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November 2008
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EVALUATING REACTION KINETIC MODELS USING WELL-DEFINED KRAFT DELIGNIFICATION DATA
Johannes Bogrena*, Harald Brelida and Hans Theliandera - Presented at ISWFPC 2007

Keywords: Kraft delignification, wood meal, softwood, reaction kinetics, modelling, three- phase model, continuous distribution of reactivity model.

ABSTRACT
A three-phase model and a continuous distribution of reactivity model have been fitted to well-defined data obtained from kraft delignification of Scots pine wood meal. The performance of the two models, when fitted to the data, is fairly similar.

However, the continuous distribution of reactivity model is a simpler model to use and renders a more probable description of the chemical and physical mechanisms during delignification than the three-phase model. The continuous distribution of reactivity model is recommended for modelling the reaction kinetics of delignification.

1. INTRODUCTION
Knowledge regarding the chemistry and physics behind the reaction and dissolution kinetics of lignin during kraft pulping is not complete. The kinetics of certain bond cleavages of model compounds have been investigated and the results have been used to interpret the kinetics of delignification1. The use of model compound experiments to directly explain the kinetics of delignification is doubtful mainly because of the complexity of the heterogeneous, polymer degradation and dissolution characteristics of delignification2. Therefore, the application of data from model compound experiments on delignification kinetics could be misleading in the understanding of the mechanisms.

One way of analysing reaction kinetics is to fit models to experimental data, and then interpret the parameters and construction of the model in order to obtain a physical and chemical explanation for the reaction kinetic pattern. This technique has been utilised for a long time in this field of research, but has not yet rendered a model that is widely accepted.

One of the first attempts to model the reaction kinetics of delignification was done by Vroom3 who assumed delignification to proceed according to nth-order kinetics. The model is known as the H-factor model, and the H-factor is a parameter describing the combined effect of time and temperature. One drawback of the H-factor model is that it does not consider the effect of chemical concentration in the cook.

The most common way of modelling reaction kinetics in a more generalized way has been to assume that delignification follows first order kinetics. To obtain a proper fit to experimental data, lignin has been divided into three different subspecies reacting according to different kinetics. The three types of lignin are denoted initial, bulk, and residual phase lignin, and are either defined as being a result of the cook or already present in wood, i.e. reacting in series or in parallel, respectively. The assumption that lignin consists of three different subspecies already present in wood has, for instance, been used by Chiang et al.4 when modelling the whole kraft cook, and by Lindgren and Lindström5 in their model of the bulk and residual phase.

One of the drawbacks of this kind of model is difficulty in estimating the amount of lignin reacting in each phase, which leads to an increased number of parameters that need to be adjusted to achieve an acceptable fit of the model6,7. Another approach to modelling delignification kinetics is to use a time-dependent rate constant which leads to the physical interpretation that lignin consists of an infinite number of subspecies reacting with a continuous distribution of activation energies.

This type of model was first introduced in kraft cooking kinetics by Montané et al.8 who modelled the data by Chiang et a14. One great advantage of a model that uses a time -dependent rate constant is that it only consists of one differential equation, compared to three in case of the three phase model.

Nevertheless, the dependence of chemical concentration on kinetics has not yet been incorporated into the model, which decreases its usefulness. In this paper, a chemical concentration dependence will be introduced, and the model will be denoted continuous distribution of reactivity model, because of the physical interpretation of the time-dependent rate constant.

The aim of the present study was to evaluate the two mentioned models by fitting them to well-defined kraft delignification data obtained over a wide range of chemical concentrations, temperatures, and degrees of delignification. The models are to only describe reaction kinetics and not the combined kinetics of reactions and mass transport, which has normally been the case in kraft delignification studies.

The models then serve as a foundation for a physical and chemical explanation of the pattern of the reaction kinetics of delignification. The focus of this paper is to find an appropriate model rather than achieving the best fit to the experimental data by introducing and adjusting additional parameters.

