Abstract
We present a method for pulp particle characterization based on a truncated lognormal mixture (TLM) model, as motivated by size statistics of organisms. We use an optical fiber analyzer to measure the length–width distribution of kraft-cooked roundwood or sawmill sources, of chemi-thermomechanical pulp (CTMP) samples from roundwood or sawmill sources, and the same CTMP samples after kraft post-processing. Our results show that bimodal TLMs capture salient features of the investigated pulp particle distributions, by decomposition into a large-particle and a small-particle fraction. However, we find that fibers from sawmill sources, which have not undergone mechanical treatment, cannot be described by TLM, likely due to non-random sampling. Within the confines of our dataset, the TLM characterization predicts laboratory sheet properties more effectively than conventional averaging methods for pulp particle size distributions. The TLM characterization is intended as a tool for controlling the pulp production process towards higher product quality, uniformity, and energy efficiency, pending further mill trials for validation.
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Introduction
The pulp particle geometry is a crucial factor in the development of paper properties. With the advent of online optical methods for particle size measurement, large amounts of particle size distribution data are now available in real-time. However, these data are not fully utilized, as only the mean values of the distributions are often considered. To achieve more uniform quality, a better product, and lower energy consumption in the pulping process, it is necessary to characterize the details of the measured particle size distributions. Presenting a novel approach for understanding pulp particle size distributions, this research aims to offer insights that could eventually lead to more strategic control in pulp production processes.
In nature, the geometry of wood fibers, such as their length and width, follows a particular distribution. Herein, we wish to base a statistical model of pulp particles on this natural wood fiber distribution. Consider the lognormal distribution, which is observed in many quantities in nature (Limpert et al. 2001). Moreover, strictly positive quantities with a fair fit to the normal distribution, typically display an equally fair fit to the lognormal distribution (Limpert et al. 2001). The reason for the ubiquity of the lognormal distribution is that it represents a random variable constructed from a product of many independent random variables, cf. the Central Limit Theorem. For instance, consider the mass of a pumpkin, which depends on the presence of certain genes, as well as the availability of water, minerals, and nutrients in the soil. The mass of the pumpkins becomes lognormally distributed as a result of the product of these essential factors (Sinnott 1937). This principle applies broadly, as evidenced by the lognormal distribution of scientific output from different individuals, which depends on a range of necessary mental attributes (Shockley 1957).
The fiber length distribution of wood fibers has previously been reported as approximately lognormal in several publications (Yan 1975; Dodson 1992; Kropholler and Sampson 2001). Recently, the length and width marginal distributions of fractionated pulp particles have been identified as lognormal (Brandberg et al. 2022) using the Akaike information criterion (Akaike 1974). The lognormal distribution is also frequently manifested in paper products. For instance, the free fiber length distribution in foam-formed fiber networks closely follows a bimodal lognormal distribution (Alimadadi et al. 2018), and fiberboard fiber and wood particle length distributions also fit the lognormal distribution, with some exceptions (Lu et al. 2007). However, fiber length is not the only dimension of the pulp particle; length, width and wall thickness form a multivariate distribution. Such a distribution can be approximated using the copula approach, that is by selecting marginal distributions of each fiber dimension, and a copula for describing the correlations between these dimensions (Brandberg et al. 2022).
Nonrandom fluctuations in the fiber dimensions are present in the tree. Fiber length and width increase with distance from the pith in Norway spruce (Sarén et al. 2001). Moreover, the distributions of fiber width and the cell wall thickness within a single Norway spruce log are distinctly bimodal owing to the different coarseness of earlywood and latewood (Havimo et al. 2008, 2009). Herein, however, we consider raw materials of the pulping process, which include various parts of the tree, and variations across entire forests and countries. Consequently, we assume that local fiber size variations within a tree become indistinguishable when convolved with the variations across a species of tree.
Since the lognormal distribution is ubiquitous in nature (Limpert et al. 2001), it is reasonable to expect that the length and width dimensions of wood fibers follow lognormal distributions across a species of tree. When processed into pulp, however, new fractions of fiber fragments and mechanically treated fibers are formed, which modifies the original distribution. Also, optical characterization of pulps is restricted by the resolution of the equipment. Consequently, only a truncated distribution of geometric particle properties can be measured.
