ¡Thermodynamic theories ¡Gas solubility in liquids (Henry’s constant) l Mass transfer coefficients k. Li, k. Gi ¡Correlation equations
Numerical aspects l CSTR – non-linear equations ¡Newton-Raphson method l Reactors with plug flow (concurrent) ¡Runge-Kutta-, Backward difference -methods l Reactors with plug flow (countercurrent) and reactors with axial dispersion (BVP) ¡orthogonal collocation
Examples l Production of Sitostanol ¡A cholesterol suppressing agent ¡Carried out through hydrogenation of Sitosterol on Pd catalysts (Pd/C, Pd/Zeolite) l Production of Xylitol ¡An anti-caries and anti-inflamatory component ¡Carried out through hydrogenation of Xylose on Ni- and Ru-catalysts (Raney Ni, Ru/C)
Exemple: from cholesterol tol sitostanol
Reaction scheme A superficially complicated scheme
From laboratory scale to industrial scale Slurry, three-phase reactor Lab reactor, 1 liter, liquid amount 0. 5 kg Large scale reactor, liquid amount 8080 kg Simulation of large-scalle reactor based on laboratory reactor
Catalytic reactor l Semi-batch stirred tank reactor ¡ Well agitated, no concentration differences appear in the bulk of the liquid ¡ Gas-liquid and liquid-solid mass transfer resistances can prevail ¡ The liquid phase is in batch, while gas is continuously fed into the reactor. ¡ The gas pressure is maintained constant. ¡ The liquid and gas volumes inside the reactor vessel can be regarded as constant
Mathematical modelling Reaction, diffusion and catalyst deactivation in porous particles Particle model Rates
Model implementation , where Dei=( p/ p)Dmi Boundary conditions
Catalytic reactor, mass balances Liquid phase mass balance Liquid-solid flux Gas-liquid flux
Numerical approach l PDEs discretizied with finite difference formulae l The ODEs created solved with a stiff algorithm (BD, Hindmarsh)
Rate equations Surface reaction, rate determining Essentially non-competetive adsorption of hydrogen and organics
Kinetics in laboratory scale Concentrations as a function of reaction time
Kinetics in plant scale Concentration of organics Hydrogen concentration in liquid phase
Comparison of lab and plant scale Laboratory Factory
Hydrogen concentration in liquid phase in plant scale
Hydrogenation of Xylose
Modelling results Xylose hydrogenation Heavy mass transfer Moderate mass transfer Effect of external mass transfer Heavy mass transfer and Moderatly deactivated Light mass transfer
Gas-liquid reactors
Gas-liquid reactors l Non-catalytic or homogeneously catalyzed reactions ¡Gas phase ¡Liquid phase ( + homogeneous catalyst) l Components i gas phase diffuse to the gas -liquid boundary and dissolve in the liquid phase l Procukt molecules desorb from liquid to gas or remain in liquid
Gas-liquid reactions l Synthesis of chemicals l Gas absorption, gas cleaning l Very many reactor constructions used, depending on the application
Gas-liquid reaction: basic principle
Gas-liquid reactor constructions l Spray column l Wetted wall column l Packed column l Plate column l Bubble columns l Continuous, semibatch and batch tank reactors l Gas lift reactors l Venturi scrubbers
Gas-liquid reactors - overview
Tank reactor
Gas-liquid reactors l Packed column ¡Absorption of gases ¡Countercurrent principle: gas upwards, liquid downwards ¡Column packings lenable a large gas-liquid contact area lmade of ceramics, plastics or metal lgood gas distribution because of packings lchanneling can appear in liquid phase; can be handled with distribution plates ¡Plug flow in gas and liquid phases
Gas-liquid reactors l Plate column ¡ Absorption of gases ¡ Countercurrent ¡ Various plates used as in distillation, e. g. l Bubble cap l Plate column l Packed column ¡ Absorption of gases ¡ Countercurrent ¡ A lot of column packings available; continuous development
Bubble column Gas-lift -reactor
Bubble column – design examples
Bubble column
Packed column
Packings
Plate column
Gas-liquid reactors l Gas scrubbers ¡Spray tower l. Gas is the continuous phase l. In shower ! ¡ Venturi scrubber l. Liquid dispergation via a venturi neck l For very rapid reactions
Spray tower
Venturi scrubber
Gas-liquid reactors l Selection criteria ¡Bubble columns for slow reactions ¡Sckrubbers or spray towers for rapid reactions ¡Packed column or plate column if high reatant conversion is desired ¡
Mass balances
Gas-liquid reactors Mass balances l Plug flow l Liquid phase l Gas phase l av =gas-liquid surface area/reactor volume l L = liquid hold-up
Gas-liquid reactions Mass balances l Complete backmixing l Liquid phase l Gas phase l av =gas-liquid surface area/reactor volume l L = liquid hold-up
Gas-liquid reactors Mass balances l Batch reactor l Liquid phase l Gas phase l av =interfacial area/reactor volume l L = liquid hold-up
