Chip scale modelling of the kraft pulping process by considering the heterogeneous nature of the lignocellulosic feedstock

This article focuses on a multiscale modelling approach to describe the delignification of softwood during the kraft pulping process. A framework for modelling the lignocellulosic feedstock on a fibre scale which considered the fundamental chemical components of wood as a distributed variable is re-assessed and extended to include chip-level phenomena such as diffusion limitations and initial component distributions within a soft-wood chip mixture. A new description of the wood chip is presented using a finite volume discretisation along one spatial dimension by simultaneously considering the anisotropic structural differences of the wood. Additionally, based on literature data, a distinction between the softwood chips' early-and latewood regions with their differences in densities and chemical composition is suggested. The presented model framework uses published sub-models for kinetics, diffusion etc. The validation and estimation of the remaining parameters are conducted from experimental data that quantifies the kappa number distribution of individual softwood fibres after kraft pulping. The investigation hypothesises a Gaussian distribution for the initial chemical component distribution within wood chips from a well-defined region. In contrast, a Log-normal distribution is used to describe the initial chemical distribution within a softwood chip mixture. The established sub-models for the kraft pulping process's kinetics and mass transfer phenomena could not predict the experimental data satisfactorily. However, modifying the sub-models by including a change in lignin reactivity and a temperature dependency of the lignin reactivity decline during the delignification progress could predict the essence of the observed experimental kappa number distribution.


Introduction
The transformation from the currently dominating petroleum-based economy toward a more sustainable economy based on renewable resources poses one of the greatest concerns of the 21st century.With researchers tackling the challenge from all frontiers, the collaboration between experimental observations and theoretical modelling of processes based on renewable feedstocks can offer valuable tools for process insights.
In recent years awareness has arisen regarding the highly heterogeneous and variable nature of lignocellulosic feedstocks (Yan et al., 2020;Oyedeji et al., 2020;Kenney et al., 2013;Balakshin et al., 2020).Due to the biomass feedstock's complex hierarchical structure, classical chemical engineering model approaches must be extended to multiscale modelling approaches tailored to the feedstock in question (Ciesielski et al., 2020).
Properties such as the distributed chemical composition and anisotropic structural features of different regions within the feedstock need to be addressed to portray biorefinery processes with their nuances accurately.In light of the biomass raw material knowledge, new and established biorefinery processes must be reviewed and updated regarding their theoretical modelling approaches.
The pulp and paper industry represents one major stakeholder in the bioeconomy, with the kraft pulping process as its predominant chemical pulping technology.The process aims to delignify wood fibres into pulp while retaining the carbohydrates using an inorganic chemical mixture of sodium hydroxide and sodium sulfide called white liquor.Traditionally, this chemical pulping process produces cellulose derivatives and paper as its products.However, the process offers the possibility of an integrated multi-product biorefinery network with the further valorisation of lignin (Mongkhonsiri et al., 2018).The broad literature knowledge on the kraft pulping process, ranging from laboratory-scale investigation to the large-scale digester, makes it an interesting benchmark process with many shared characteristics with other biorefinery processes.
The utilised feedstock comprises different wood species, roughly divided into soft-and hardwoods.The hierarchal structure of the wood includes several layers on a fibre scale, namely, the secondary wall, primary wall, and middle lamella, with the main fundamental chemical components being lignin, hemicellulose and cellulose.Moreover, the wood can be distinguished into earlywood (low-density region), latewood (high-density region), sapwood (younger region) and heartwood (mature region) (Sjöström, 1993).Each region has its characteristics regarding chemical composition and physical properties, which differ in average values and are distributed within the same region (Bertaud and Holmbom, 2004;Scott et al., 1969;Saka et al., 1982;Wood and Goring, 1971;Agarwal, 2006).