In optical fiber analyzers, a dilute suspension of pulp flow between two co-planar glass plates, while micrographs are captured using an optical microscope. Individual particles are identified using digital image analysis. For each pulp particle, the analyzer reports a vector of geometric properties \({\varvec{x}} = [x_1\ \cdot \cdot \cdot \ x_d]^{\top }\), where we take \(x_1 = L\) to be the contour length of the particle, \(x_2 = W\) is the width of the particle, and the other properties may be curl, fibrillation and more. The pulp particle geometries follow some multivariate distribution \(\psi ({\varvec{x}})\), but for practical reasons, the reported particle samples are drawn from a conditional probability density, \({\hat{\psi }}({\varvec{x}}) = \psi ({\varvec{x}}\;|\;x_1 \ge \ell _1,\ x_2 \ge \ell _2)\), where \(\ell _1\) is a preset minimum contour length, and \(\ell _2\) is a lower limit needed to eliminate particles whose width is close to the resolution of the digital image. Hence, truncation is inherent to the measurements.
Building on the ubiquity of lognormal distributions in nature and truncation in optical measurements, this study has a dual aim. First, we seek to rigorously characterize the truncated, multivariate distributions observed in CTMP particles, with particular attention to their potential lognormal origins and the effects of mechanical treatment. Second, this leads us to the core hypothesis of our work: that truncated mixtures of multivariate lognormal distributions can effectively encapsulate the geometric intricacies of CTMP particles. By exploring this hypothesis, we aim to provide insights that could inform future work aimed at optimizing control strategies in pulp production, although full-scale operational data will be needed for definitive conclusions.
In this work, we consider chemi-thermomechanical pulp (CTMP) made from spruce, which is an important material, especially as a middle layer in cardboard. In this study, we characterize CTMP produced from roundwood chips, sawmill chips, and a 50/50 mixture of both. Additionally, we investigate CTMP that has undergone post-treatment in a kraft pulp cooking process, as well as fibers from wood chips that have been processed in a kraft-cooking process. By doing so, we can compare CTMP with fibers that have not been mechanically treated, and better understand the effects of different processing methods on pulp characteristics. Our characterization approach reduces the complex length–width distribution of CTMP particles to a few key parameters.
Theory
Because truncated domains are inherent to pulp particle characterization, it is necessary to define a truncated version of the multivariate lognormal distribution.
Truncated multivariate lognormal distribution
First, we review the multivariate normal distribution with \({\varvec{\mu }}\in {\mathbb {R}}^d\) the vector of mean values, and \({\varvec{\Sigma }}\in {\mathbb {R}}^{d \times d}\) the covariance matrix, which is symmetric, \({\varvec{\Sigma }}^{\top }= {\varvec{\Sigma }}\), and positive definite, \({\varvec{\Sigma }} \succ {\varvec{0}}\). Its probability density function (PDF) is,
where \({\varvec{\xi }}\in {\mathbb {R}}^d\) is a random-variable vector of dimension d. Its multivariate, cumulative distribution function (CDF) evaluated on a rectangular domain is,
with \({\varvec{\lambda }}\in {\mathbb {R}}^d\) the lower limits and \({\varvec{\upsilon }}\in {\mathbb {R}}^d\) the upper limits. The CDF is not available in closed form but can be evaluated numerically using mvncdf in Matlab (MathWorks 2023). The PDF of the truncated, multivariate normal distribution is then
Here, if \({\varvec{a}}\in {\mathbb {R}}^d\) and \({\varvec{b}}\in {\mathbb {R}}^d\), then \({\varvec{a}}<{\varvec{b}} = \bigwedge _{i=1}^d (a_i < b_i)\), with analogous definitions for other vector inequalities. Henceforth, a superposed circumflex indicates a truncated distribution.
The truncated, multivariate lognormal distribution is obtained through a change of variables \({\varvec{x}} = \exp ({\varvec{\xi }})\), so that \({\varvec{\xi }}({\varvec{x}}) = \ln ({\varvec{x}})\). The Multivariate change of variables theorem gives the multivariate lognormal PDF,
where \({\varvec{\ell }} = \exp ({\varvec{\lambda }})\), \({\varvec{u}} = \exp ({\varvec{\upsilon }})\), and where we used that \({\varvec{x}} > {\varvec{0}}\).