2. EXPERIMENTAL
The experiments were conducted using wood meal in order to minimise the effect of mass transport resistance on the reaction kinetics. The substrate used was sapwood from Scots pine (Pinus silvestris). All cooks were performed in autoclaves with a high liquor-to-wood ratio (200:1) rotated in a polyethylene glycol bath. The temperature of the bath used during cooking was constant, however, the temperature rise of the liquor inside the autoclaves in the beginning of the cook was measured, and used in the fitting of models.

The time to reach cooking temperature was approximately 25 minutes, regardless of maximum temperature. Hydroxide and hydrosulphide ion concentrations were varied between 0.1-0.75 mol/kg solvent and 0-0.5 mol/kg solvent, respectively, and temperature levels were varied from 108ºC to 168ºC. In four cooking series, the sodium ion concentration was increased by the addition of sodium chloride, thus yielding a total concentration of 2.90 mol Na+/kg solvent. After cooking, the content of the autoclaves was transferred to glass filters and washed with 0.5 l filtrate and 1 l of deionised water.

The solid residue was analysed for yield, Klason lignin, and composition of carbohydrates. A more thorough description of the experimental procedures is described elsewhere9.

3. DEFINITIONS
In this section, the chosen models will be explained and defined.

3.1 Three-phase model
In a three-phase model, lignin (L) is assumed to consist of three different subspecies (see equation 1). The second assumption is that the three subspecies are already present in wood, i.e. three differential equations are needed to describe the kinetics (equations 2-4).

Furthermore, the dependence of chemicals is assumed to be described by power law equations. The three differential equations are then solved simultaneously, and considering the temperature rise profile, the results are summarized (equation 1) and fitted to the experimental data by adjusting the, in total, 17 parameters.

kraf_1to4(1)


(2)



(3)



(4)




3.2 Continuous distribution of reactivity model
In the continuous distribution of reactivity model, delignification is assumed to proceed according to first order kinetics with a rate constant that is time-dependent (equation 5). This rate constant can be described by the two parameters ß and g as in equation 6.

kraf_56(5)



(6)



To be able to use the Kohlrausch relaxation function, which can be used for describing a distribution of exponential decays, the effective lifetime t0 must be defined according to equation 710.


kraf_7(7)


Hence, the mean lifetime kraf_pie is

kraf_8(8)

where g denotes the gamma function and is defined as kraf_fig1. The mean lifetime can also be expressed as an Arrhenius expression and a preexponential factor, which is a function of the chemical concentrations (equation 9).

kraf_9(9)

Equation 10 can be derived from equations 6-9. Equation 10 is the basis of the model fitted to the experimental data in this study.

kraf_10(10)

3.3 Residuals
The two different models have been fitted to the experimental data using the nonlinear least square solver lsqnonlin in MATLAB, thus minimising the sum of two different squared residuals . The first residual, R1 (equation 11) provides the deviation between the actual lignin values and the values obtained by the model, while the second residual, R2 (equation 12) gives the product between the lignin values and the deviation between the actual cooking time, and the one predicted by the model at the experimentally determined lignin value.

kraf_1112
(11)


(12)


4. RESULTS AND DISCUSSION
In figures 4.1 and 4.2, the two models evaluated in this study are compared with some of the experimental data used for the fitting of the models. It can be seen that the temperature dependence of the reaction kinetics of lignin can very well be explained by both models, despite the wide range of cooking temperatures studied, ranging from 108ºC to 168ºC.

When the fitting of the models was performed, it was noticed that the use of an ordinary residual, R1 (equation 11), resulted in a rather poor fit at low lignin contents when results from all cooking temperatures were used in the fitting. An explanation for this behaviour is that the majority of data is at a low degree of delignification, and the delignification rate in the beginning of the cook is high. This results in greater importance of a proper fit at high lignin contents in order to minimise the sum of the squared residuals.

Two methods were used to compensate for this; use of the same residual, but less data at low degrees of delignification by discarding all data from T=108ºC in the fitting (first column in tables 4.1 and 4.2).