Classical models for the kraft pulping process are limited to viewing the feedstock as a homogenous reactant with a distinction in its fundamental chemical components.The main focus of these models lies in the description of the kinetics and mass transfer phenomena on average (Agarwal and Gustafson, 1997;Gustafson et al., 1983;Smith, 1974;Wisnewski et al., 1997;Christensen, 1982).Much work on the molecular level has been done regarding lignin and carbohydrate kinetics.Nonetheless, kinetics are still the subject of research since none of the models captures the whole picture of the complex chemical interactions within the wood.The proposed kinetic model structures include a three-stage delignification approach, parallel reactions of wood components with different reactivities, a continuous distributed reactivity model and detailed kinetics for the distinct solid components (Gustafson et al., 1983;Christensen, 1982;Andersson et al., 2003;Bogren et al., 2008;Nieminen et al., 2014aNieminen et al., , 2014b;;Fearon et al., 2020aFearon et al., , 2020b)).
The effect of mass transfer on a wood chip level has also been investigated.Here the models cover the mass transfer within the wood chip from one-dimensional approaches of a critical dimension with average concentration profiles, full three-dimensional mass transfer models, to models which include phenomena such as the Donnan effect resulting in exchange equilibria (Agarwal and Gustafson, 1997;Grénman et al., 2010;Kuitunen et al., 2013;Simão et al., 2011Simão et al., , 2008;;McKibbins, 1960).Some experimental data suggest that the mass transfer phenomena of dissolved lignin also depends on the macromolecular size and different fractions of the dissolved lignin are found within, e.g. the entrapped liquor phase of the wood chip and the free liquor phase in the digester (Brännvall and Rönnols, 2021;Ghaffari et al., 2022).Ultimately, these models are combined into a full-scale model on a digester level to predict the cooking process during kraft pulping (Wisnewski et al., 1997;Michelsen, 1995;Fernandes and Castro, 2000;Bhartiya and Doyle, 2004;Pourian and Dahlquist, 2011;Silva and Biscaia, 2003).
Recently, research efforts have been put into the multiscale modelling of the kraft pulping process, considering the subtleties of the physical/chemical phenomena (Choi and Kwon, 2020, 2019a, 2019b;Il Kwon et al., 2021;Son et al., 2020) and the inclusion of the distribution character for fundamental chemical components (Bijok et al., 2022) during chemical pulping.
This article aims to review and discusses published model approaches focusing on the distributed character of the fundamental chemical components within a softwood chip mixture.A new discretisation method is presented to describe the three-dimensional wood chip in a one-dimensional mathematical way.Additionally, to further entangle the wood feedstock properties, a distinction between earlyand latewood regions is suggested to consider differences between both regions.Ultimately, the model is tested against experimental data for the delignification of individual softwood fibres in terms of kappa number distribution, a measure of the residual lignin content in the pulp (Ming, 2007).

Model hypotheses and equations
The model framework for a batch reactor assumes a threephase system, the solid wood components, the entrapped liquor phase prevailing in the porous structure of the wood chip (lumen and cell wall), and the free liquor phases (Fig. 1).The solid and entrapped liquor phase represents the wood chip which is in mass transfer with the free liquor phase presented in the digester.The chip geometry produced by industrial chippers have typically a parallelepipedal shape, cut in the longitudinal tree growth direction.The model approximates the parallelepipedal shape by a rectangular wood chip discretised into k arbitrary volume compartments, as illustrated in Fig. 2. Instead of describing the mass transfer as mathematically three-dimensional (Grénman et al., 2010;Liu et al., 2014) or using an average concentration profile over one dimension (Simão et al., 2008), the mass transfer is described by a onedimensional finite volume discretisation along several volume compartments.Each wood chip direction contributes to the mass transfer according to its defined directional properties.The respective volume compartment is assumed to be fully mixed and evaluated at the discretisation grid point centre.
Depending on a discretisation parameter, the wood chip can be discretised in several ways, including equally sized distances centres of the volume compartments or equal volumes.The supporting material provides details on the discretisation schema (supporting material A).