The likelihood of a set of n lognormally distributed samples \({\varvec{x}}^*_1, {\varvec{x}}^*_2,..., {\varvec{x}}^*_n\), truncated so that \({\varvec{\ell }}<{\varvec{x}}^*_i<{\varvec{u}}\), is given by
For maximum likelihood estimation (MLE), one seeks
where Eq. (4) yields,
Pulp particle geometry distribution
The pulp particle geometry follows a multivariate distribution \(\psi ({\varvec{x}})\), where \(x_1=L\) is the fiber contour length and \(x_2 = W\) is the fiber width, and so on. Due to the window of observation, however, only a truncated distribution \({\hat{\psi }}({\varvec{x}}; {\varvec{\ell }}, {\varvec{u}})\) is experimentally accessible. The pulp may contain fractions of different character, such as shives, fibers, and fines, which implies that \({\hat{\psi }}\) may be multimodal with N modes,
where \({\varvec{c}} \ge {\varvec{0}}\) is a vector such that \(\sum _{p=1}^N c_p=1\). We call this \({\hat{\psi }}\) a truncated lognormal mixture (TLM), inspired by the Gaussian-mixture concept. Each choice of N represents a different pulp particle model with
free parameters, including \(N-1\) elements of \({\varvec{c}}\), the vectors \({\varvec{\mu }}^{1},...,{\varvec{\mu }}^{N}\), and the lower triangles of \({\varvec{\Sigma }}^{1},...,{\varvec{\Sigma }}^{N}\).
The logarithm of the likelihood of a set of particle geometry samples \({\varvec{x}}^*_i\), \(i=1,2,...,n\), such that \({\varvec{\ell }}< {\varvec{x}}^*_i < {\varvec{u}}\), drawn from the TLM is
where the logarithm of the sum is evaluated using the method in Appendix A to avoid numerical overflow, with \(\ln ({\hat{\phi }})\) given by Eq. (7). The maximum likelihood for the observed samples is found by solving the optimization problem
In this work, problem (11) is solved using the Nelder–Mead simplex algorithm, i.e. fminsearch in Matlab (Lagarias et al. 1998; MathWorks 2023), with a penalty step function to enforce \({\varvec{c}} \ge {\varvec{0}}\) and \({\varvec{\Sigma }}^{p} \succ {\varvec{0}}\).
Within this modeling framework, the parameter values \({\varvec{\mu }}^{1}, \ldots , {\varvec{\mu }}^{N}\) and \({\varvec{\Sigma }}^{1}, \ldots , {\varvec{\Sigma }}^{N}\) are independent of the truncation limits, \({\varvec{\ell }}\) and \({\varvec{u}}\). Therefore, when fitting these parameters using maximum likelihood estimation, the expected values remain consistent, irrespective of the chosen truncation limits. This consistency does not extend to \({\varvec{c}}\), which describes the number fractions for truncated distributions.
Marginal distributions
Illustration of bimodal, i.e. \(N=2\), TLM distribution typical to roundwood CTMP. Triangles indicate \(\exp ({\varvec{\mu }}_1)\) and \(\exp ({\varvec{\mu }}_2)\), and dots indicate samples drawn from the TLM. Shaded regions show the truncation. The theoretical (line) and sampled (histogram) marginal distributions are illustrated on the side walls of the plot
It is often of interest to calculate the marginal distributions of multivariate distributions; the qth marginal distribution of \({\hat{\psi }}\) is defined as,
where \({\hat{\psi }}_q(x_q) = {\hat{\psi }}_q(x_q; {\varvec{c}},{\varvec{\mu }}^{1},...,{\varvec{\mu }}^{N},{\varvec{\Sigma }}^{1},...,{\varvec{\Sigma }}^{N}, {\varvec{\ell }}, {\varvec{u}})\). It is easy to show by direct integration that, if \({\varvec{\Sigma }}^1,...,{\varvec{\Sigma }}^N\) are all diagonal, then the marginal distributions of \({\hat{\psi }}\) are univariate TLMs, \({\hat{\psi }}_q(x_q) = {\hat{\psi }}_q(x_q; {\varvec{c}},\mu _q^{1},...,\mu _q^{N},\Sigma ^{1}_{qq},...,\Sigma ^{N}_{qq}, \ell _q, u_q)\). However, if the random variables are correlated in any way, then the marginal distributions are not univariate TLMs.