Gas-liquid reactors - Gas-liquid film l Fluxes in gas-liquid films ¡Nb. Li Nb. Gi l Two-film theory ¡Chemical reaction and molecular diffusion proceed simultaneously in the liquid film with a thickness of d. L ¡Only molecular diffusion in gas film, thickness d. G l Fick’s law can be used:
Gas-liquid reactors Gas film l Gas film, no reaction l Analytical solution possible l The flux depends on the mass transfer coefficient and concentration difference
Gas-liquid reactors Liquid film l Diffusion and reaction in liquid film: l Boundary conditions:
Gas-liquid reactors Liquid film l Liquid film ¡Equation can be solved analytically for isothermas cases for few cases of linear kinetics; in other case numerical solution should be used
Reaction categories l Physical absorption ¡No reaction in liquid film, no reaction in liquid bulk l Very slow reaction ¡The same reaction rate in liquid film and liquid bulk – no concentration gradients in the liquid film, a pseudohomogeneous system l Slow reaction ¡ Reaction in the liquid film negligible, reactions in the liquid bulk; linear concentration profiles in the liquid film
Reaction categories l Moderates ¡Reaction in liquid film and liquid bulk l Rapid reaction ¡ Chemical reactions in liquid film, no reactions in bulk l Instantaneous reaction ¡Reaction in liquid film; totally diffusion-controlled process
Concentration profiles in liquid film
Enhancement factor l Real flux/flux in the presence of pure physical absorption l EA ³ 1
Gas-liquid reactors - very slow reaction l No concentration gradients in the liquid film l Depends on the role of diffusion resistance in the gas film
Gas-liquid reactors - slow reaction l Diffusion resistance both in gas- and liquid- film retards the adsorption, but the role of reactions is negligible in the liquid film
Gas-liquid reactors - moderate in liquid film l Chemical reactions in liquid film Reaction in liquid film No reaction in gas film l The transport equation should be solved numerically
Moderate in the liquid film l Transport equation can be solved analytically only for some special cases: ¡ isothermal liquid film – zero or first order kinetics Approximative solutions exist for rapid second order kinetics
Moderate… l Zero order kinetics
Moderate… l First order kinetics l Hatta number Ha=ÖM (compare with Thiele modulus)
Rapid reactions l Special case of reactions with finite rate l All gas components totally consumed in the film; bulk concentration is zero, cb. LA=0
Instantaneous reactions l Components react completely in the liquid film l A reaction plane exists l Reaction plane coordinate
Instantaneous reactions l Enhancement factor: l Flux at the interface: l Coordinate of the interface:
Instantaneous reactions l Flux l Only diffusion coeffcients affect ! l For simultaneous reactions can several reaction planes appear in the film
Fluxes in reactor mass balances l Fluxes are inserted in mass balances l For reactants: l For slow and very slow reactions: (no reaction in liquid film)
General approach l We are left with the model for the liquid film:
Solution of mass balances l Numerical strategy: ¡Algebraic equations l. Newton-Raphson method ¡Differential equations, initial value problem (IVP) l. Backward difference- and SI Runge-Kutta-methods ¡Differential equations, BVP lorthogonal collocation or finite differences
Number of equations l N = number of components in the system l N eqs for liquid phase; N eqs for gas phase l N eqs for the liquid film l Energy balances ¡ 1 for gas phase ¡ 1 for liquid phase l 3 N+2 equations in total
Mass transfer coefficients l Flux through the gas film l Partial pressures often used: l Ideal gas law gives the relation:
Gas-liquid equilibria l Definition l For sparingly soluble gases: l Relation becomes l KA from thermodynamics; often Henry’s constant is enough
Simulation example l Chlorination of p-kresol ¡p-cresol + Cl 2 -> monocloro p-kresol + HCl ¡monocloro p-kresol + Cl 2 -> dichloro p-kresol + HCl l CSTR ¡Newton-Raphson-iteration l Liquid film ¡Orthogonal collocation
Chlorination of para-cresol in a CSTR
Fluid-solid reactions l Three main types of reactions: ¡ Reactions between gas and solid ¡ Reactions between liquid and solid ¡ Gas-liquid-solid reactions
Fluid-solid reactions l The size of the solid phase ¡Changes: l. Burning oc charcoal or wood ¡Does not change: loxidation av sulfides, e. g. zinc sulphide --> zinc oxide
Reactors for fluid-solid reactions l Reactor configurations ¡Fluidized bed ¡Moving bed ¡Batch, semibatch and continuous tank reactors (liquid and solid, e. g. CMC production, leaching of minerals)
Processes and reactors