Zooming into one of the volume compartments, each compartment is a build-up of an array of fibres.The model distinguishes between earlywood (low-density region) and latewood (high-density region) fibres with different initial fundamental chemical component concentration distributions (see Fig. 3).No distinction between the different fibre layers has been made yet.The fibres have an average concentration, including the secondary wall, compound middle lamella, and cell corner portion.
The model assumes that the fibre volume stays constant throughout the cooking process, and the wood chips are cut homogeneously without having fissures and cracks resulting from industrial chipping processes.Although this assumption does not fully reflect reality, it is used to simplify the model, concentrating on the wood chip's distributed character and regions rather than the mechanical/physical properties.
The solid phase mass balance is given for the fundamental chemical components of the individual early-and latewood fibres.The chemical components within each fibre are treated as distributed variables with different initial concentrations: (2) A more comprehensive description of treating the solid components of the wood fibres in a distributed manner can be found in an earlier published article (Bijok et al., 2022).
The mass balance for the entrapped liquor phase is established for each volume compartment k of the wood chip as: (3) where N A j k direction k , , direction refers to the mass transfer in or out of the volume compartment surface of the respective direction and R e k , j is the reaction rate of the liquid component e j .The anisotropic structural differences of the wood chip are considered by the effective capillary cross-section area ECCSA direction k , of the respective volume compartment.The reaction rates of the entrapped liquid components are related to the reaction rates of the solid components through a stoichiometric coefficient matrix: Since the fibre volume is assumed to be constant throughout the cooking, a bulk volume flow from the free liquor to the void created by the reacted solids is added (similar to the approach used in Wisnewski et al (Wisnewski et al., 1997).): the ECCSA direction k , from compartment k of the respective di- rection is used.For the outlet flux at k 1 2 , the structural properties of compartment k 1 is used.Mass transfer be- tween the volume compartments is calculated according to Fick's laws of diffusion: Boundary conditions for the mass transfer are as follows.For the mass transfer at the most outer shell (interface between free and entrapped liquor), the boundary condition is: where K ol direction , refers to an overall mass transfer coefficient, including the external film and internal diffusion resistance between the first grid point (centre of the outmost volume compartment) and the free liquor phase.At the inner volume compartment, the flux vanished due to the symmetry of the wood chip (see Figure 5 and Figure 6): The free liquor phase mass balance is presented as follows: Where V f is the free liquor volume of the digester and N A , , ECCSA j direction direction direction describes the mass transfer between the free liquor and the wood chip's entrapped liquor phase, considering the anisotropic properties at the chip surfaces.

Case study: delignification of douglas fir softwood chips during kraft pulping
This chapter introduces the constitutive model equation for the softwood kraft pulping process.Afterwards, the presented model is tested against experimental findings from the literature regarding the non-uniform delignification expressed in the kappa number distribution of individual fibres (Ming, 2007).
The kinetics are described by the Purdue model (Wisnewski et al., 1997) with the proposed extension of a yield-dependent effectiveness factor for the lignin fractions to consider changes in lignin reactivity during delignification (Bijok et al., 2022).The reaction rate for the solid component s i of the concentration class n within the volume compart- ment k is given by: are the effective alkali and hydrosulfide concentrations in the entrapped liquor phase.The yield-dependent effectiveness factor is calculated as: With a species-dependent reactivity part e f s i 0, and a yielddependent part, including the parameter to describe the decline in reactivity for a solid component.In this article, only the lignin will be assumed to change in reactivity.The solid carbohydrate components react according to the original Purdue model with an effectiveness factor equal to one.