One use of the marginal distribution is ranking of the goodness-of-fit using the one-sample Kolmogorov–Smirnov (KS) statistic (Kolmogorov 1933; Smirnov 1944). This requires the marginal CDF
The KS statistic of the qth random variable is defined as,
where \({\hat{\Psi }}_q^*(x_q)\) is the empirical marginal CDF.
The marginal distributions and corresponding CDFs generally depend on all elements of the vectors \({\varvec{\ell }}\) and \({\varvec{u}}\), and the analytical form of those marginal distributions is difficult to access. It is, however, possible to draw samples from the TLM by applying the exponential function to samples drawn from multivariate Gaussian distributions, i.e. by using mvnrnd in Matlab (MathWorks 2023). Hence, it is straight-forward to construct the marginal distributions and CDFs using a Monte-Carlo approach with a posteriori truncation (Fig. 1).
Methods
Raw materials and pulp preparation
We consider data from previously published mill trials with different assortments of spruce wood at the CTMP plant in Skoghall papermill (Stora Enso) producing board grades. (Ferritsius et al. 2018, 2022) Briefly, the wood raw materials are three assortments of Norway spruce (Picea abies L. Karst.): roundwood chips, sawmill chips, or a 50/50 mixtures of roundwood and sawmill chips. The chip refiner is a Valmet CD 82 single disc. Each one of the chip assortments are run at three different levels of specific energy input by adjusting the flow rate of dilution water to the flat zone of the refiner.
CTMP from each chip assortment are produced for 4 h. Composite samples of screened chips going to the presteaming bin are collected before CTMP samples are taken. The refiner runs for 1 h with fixed process settings. Then, a composite CTMP sample is collected from the latency chest. Subsequently, the process settings are adjusted for the next specific energy input, and so on.
The chip samples from the mill trials are digested in a laboratory kraft cook to a yield of 49 %. In addition, a fraction of the CTMP samples is post-processed in the laboratory kraft cook. The CTMP, cooked CTMP, and cooked chip samples from different material sources are compiled in first three columns of Table 1.
Importantly, it should be noted that the scope of our study is limited to selected samples. These serve as distinct yet illustrative examples to demonstrate the potential utility of our novel characterization method. Therefore, while our findings offer insights into the method’s capabilities, they should not be interpreted as universally applicable.
Characterization methods
After hot disintegration according to ISO 5263-3, the pulp samples are analyzed according to ISO 16,065-2 using the FiberLab™ optical analyzer (Valmet Automation Oy, Kajaani, Finland). Fiber length is measured with 10 \(\mu\)m resolution, and fiber width is measured with 1.5 \(\mu\)m resolution (FiberLab 2008). It is important to note that the FiberLab analyzer imposes inherent truncation limits on particle length between \(\ell _1 = 200\) \(\mu\)m and \(u_1 = 8900\) \(\mu\)m. Measurements falling outside these length parameters are automatically discarded by the analyzer. In our analysis, a lower width truncation limit of \(\ell _2 = 10\) \(\mu\)m set, below which measurements are discarded. A justification for selecting this particular truncation limit can be found in Appendix B. We also introduce the variable \(\gamma\) to quantify the fraction of excluded particles due to this width constraint (Table 1).
Laboratory sheets are prepared in compliance with ISO 5269-1 to facilitate mechanical testing. The apparent density of the sheets is assessed following the procedures outlined in ISO 534 (10 samples). Tensile attributes, including tensile index (TI), tensile stiffness index (TSI), strain-at-break, and tensile energy absorption (TEA), are evaluated using the methods prescribed in ISO 1924-3 (20 samples). The Scott-Bond energy is determined in accordance with the Tappi T 569 standard (6 samples), and the short span compression test is executed following the guidelines of ISO 9895 (10 samples).
AI language support
We use AI language support provided by OpenAI’s ChatGPT to assist in the writing and editing of this article. We declare that the use of AI language support was for purpose of improving the clarity and accuracy of the language used in the article.