Fluid-solid reaction modelling l Mathematical models used ¡ Porous particle model l Simultaneous chemical reaction and diffusion throughout the particle ¡ Shrinking particle model l Reaction product continuously removed from the surface ¡ Product layer model (shrinking core model) l A porous product layer is formed around the non-reacted core of the solid particle ¡ Grain model l The solid phase consists of smaller non-porous particles (rasberry structure)
Fluid-solid reactions l Solid particles react with gases in such a way that a narrow reaction zone is formed l Shrinking particle model can thus often be used even for porous particles l Grain model most rrealistic but mathematically complicated
Product layer
Product layer Concentration profiles in the product layer
Shrinking particle
Grain model
Fluid-solid reactions l Particle with a porous product layer ¡Gas or liquid film around the product layer ¡Porous product layer ¡The reaction proceeds on the surface of nonreacted solid material l. Gas molecules diffuse through the gas film and through the porous product layer to the surface of fresh, non-reacted material
Fluid-solid reactions l Reaction between A in fluid phase and B in solid phase l R=reaction rate, A=particle surface area l Generated B= Accumulated B
Fluid-solid reactions l Diffusion through the porous product layer (spherical particle) l Solution gives NA=De. A(dc. A/dr):
Fluid-solid reactions l Fick’s law is applied for the diffusion in the product layer gives the particle radius l Surface concentration is obtained from
Fluid-solid reactions l For first-order kinetics an analytical solution is possible l Four cases – rate limiting steps ¡Chemical reaction ¡Diffusion through product layer and fluid film ¡Diffusion through the product layer ¡Diffusion through the fluid film
Fluid-solid reactions l Reaction time (t) and total reaction time (t 0 ) related to the particle radius (r) l Limit cases ¡Chemical reaction controls the process – Thiele modulus is small -> Thiele modulus small ¡Diffusion through product layer and fluid film rate limiting -> Thiele modulus large
Reaktorer med reaktiv fast fas ¡Diffusion through the product layer much slower than diffusion through the fluid -> Bi. AM=¥ ¡Diffusion through fluid film rate limiting -> Bi. AM=0
Fluid-solid reactions l Shrinking particle ¡Phase boundary ¡Fluid film around particles l Product molecules (gas or liquid) disappear directly from the particle surface l Mass balance In via diffusion through the fluid film + generated = 0
Fluid-solid reactions l First order kinetics ¡Surface reaction rate limiting ¡Diffusion through fluid film rate limiting l Arbitrary kinetics ¡A general solution possible, if diffusion through the fluid film is rate limiting
Semibatch reactor l An interesting special case ¡Semibatch reactor l. High throughflow of gas so that the concentrations in the gas phase can be regarded as constant; used e. g. in the investigation of gas-solid kinetics (thermogravimetric equipment) l. Complete backmixing locally l simple realtions between the reaction time and the particle radius obtained
Reaction time and particle radius l Thiele modulus, φ=-νAk. R/De. A and Biot number, Bi. M=k. GAR/De. A l Special cases – large Thiele modulus φ; l control by product layer and fluid film
Fluid-solid reactions l Product layer model l Large Thiele modulus, φ=-νAk. R/De. A and large Bi - control by product layer l Large Thiele modulus, φ=-νAk. R/De. A and small Bi - control by film
Fluid-solid reactions l Product layer model l Small Thiele modulus, φ=-νAk. R/De. A and large Bi - control by chemical reaction
Fluid-solid reactions l Shrinking particle model l Small Bi - control by film diffusion l Large Bi - control by chemical reaction
Packed bed ¡Packed bed – operation principle l. Gas or liquid flows through a stagnant bed of particles, e. g. combustion processes or ion exchangers l. Plug flow often a sufficient description for the flow pattern l. Radial and axial dispersion effects neglected
Simulation of a packed bed
Mechanistic modelling of kinetics and mass transfer for a solid-liquid system: Leaching of zinc with ferric iron Tapio Salmi, Henrik Grénman, Heidi Bernas, Johan Wärnå, Dmitry Yu. Murzin Laboratory of Industrial Chemistry and Reaction Engineering, Process Chemistry Centre, Åbo Akademi, FI-20500 Turku/Åbo, Finland
Reaction system Zn. S(s) + Fe 2(SO 4)3 ↔ S(s) + 2 Fe. SO 4 + Zn. SO 4 SEM
Experimental system l Isothermal batch reactor l Turbine impeller l Ultrasound input l SIA – analysis of Fe 3+ Experimental data of Bernas (Markus) & Grénman Markus et al, Hydrometallurgy 73 (2004) 269 -282, Grénman et al, Chemical Engineering and Processing 46 (2007) 862 -869
Multi-transducer ultradound reactor 6 transducers Generator (0 -600 W) 20 k. Hz Reactor pot inserted A time-variable power input
Experimental results - Stirring speed T = 85°C , Sphalerite : Fe 3+ = 1. 1: 1 The effect of the stirring speed on the leaching kinetics.