The Purdue kinetics consists of five solid components.The total lignin is divided into 20% high reactive and 80% low reactive lignin.The carbohydrates are divided into cellulose, galactoglucomannan and xylan.To account for early-and latewood differences, these solid components remain the same with an extension that assumes a further division of the kinetics into high reactive lignin earlywood (s 1 ), low reactive lignin earlywood (s 2 ), cellulose earlywood (s 3 ), galactoglucomannan earlywood (s 4 ), xylan earlywood (s 5 ), high reactive lignin latewood (s 6 ), low reactive lignin latewood (s 7 ), cellulose latewood (s 8 ), galactoglucomannan latewood (s 9 ) and xylan latewood (s 10 ), assuming that the kinetics for the early-and latewood components follow the same reaction rates.At this point, the only differences between early-and latewood are reflected in the initial fundamental chemical component distribution and densities.The Arrhenius kinetic coefficients are calculated as: Where R is the ideal gas constant, T is the temperature, A s 1, i , A s 2, i are the pre-exponential factors and E s 1, i , E s 2, i are the activation energies of the solid components.
The reaction rates for the four liquid components, active effective alkali (e 1 ), active hydrosulfide (e 2 ), dissolved lignin (e 3 ) and dissolved carbohydrates (e 4 ), are given through the stoichiometric coefficient matrix: The diffusion within the porous structure of the wood chip is described by an Arrhenius-type equation presented by McKibbins (McKibbins, 1960): Here T is the temperature, D direction k 0, , are pre-exponential constants and E diffusion direction , are activation energies of the respective wood chip direction.The diffusion model presented by McKibbins was developed for the active alkali diffusion within cooked Spruce wood chips.
To account for the experimental data set used in this article with a different softwood species, a multiplier c direction 1, is added to adjust the diffusion coefficient correlation.
The model assumes equal diffusivities for all liquid components.This assumption simplifies the diffusion of the dissolved solid components to a great extent.However, since the Purdue kinetics are considered irreversible, possible precipitation of the dissolved solid components on the fibres and/or condensation reactions are neglected.
The overall mass transfer from the free liquor to the outmost volume compartment of the wood chip is calculated according to the film theory as follows: Here direction is the distance between the surface of the wood chip and the grid point of the volume compartment in each direction, respectively.Eq. 16 implies the assumption that external mass transfer is neglectable in a well-mixed digester compared to internal diffusion limitation.
The porosity of compartment k is calculated as an average porosity derived from the solid concentrations and changes with the extent of the reaction according to: Here wood is the cell wall density of the solid wood material and the concentrations of the solid components A linear correlation presented by Bäckström calculates the changes in ECCSA developed for Pine softwood (cited by (Sixta, 2006)): Here m direction are empirical parameters, ECCSA direction ,k 0 are the initial effective capillary cross-section areas and yield total k , is the total solid component yield of the respective volume compartment.
The total yield for each volume compartment is calculated as: Additionally, yields for the individual solid component are calculated according to: The kappa number distribution is calculated with the following correlations (Wisnewski et al., 1997): , , (22) The model distinguishes between the early-and latewood fibres regarding the kappa number distribution.The average kappa number is calculated as a weighted average overall dimension of the model (earlywood/latewood percentage, concentration class distribution and fraction of volume compartment occupying the total chip volume).
For the fundamental chemical component distribution within the wood chip, two different distributions are tested, the Gaussian distribution and a Log-normal distribution: Where µ is the mean and is the standard deviation of the respective distribution.Unknown model parameters are deducted from Ming's kappa number distribution dataset (Ming, 2007).
The estimated parameters are kept to a minimum with some physical/chemical interpretation.Only the standard deviations of the early-and latewood lignin distribution are estimated along with one kinetic parameter and diffusion coefficient multiplier.The carbohydrate concentration distribution follows directly from the lignin distribution, assuming that the overall wood density is constant.The supporting material (supporting material B) provides the correlation between lignin and carbohydrate distribution used in this model.
The density for Douglas fir's early-and latewood are chosen to be 316 kg/m3 and 753 kg/m3, respectively (Vargas-Hernandez and Adams, 1991).There is limited information on the exact fraction of the chemical components within each region.Bertaud and Holmbom have measured the chemical composition of early-and latewood in Norway Spruce, with a difference in lignin content ranging between two and three per cent and higher galactoglucomannan and cellulose content in the latewood compared to earlywood (Bertaud and Holmbom, 2004).Zhang et al. (2014) have measured differences in early-and latewood lignin content for Douglas fir ranging from five to six per cent.Taking these two studies and the average chemical composition of Douglas fir wood (Sixta, 2006) as a basis, the model uses the fundamental chemical components fraction listed in.