Results
a , Empirical length–width histogram (bars) and maximum-likelihood TLM (surface) for a roundwood CTMP. b , Logarithmically equidistant contours of the TLM (lines) fit to the length–width particle distribution of a roundwood CTMP. Data points that originate from the larger-particle fraction with more than 50 % probability are indicated by red markers. Data points in the 0.1 percentile of the likelihood (‘\(\times\)’ symbols) are considered as outliers
A bimodal TLM, \({\hat{\psi }}({\varvec{x}}; {\varvec{c}},{\varvec{\mu }}^1,{\varvec{\mu }}^2,{\varvec{\Sigma }}^1,{\varvec{\Sigma }}^2)\), with \(k=11\) free parameters, is fit to length–width data of pulps, resulting in a small-particle fraction, \(p=1\), and a large-particle fraction, \(p=2\) (Fig. 2). The length–width histogram is compared to the maximum-likelihood distribution \({\hat{\psi }}\) for a roundwood CTMP in Fig. 2a. Also, contours of \({\hat{\psi }}\) are shown with a scatter plot of all observations \({\varvec{x}}^*_i\) for the same roundwood CTMP in Fig. 2b. The TLM parameters obtained for all pulp samples are compiled in Table 1.
Particle classification and outliers
The probability that an observation \({\varvec{x}}^*_i\) originates from fraction p is
Particularly, the probability that \({\varvec{x}}^*_i\) originates from the large-particle fraction is, \(P_{\textrm{large}}({\varvec{x}}^*_i) = P_2({\varvec{x}}^*_i)\) in the bimodal case. This probability can be used to categorize individual particles, and particles with \(P_{\textrm{large}} \ge 0.5\) are indicated by red markers in Fig. 2. We also define a third category of outliers, taken as the \(\alpha = 0.1\) % percentile of the likelihood. These objects are the most unlikely to be observed under the assumption of a bimodal TLM, and are indicated by ‘\(\times\)’ symbols in Fig. 2. Examples of possible outliers are shives or aggregates.
We exclude outliers and construct the marginal distributions of the observed values and the TLM for the roundwood CTMP in Fig. 3a, b. The agreement of the TLM marginal distribution with the empirical marginal distribution is excellent for the particle length and width in this particular example.
Comparison of TLMs from different pulps
a Markers grouped to the lower left represent \(({\bar{L}}^1,{\bar{W}}^1)\), while those grouped in the upper right represent \(({\bar{L}}^2,{\bar{W}}^2)\). The area of the markers is drawn proportional to the corresponding number fraction \(c_p\). b Number fraction \(c_2\) of the large-particle fraction, for groups of different raw materials and different treatments
We proceed to consider bimodal TLMs fit for each of the CTMP samples, and to each one of the kraft post-processed samples. For convenience, we introduce the logspace-average length \({\bar{L}}^p = \exp (\mu ^p_1)\) and width \({\bar{W}}^p = \exp (\mu ^p_2)\) for fractions \(p=1,2\). The fit values of \({\bar{L}}^p\) and \({\bar{W}}^p\) are consistently reproducible for different pulps produced with the same raw material and processing conditions within our dataset (Fig. 4a). Pulps made from sawmill chips have wider large particles than roundwood-based pulp, and the width is reduced by kraft post-processing due to loss of lignin. For the CTMP samples, neither \({\bar{L}}^p\) nor \({\bar{W}}^p\) shows significant dependence on specific energy (not shown). The number fraction \(c_2\) of large particles is averaged across pulp samples with the same raw material and processing conditions, and plotted in Fig. 4b. We observe that there is a relatively larger fraction of large particles in cooked wood chips, which suggests fiber length reduction in the CTMP process. Figure 4b also shows that wood-chip samples are roughly equally divided between small and large particle fractions. This could be seen as conflicting with the initial assumption that the unprocessed fiber in the raw material is lognormal distributed. However, it has been previously established that the native fiber length–width distribution of Norway spruce does not include the small-particle component observed in this study (Havimo et al. 2008). This implies that even the cooked chips contain a substantial number fraction of debris.
We study the large-particle fraction, \(p=2\), of CTMP and kraft post-processed CTMP in more detail. The logspace-average length and width are plotted in Fig. 5, together with the elliptical region in logspace, which represents one standard deviation. These ellipses give a visual representation of the covariance matrix \({\varvec{\Sigma }}^2\). The covariance matrices are also robustly reproduced for pulps produced with the same raw material and processing conditions.