Experimental results T = 85°C, C 0 Fe(III) = 0. 2 mol/L The effect of the zinc sulphide concentration on the leaching kinetics.
Experimental results T = 95°C, Sphalerite : Fe 3+ = 1. 1: 1 The effect of the ferric ion concentration on the leaching kinetics.
Experimental results T = 95°C, Sphalerite : Fe 3+ = 1. 1: 1 The effect of sulphuric acid on the leaching kinetics.
Experimental results - Temperature effect Sphalerite : Fe 3+ = 1. 1: 1 The effect of temperature on the leaching kinetics.
Experimental results - Ultrasound effect T = 85°C Stirring rate 350 rpm The effect of ultrasound on the leaching kinetics.
Reaction mechanism and rate equations l Surface reaction l Stepwise process ( first reacts one Fe 3+, then the second one! ) l Rough particles
Three-step surface reaction mechanism Zn. S(s) + Fe 3+ ↔ I 1 (I) I 1+ Fe 3+ ↔ I 2 (II) I 2 ↔ S(s) + 2 Fe 2+ + Zn 2+ (III) Zn. S(s) + 2 Fe 3+ ↔ S(s) + 2 Fe 2+ + Zn 2+ rates of steps (I-III) c. I 1, c. I 2 and c. I 3 = surface concentrations of the intermediates.
Development of rate equations Pseudo-steady state hypothesis Rate equation Back-substitution of a 1…. a-3 gives D = k-1 k-2+k-1 k+3+k+2 k+3 c. Fe. III
Rate equations Final form where β = (k-1 k-2+k-1 k+3)/(k+2 k+3) An alternative rate equation NOT VALID FOR THIS CASE!
Area & Shape factor Development of a general approach The surface area (A) can be expressed with a generalized equation n = amount of solid n 0= initial amount of solid Shape factor (a=1/x)
Area & Shape factor Reaction order can vary between 0 and 1!
Mass balance for batch reactor , where γ=(k 1σM / x 0 Zn. S)
Parameter estimation Nonlinear regression applied on intrinsic kinetic data
Intrinsic kinetics - Model fit T = 85°C The effect of the ratio sphalerite : Fe. III on the kinetics
Intrinsic kinetics - Model fit Temperature effect on the kinetics.
Mass transfer limitations in Batch reactor where ri=νir The mass transfer term (NLis) is described by Fick’s law β’=β/ci, γ’=(-νik 1 ci/k. Li), y=ci*/ci The solution becomes
Liquid-solid mass transfer coefficient General correlation where z=c. Zn. S/c 0 Zn. S. The index (i) refers to Fe(III) and Fe(II)
Correlations in rate equation b’=b(ε d 04/ υ 3)1/6(υ/Di)1/3 IF b’z 2/9 >> 2 under stirring, γ’=-νFe. III x 0 Zn. S γ c. Fe. IIIz 1/9/(σMb’’Fe. III) The surface concentration: The rate:
Determination of mass transfer parameter (ω)
Modelling of kinetics and mass transfer External mass transfer limitations – modelling of individual mass transfer parameters at different agitation rates.
Modelling of kinetics and mass transfer External mass transfer limitations – modelling of individual mass transfer parameters at different ultrasound inputs.
Mass transfer parameter Normal agitation Ultrasound
The real impact of mass transfer limitations The difference in the model based surface concentrations and measured bulk concentrations of Fe 3+ at different stirring rates.
The real impact of mass transfer limitations The difference in the model based surface concentrations and measured bulk concentrations of Fe 3+ at different ultrasound inputs.