Table 1.The starting point is Douglas fir's average fundamental chemical component fraction.From there, the lignin fraction of the earlywood is increased by two per cent, and the lignin fraction of the latewood is decreased by two per cent, for a total lignin difference of four per cent.The two per cent of the solid component density is redistributed to one per cent cellulose and galactoglucomannan fraction, respectively.The reactions and distributions of the extractives and polysaccharides other than hemicelluloses are neglected in the model.

Results and discussion
The current model revisits an earlier published article, which has considered the fundamental chemical component as distributed variables without the distinction between earlyand latewood regions (Bijok et al., 2022).The validation was done with experimental data for the kappa number distribution from well-defined earlywood regions under conditions which resulted in the most uniform delignification (Ming, 2007).However, the model has used average values for the properties of Douglas fir softwood.
In light of the further distinction between the and latewood regions, the previously published model has re-evaluated and adjusted according to the model proposed in this article.Furthermore, additional experimental data for typical kraft pulping conditions are added to extend the model approach to a chip-level scale.
The uniform cooking experiments by Ming (Ming, 2007), using miniature chips and mild pulping conditions (summary of the experiments in the supporting material C), have been re-assessed with a distinction between early-and latewood and their differences in densities and average chemical component contents.In addition to the earlywood experiments (EW01-HA02, Table 1), two experimental points were added to include the latewood region of the wood (LW and EW/LW.Here LW refers to pulping experiments with latewood miniature chips and EW/LW having a mixture of earlywood (70%) and latewood (30%) chips.Assuming that the pulping experiments in such a setup are not influenced by diffusion limitation, only one volume compartment is considered to model the wood chip during the uniform cooking experiments (Bijok et al., 2022).
The unknown parameters for the lignin reactivity decline uniform , the standard deviations of the earlywood 0,EW,lignin and latewood 0,LW,lignin initial distributions are estimated by solving the model with the "lsqnonlin" function provided in MATLAB.A least-square objective function criterion for the kappa number average and standard deviation of the experimental data were used: The parameter optimisation is conducted with a single set of parameters (Table 2) for all experiments (Table 3), using one volume compartment and the suggested two-class concentrations discretisation based on the quadrature method of moments (Bijok et al., 2022;Gordon, 1968;McGraw, 1997).The estimated parameters are presented in Table 2.
The new parameter uniform for the reactivity decline of lignin is similar to the earlier reported parameter.Moreover, the coefficient of variation for the lignin distribution of the earlier reported initial distribution is similar to the coefficient of variation for both the early-and latewood distribution presented in this article (Bijok et al., 2022).
The comparison between the model prediction for the average kappa number and the standard deviation of the fibre distribution is shown in Table 3 and Fig. 7.The model predicts the data well for all the cooking experiments with the new parameters.The suggested initial lignin distribution based on the model and parameter optimisation is shown in Fig. 8.
The good agreement between the model prediction and experimental data supports the underlying estimation of the initial average lignin fraction within the early-and latewood regions.The initial average lignin fraction for both regions will be used for all following simulations.
In the next step, the investigation discusses two main points, the distribution of the fundamental chemical components within a wood chip mixture and kinetic-related subtleties of the feedstock.The model is tested for the delignification of different wood chip sizes under typical kraft pulping conditions.Details of the experiments can be found in Ming´s thesis (Ming, 2007), and a summary is provided in the supporting material of this article (supporting material D).
Compared to the uniform delignification experiments, where the chip size and pulping conditions were carefully selected, additional sources of heterogeneities are introduced by pulping typical-sized wood chips.One wellknown factor for non-uniform delignification during kraft pulping is the diffusion limitation within thick chips.