Goodness-of-fit
Let \(D_L\) be the KS statistic of the truncated length distributions, while \(D_W\) is the KS statistic of the truncated width distribution. For a large sample size n, a good fit is characterized by small values \(\sqrt{n}D_L\) and \(\sqrt{n}D_W\). We plot \((\sqrt{n}D_L,\sqrt{n}D_W)\) for all samples in Fig. 6a. We observe that the four cooked chip distributions exhibit the worst fits to the marginal length distribution. Instead of a lognormal small-particle fraction, the particles of chips exhibit an essentially uniform length distribution in the small-particle range, as exemplified by the sawmill chips (Fig. 6b).
The lognormality of the small-particle fraction appears to be contingent on mechanical treatment, as evidenced by a better fit to CTMP compared to cooked chips (Fig. 6a). Moreover, a possible explanation for the particularly poor fit of sawmill chip pulp is that the sawmill raw material represents conditional sampling of wood fibers; the sample is not drawn from all natural fibers of Norway spruce, but only those from mature trees, and only the most distal fibers from the pith.
Predictive capability of TLM parameters
The laboratory sheet density, TI, TSI, strain-at-break, TEA, Scott-Bond, and SCT index for uncooked CTMP are compiled in Table 2. It is of interest to predict these properties using particle size data. Therefore, we compare the predictive capability of the TLM parameters with those of conventional averages of the length and width marginal distributions.
We first define different types of weighted averages often encountered in the literature (Ferritsius et al. 2018). Let t be an element of the vector \({\varvec{x}}\) of particle properties to be averaged. For each observed particle \(i=1,...,n\), with properties \({\varvec{x}}^*_i\), \(t^*_i\) is the element of interest, and \(L^*_i\) is the particle length. We define the jth-order length-weighted average of t as
That is, \(\left\langle t \right\rangle _0\) is the arithmetic average, \(\left\langle t \right\rangle _1\) is the length-weighted average, and \(\left\langle t \right\rangle _2\) is the length-length-weighted average.
To formalize the question of whether TLM parameters, in some sense, are better predictors than weighted averages, we define the predictor variable vector of weighted averages as \({\varvec{X}}_{\textrm{avg}} = [\left\langle L \right\rangle _0,\left\langle W \right\rangle _0,\left\langle L \right\rangle _1,\left\langle W \right\rangle _1,\left\langle L \right\rangle _2,\left\langle W \right\rangle _2]\), and the TLM predictor variable vector as \({\varvec{X}}_{\textrm{TLM}} = [c_2,\mu ^1_1,\mu ^1_2,\mu ^2_1,\mu ^2_2]\). The response variable \({\varvec{Y}}\) comprises laboratory sheet density, TI, TSI, strain-at-break, TEA, Scott-Bond, and SCT index. The corresponding standardized variable vectors, \({\varvec{{\tilde{X}}}}_{\textrm{avg}}\), \({\varvec{{\tilde{X}}}}_{\textrm{TLM}}\) and \({\varvec{{\tilde{Y}}}}\), are obtained by scaling each element so that it has zero mean and unit variance. Finally, we fit each one of the standardized predictor variables to the standardized response variables in a least-squares sense, using fitlm in Matlab (MathWorks 2023). The predictive capability of weighted averages, \({\varvec{{\tilde{X}}}}_{\textrm{avg}}\) (Fig. 7a), is outperformed by that of the TLM, \({\varvec{{\tilde{X}}}}_{\textrm{TLM}}\) (Fig. 7b), for all responses, as demonstrated by comparing the \(R^2\) values (Fig. 7c) of the respective fits. This shows that the TLM characterization captures the details of the particle distribution better than any combination of conventional averages of the length–width distribution, when judged as predictors of sheet properties. We believe that the TLM description of the length–width distribution is more precise than averages formed using marginal distributions of width and length, as evidenced by better correlation with laboratory sheet properties even though \({\varvec{{\tilde{X}}}}_{\textrm{TLM}}\) has a lower dimensionality than \({\varvec{{\tilde{X}}}}_{\textrm{avg}}\).
Conclusions
The length–width distribution of CTMP particles can be described as a multivariate lognormal mixture. The distribution consists of a large-particle fraction, whose properties derive mainly from the wood fibers, as modified by the pulping process. Their lognormality can thus be understood as a result of the growth process of the plant cell. A second small-particle fraction consists of debris, which is generated in the CTMP process. The lognormality of this fraction cannot be easily explained, but we speculate that the small-particle fraction is generated by independent, random events during chipping and pulping, and in that sense their generation is statistically similar to a growth process. It is also feasible that loose fibrils and fines produced by abrasion during refining follow a gamma distribution based on refining intensity (Kerekes et al. 2023), so that lognormality is approximate for small particles.