Conclusions l A new kinetic model was proposed l A general treatment of smooth, rough and porous surfaces was developed l The theory of mass transfer was implemented in the model l Model parameters were estimated l The model works
Modelling and simulation of porous, reactive particles in liquids: delignification of wood Tapio Salmi, Johan Wärnå, J. -P. Mikkola, Mats Rönnholm Åbo Akademi Process Chemistry Centre, Laboratory of Industrial Chemistry FIN-20500 Turku / Åbo Finland, Johan. Warna@abo. fi
Typical view of Finland 338000 km 2 of which 70% forest
Papermaking l Wood chips l This is where paper making begins. l A typical wood chip measures 40 x 25 x 10 mm.
Wood l Each chip comprises water, cellulose wood fibres and the binding agent lignin. .
Pulp l To make paper, we need to first make pulp, which is the process of breaking the wood structure down into individual fibers Chips Digester
Reactions The reactions in chemical pulping are numerous. Typical pulping chemicals are Na. OH and Na. HS Part of Lignin molecule cellulose Overall process: Lignin+Cellulose+Carbohydrates+Xylanes+OH+HS -> Dissolved components
Kinetic modelling of wood delignification l Purdue model (Smith et. al. (1974) Christensen et al. 1983), 5 pseudocomponents l Gustafson et al. 1983, 2 wood components Lignin and Carbohydrate, 3 stages l Andersson 2003, 15 pseudocomponents l Very few models available!
Wood chip structure l Wood material is built up of fibres l We can expect different diffusion rates in the fibre direction and in the opposite direction to the fibres.
Existing models l The existing models for delignification of wood consider a 1 dimensional case with equal diffusion rates in all directions l Is a 2 - or 3 -dimensional model needed ?
Characteristics of our model l Time dependent dynamic model l Complex reaction network included l Mass transfer via diffusion in different directions l Structural changes of the wood chip included l All wood chips of equal size l Perfectly mixed batch reactor assumed
Mathematical model, volume element l 3 D –model for a wood chip
Mass balance for a wood chip Porosity
Boundary conditions The concentrations outside the wood chip are locally known ci=c. Li at the centre of the chip (symmetry) dci/dx=dci/dy=dci/dz=0
Reactor model Batch reactor model, ideal flow Fluxes from wood chip
Structural changes of the wood chip Generally one can state that the porosity of the chip increases during the process, since lignin and hemicelluloses are dissolved Change of porosity as a function of the lignin conversion
Kinetic models Andersson model, 12 wood pseudocomponents Purdue model (Christensen et al), 5 wood pseudocomponents Gustafsson model, 2 wood components, 3 stages Initial stage, >22% Lignin, Bulk stage , 22% > Lignin > 2% Residual stage < 2% Lignin
Diffusion models Mc. Kibbins Wilke-Chang Nernst-Haskel (infinite dillution)
Kappa number The progress of delignification is by pulp professionals described by the Kappa number L = Lignin on wood, CH = Carbohydrates on wood
Numerical approach l Discretizing the partial differential equations (PDEs) with respect to the spatial coordinates (x, y, z). l Central finite difference formulae were used to approximate the spatial derivatives l Thus the PDEs were transformed to ordinary differential equations (ODEs) with respect to the reaction time with the use of the powerful finite difference method. l The created ODEs were solved with the backward difference method with the software LSODES
Simulation results, profiles inside wood chip Kappa value centre Lignin content T=170 ºC C 0, Na. OH=0. 5 mol/l centre Porosity
The impact of 2 -D model 19 19 18. 8 18. 6 18. 8 y 18. 6 18. 2 Lignin (w%) 18. 4 18 17. 6 x 18. 4 18. 2 17. 4 18 17. 2 17 0 surface 2 4 6 x-node 8 10 12 centre 17. 8 0 surface 2 4 6 y-node Red line, different diffusion rates in x and y directions Blue line, same diffusion rates in x and y direction (Andersson kinetic model) 8 10 12 centre
Content of lignin on wood as a function of reaction time Lignin concentration (w-%) in wood chip as a function of reaction time (min) with Andersson kinetic model (left) and Purdue kinetic model (right).
Simulation software l 2 -D model for a wood chip in a batch reactor l Different kinetic and diffusion models available l Structural change model included (porosity) l Dynamic model ¡ all results can be presented as a function of reaction time l Temperature and alkali concentrationprofiles can be programmed as a function of reaction time
Conclusions l A general dynamic model and software for the description of wood delignification l Solved numerically for example cases, which concerned delignification of wood chips in perfectly backmixed batch reactors. l Structural changes and anisotropies of wood chips are included in the model. l The software utilizes standard stiff ODE solvers combined with a discretization algorithm for parabolic partial differential equations. l Example simulations indicated that the selected approach is fruitful, and the software can be extended to continuous delignification processes with more complicated flow patterns.
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