Experimental data on the non-uniformity of kraft pulps from the '90 s have already investigated this issue (Gullichsen et al., 1993).However, these experiments considered only the average kappa number across different wood chip thicknesses.The experimental investigations by Ming (Ming, 2007) have a higher resolution on a fibre level, revealing that diffusion limitation is not the only source of non-uniformity delignification since the fibres are suspected of having a distributed character for the fundamental chemical components (Agarwal, 2006;Bijok et al., 2022;Ming, 2007).
The origin of conventional cut wood chips should be kept in mind.Wood chips are cut from different regions of the tree logs where the chip's properties depend on the cutting location.Characteristics of the wood quality, such as the density, vary along a tree's length and diameter (Megraw, 1985;Larson, 1969).Additionally, these wood chips can contain reaction wood which differs in the lignin concentration compared to normal wood (Sjöström, 1993).With this in mind, the Gaussian distribution of the fundamental chemical components is assumed to be limited to a well-defined region within the wood log (e.g.within an annual ring at a specific tree height).It will not reflect the overall heterogeneity of a wood chip mixture produced during chipping.
Therefore, the Gaussian distribution is replaced by a Lognormal distribution to consider the distribution of the chemical components within a wood chip mixture produced from different chipping locations of the tree logs, assuming the same average values as in Table 1.The standard deviations of the early-and latewood region ( Log EW lignin 0, , Log LW lignin 0, ) are unknown parameters to be estimated.The percentages of early-and latewood within the wood chips are also important characteristics since the resulting kappa number of earlywood fibres differ from the latewood fibres when pulped under the same conditions (compare LW and EW04 from the uniform pulping experiments).There is limited information on the earlywood to latewood ratio within chipped Douglas fir wood.In a report by Smith (Smith, 1955), the percentage of wide-ringed Douglas fir wood from 16 butt logs, a total of 96 annual rings, has been sampled and analysed regarding their latewood percentage.The percentage ranges from 16% to 60%, with an average of 33%.Acknowledging that the percentage varies substantially and depends on various factors such as rainfall, temperature, growth location, etc.The average values of the latewood percentage are used in the model as a representative estimation.Hence, the model uses an average percentage of early-and latewood within the wood chip mixture, with 33% latewood and 67% earlywood, respectively.The higher resolution of the experimental data with some modifications of the established kraft pulping models reveals nuance during the delignification process.As argued in a previous article, the original Purdue model's kinetics cannot predict the experimental results for the uniform pulped wood chips (Bijok et al., 2022).Therefore, an addition has been proposed to account for the decline in lignin reactivity with progressing delignification, resulting from changes in the lignin structure on a chemical structure level (see Eq. 11).
The parameter uniform was optimised for the temperature of 130 • C used in the uniform pulping experiments.However, preliminary simulations of the experiments with the typically pulping conditions and wood chip sizes indicated that the decline in lignin reactivity must be a function of the temperature.Qualitatively the decline in lignin reactivity needs to decrease with increasing temperature to predict the experimental data with the model.
Similar observations have been reported in previous experimental research, where the authors suggested that the residual lignin decreases by increasing the temperature (Kleinert, 1966;Axegard and Wiken, 1983).To account for the temperature dependency, the decline parameter of the yield-dependent effectiveness factor is extended to an Arrhenius-like equation: Where describes the temperature orthogonalised around the temperature T 0 of 130 • C to be consistent with the uni- form pulped wood chips parameter.Hence, the pre-exponential constants equals the value of uniform given in  ), one kinetic parameter E reactivity and one diffusion parameter c 1 .The optimisation is conducted with a single set of parameters (Table 4) for all typical pulping experimental points provided by the Ming thesis (Table 5) (Ming, 2007).Test simulations suggested discretising the wood chip into twelve-volume compartments with equal distances between grid points and three concentration classes based on the quadrature method of moments (Gordon, 1968;McGraw, 1997).