Ideally, the marginal distributions of a lognormal mixture are also lognormal mixtures. However, the marginal distributions of a TLM are not generally TLMs due to effects of the truncation. Therefore, for any measurement with length and width truncation, its marginal length or width distributions will deviate from lognormal-mixture behavior. The bimodal nature of CTMP, together with truncation due to limited resolution, may explain why previous investigations have only reported approximate lognormality for the marginal distributions.
The TLM concept is based on the principle that plant cell growth is controlled by many independent factors, and that chipping and mechanical treatment similarly disintegrate fibers in a random process. However, it should be noted that the TLM model may not accurately describe all pulp types. For instance, the small-particle fraction in cooked chips does not conform to a lognormal length-width distribution. Additionally, deviations from lognormality in the large-particle fraction of sawmill chips may occur due to factors such as selective sampling.
In this specific context of CTMP, our limited dataset suggests that a bimodal TLM model provides a reasonable approximation for measured length–width distributions, and the TLM parameters correlate well with laboratory sheet properties (Fig. 7c). These observations suggest that TLM pulp characterization, together with e.g. curl and fibrillation distributions, could potentially inform more advanced control strategies for the CTMP process. The bimodal TLM serves as a compact representation of the pulp particle distribution, and these parameters, pending further testing, could offer insights for future improvements in both automated and manual control systems.
Data availability
The datasets generated during the current study are tabulated in the article. Additional data can be made available from the corresponding author upon reasonable request.
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Acknowledgments
The authors thank Stora Enso for support in facilitating mill trials and experimentation.
Funding
Open access funding provided by Mid Sweden University. The work has been carried out within the Strategic Innovation Program for Process Industrial IT and Automation, a joint initiative by Vinnova, Formas, and the Swedish Energy Agency, as Vinnova grant agreement No. 2022-03,597. The work was done within the research profile Neopulp financed by the Knowledge foundation. S. B. Lindström thanks Svenska Cellulosa AB (SCA) for financial support.
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SL and JP wrote first draft of the manuscript. OF conducted physical experiments. OF, RF, and SL analyzed and interpreted data. All authors reviewed and revised the manuscript for important intellectual content.
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Appendices
Appendix A: Approximation of logarithm
To compute the sum \(\ln \left( \sum _{i=1}^n a_i \right)\) for a sequence \(\{a_i\}_{i=1}^n\), such that \(0 < a_i \le a_1\), when only the logarithms \(\ln (a_i)\) are numerically accessible, we rewrite
Crucially, since \(\ln (a_i) - \ln (a_1) \le 0\), \(i=2,...,n\) the exponentials will not create overflow, and underflow is typically negligible in this calculation.
Appendix B: Truncation limit effect
Ideally, if the fiber analyzer accurately measures all particles, and these particles adhere to a bimodal lognormal mixture distribution, the parameters \({\varvec{\mu }}^p\) should remain independent of truncation limits. However, in practice, these parameters do exhibit some level of dependence on truncation limits. To investigate this further, we examined how fit TLM parameters depend on the particle width lower truncation limit, denoted as \(\ell _2\), which we can control. We used the roundwood CTMP sample whose length–width distribution is depicted in Fig. 2a, b for this investigation. As demonstrated in Fig. 8, the parameters for large-particle length and width, \(\mu _1^2\) and \(\mu _2^2\), show weak dependence on \(\ell _2\) for values greater than or equal to 10 \(\mu\)m. This informed our decision to set \(\ell _2=10\) \(\mu\)m in this study. Conversely, the parameter describing small-particle width shows a somewhat stronger dependence on \(\ell _2\), suggesting reduced accuracy for parameters related to the small-particle fraction.
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Lindström, S.B., Persson, J., Ferritsius, R. et al. Multivariate lognormal mixture for pulp particle characterization. Cellulose 31, 1843–1854 (2024). https://doi.org/10.1007/s10570-023-05686-8
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DOI: https://doi.org/10.1007/s10570-023-05686-8