The model is solved with the "lsqnonlin" function provided in MATLAB.A least-square objective function for the kappa number average, standard deviation and skewness of the experimental distribution is used (Eq.25, Eq. 26 and Eq.29).The skewness was included in the optimisation to consider the shape of the distribution.The estimated parameters are presented in Table 4 and the lignin distribution for the early-and latewood concentrations proposed by the model's optimisation is shown in Fig. 9.
The comparison between the experimental data for the typical pulped wood chips and the model prediction is shown in Fig. 10 and Table 5.Given that the feedstock is subject to a certain degree of randomness due to its origin and the experimental data may underlie scattering, the model reasonably predicts the average kappa number and the standard deviation of the measured distribution, with some scattering around the experimental data points.
Figs. 11-13 compare the shape of the kappa number distribution for the three pulping temperatures and different chip sizes.The model prediction and experimental data points are sorted in predefine kappa number classes and normalised by the class width for better comparison.The chosen assumptions and sub-models capture the kappa number distributions semi-quantitively.
As seen in Fig. 10, the model-predicted average kappa number and standard deviation agree reasonably well with the experimental data.However, the prediction for the N10 and W10 chips start to deviate from the experimental noticeable.Additionally, the model does not capture the skewness of the kappa number distributions well, with the trend of decreasing accuracy for larger chips pulped at higher temperatures (see Table 5 and Fig. 14).This observation cannot be explained simply by the scattering of experimental data or the feedstock's inherent randomness.Mattsson et al. (2017) argue that the rate governing step during delignification is the solubility and/or mass transport in the cell wall of lignin.Related experimental observations regarding lignin mass transfer have been reported by other researchers (Brännvall and Rönnols, 2021;Ghaffari et al., 2022).Additionally, experimental work on kraft pulped fibre surface lignin show a difference in the lignin concentration in the fibre bulk and surface of pulped and washed fibres (Li, 2003;Li and Reeve, 2002).Considering these findings, it can be hypothesised that the mass transfer of lignin and possible condensation reaction and/or precipitation phenomena can impact the kappa number distribution.Further research on a molecular reaction level and the mass transfer phenomena on a cell wall level is needed to include condensation reactions and/or precipitation phenomena.As for now, the model considers two different lignin fractions (high and low reactive lignin), likely resulting from topochemical delignification differences between the secondary wall and compound middle lamella lignin (Saka et al., 1982;Goring and Whiting, 1981) in a lumped kinetic approach, neglecting the divers internal structures of the macromolecule.A kinetic model approach which describes the lignin on an interunit/inter bonding level with its heterogeneous variety, instead of a lumped kinetic approach, would portray the delignification process more accurately.Moreover, irregular industrial chipping processes can result in uneven mass transfer within the wood chip, resulting in an additional source of non-uniform delignification that hasn't been included in the model.A prior and subsequent thorough feedstock characterisation can shed more light on the raised phenomena.
Nonetheless, with the given resolution for modelling the kinetics of the chemical components, the diffusion within the wood chips and the chosen feedstock parameters, the model captures the essence of the observed kappa number distribution for typically pulped wood chips.

Conclusions
This article proposed an approach for modelling lignocellulosic feedstocks that continues to build on an earlier published model, which has considered the fundamental chemical components as distributed variables.The kraft pulping process is used as a case study to illustrate the considerations regarding biomass feedstock modelling.The model was tested and validated with experimental data from the literature for the delignification of Douglas fir softwood under typical kraft pulping conditions in terms of kappa number distribution.
A new description of the wood chip is proposed.The wood chip was discretised along one dimension into finite volume compartments with mass transfer contributions from all three spatial directions, according to their structural properties.
The investigation in this article highlighted some further directions in modelling biomass feedstocks.Based on published experimental literature, the current model extended the view of the biomass feedstock by distinguishing the wood chips into early-and latewood regions with differences in their distribution and average values of the fundamental chemical component concentrations.Published kraft pulping sub-models for kinetics, diffusion etc., are incorporated into the model framework.The unmodified sub-models were not able to predict the experimental data satisfactorily.Therefore, a set of assumptions and adjustments based on experimental observations have been discussed and implemented, which allowed the model to predict the experimental kappa number distributions semi-quantitative.It can be concluded that the published sub-models for the kraft pulping process experience limitations when higher-resolution experimental data need to be predicted.It is suggested that further modelling efforts should concentrate on transforming the current lumped kinetic approach for the different chemical components into a more detailed description, e.g. an inter-unit/bonding level for the lignin component.

iii
Non-reactive fraction of solid component i [-] OHL Stoichiometric coefficients for the consumption of effective alkali [kg OH/kg lignin] HSL Stoichiometric coefficients for the consumption of hydrosulfide [kg HS/kg lignin] OHC Stoichiometric coefficients for the consumption of effective alkali [kg OH/kg carbohydrate] direction Distance between discretisation gird points of the respective direction [m] Concentration of solid component i in volume compartment k of concentration class n [kg/m 3] Parameter to adjust diffusion corelation [-] D direction Diffusion coefficient of the liquid component in the respective direction [m 2 /min] Activation energy for the reaction of solid component i [kJ/(mol K)] E diffusion direction , Activation energy for the liquid component [cal/mol] E reactivity Activation energy for the power of the yield dependent effectiveness factor correlation [kJ/(mol K)] ECCSA yz,k , ECCSA xz,k , ECCSA xy,k Effective capillary cross-section area of volume compartment k in the respective direction [-] e f s i k , Effectiveness factor of solid component i in volume compartment k [-] e f s i 0, Species-dependent reactivity part of the effectiveness factor for the solid component i [-] K ol j , direction Overall mass transfer coefficient of liquid component j in the respective direction [m/min] Flux of liquid component j in the respective direction [kg/(m 2 min)] n fibers Number of fibres within the wood chip [-] n chips Number of wood chips within the digester [-] p s n k , , Fraction of fibres in volume compartment k having the fundamental chemical component concentration of solid component i in concentration class n [-] R Universal gas constant 0.0083144 [kJ/(mol K)] R e j Reaction rate of entrapped liquid component j in volume compartment k [kg/(m 3 min)] Total solid reaction rate of solid i in volume compartment k [kg/(m 3 min)] T Temperature [K] V b,e ,k j Bulk flow from free to entrapped liquor of liquid component j from volume compart-

Where
reaction rate of the solid component s i within the con- centration class n of the volume compartment k.Additionally, a total solid reaction rate for the respective volume compartment is calculated, where the individual reaction rates R s n k , , i of the concentration classes are weighted by their fraction of fibres p s n k , , i within the volume compartment k and summed up to the total reaction rate of the solid component s i :

Figs. 4 Fig. 2 -Fig. 3 -
Figs. 4-6 illustrates the discretisation of the volume compartments within the wood chip, exemplarily for the xcoordinate direction.For the inlet flux at the location + k 1 2 , e f s i k , is the effectiveness factor, s i is the unreactive fraction of the solid component, s n k , , i 0 are the initial concentrations of the respective concentration class and e k

Fig
Fig. 4 -Mass transfer within the volume compartment in x coordinate direction.Fig. 5 -Boundary condition at the outer volume compartment in x coordinate direction.
s n k , , i are weighted by the percentage of fibres within the concentration classes p s n k , , i of the respective volume compartment.Afterwards, a summation of all solid components and concentration classes is conducted to calculate the average porosity of the volume compartment k.

Fig. 7 -Fig. 8 -
Fig. 7 -Model predictions against experimental data (Ming, 2007) for the average kappa number (left) and the standard deviation (right) of the fibre kappa number distribution for uniform pulping conditions.

Fig. 9 -
Fig. 9 -Initial lignin distribution for early-and latewood lignin according to the presented model and parameter optimisation for typical kraft pulping conditions.

Table 2 and
E reactivity is a kinetic parameter which needs to be estimated.The final set of parameters for the optimisation includes the standard deviation of the early-and latewood initial distribution ( Log EW lignin