The second method for decreasing the impact of experimental values at low degrees of delignification was to use equation 12 as the residual in the fitting (second column in tables 4.1 and 4.2). In this way, the data at high degrees of delignification become more important, because of the risk of large deviation between the cooking time and the time predicted by the model to reach a certain degree of delignification. The total four fittings of models to data are presented in tables 4.1 and 4.2. The residual R2 was used in the fitting of the models presented in the figures throughout this paper, unless otherwise stated.

kraf_41
Figure 4.1 Comparison of the threephase model with experimental data: OH-=0.26 mol/kg solvent, HS-=0.26 mol/kg solvent.


kraf_42
Figure 4.2 Comparison of the continuous distribution of reactivity model with experimental data: OH-=0.26 mol/kg solvent, HS-=0.26 mol/kg solvent.

One of the challenges when constructing a model using the concept of continuous distribution of reactivity is to incorporate the dependence of chemicals present in the liquor. It is well known from threephase models that the dependence of the chemicals varies during the cook4, 5, 7, 11, and therefore a conventional power-law dependence is not suitable for this task. Instead, a modified power-law was used (see equation 13) to compensate for variations in the dependence of chemical concentrations on the rate of the delignification during the kraft cook.

A clear example of these variations is shown in figures 4.3 and 4.4 where the data from two cooking series with different chemical compositions in the liquors intersect. This is only possible if the chemical concentration dependence varies throughout the cook. It can be concluded that the two models perform similarly in these two cooking series.

kraf_13(13)
Another measurement of the performance of the models can be found in tables 4.1 and 4.2 where the sum of the squared residuals ( å Rn2 ), and the coefficient of determination (R2) are presented.

kraf_43
Figure 4.3 Comparison of the three-phase model with experimental data at two different chemical concentrations: T=154ºC. Unit of concentration is given as mol/kg solvent.

kraf_44

Figure 4.4 Comparison of the continuous distribution of reactivity model with experimental data at two different chemical concentrations: T=154ºC. Unit of concentration is given as mol/kg solvent.

When R1 (equation 11) is used in the optimisation, the three-phase model performs somewhat better than the model using continuous distribution of reactivity.

However, when all experimental data are used and R2 (equation 12) is used in the optimisation, the performance of the models is the opposite. It is difficult to compare the models using different optimisation residuals. The coefficient of determination (R2) describes the proportion of variability in lignin content that can be described by the model without considering the variation in time as in R2.

However, when the models are compared to experimental data and figures showing the residuals at corresponding lignin values are scrutinised, it is seen that the models optimised with residual R2 perform considerably better than the ones optimised with residual R1. This is mainly an effect of an even distribution of the importance of data throughout the cook. An example of this effect is shown in figure 4.5 where the performance of the model optimised with a normal residual, R1, is poorer at low lignin contents than when residual R2 is used in the optimisation.

kraf_45
Figure 4.5 Comparison of the continuous distribution of reactivity model when the two residuals, R 1 and R2, are used in the optimisation: T=154ºC, OH-=0.26 mol/kg solvent, HS-=0.26 mol/kg solvent.

kraf_tab1
Table 4.1 Determined parameters in the three-phase model (equations 1-4): Lt=0=0.97.

kraf_tab2
Table 4.2 Determined parameters in the continuous distribution of reactivity model (equations 10 and 13): Lt=0=0.97.

4.1 Interpretation of the models
One of the aims of this paper was to find a model that can act as a foundation for the explanation of the reaction kinetics pattern of delignification. When the values of the parameters from the fitting of the models are analysed a few rather peculiar values can be found. For example, the negative values of a1 and b1 mean that it is negative for the reaction rate of L1, often called the initial phase lignin, that active cooking chemicals are present.

Furthermore, the positive value of c1 suggests that the removal of L1 is promoted by a high sodium ion concentration. Findings from studies focusing on the initial phase show that the exponents a1 and b1 are practically zero12, 13. The value of the exponents a1 and b1 is not a result of a lignin subspecies reacting according to these kinetics, but is rather a consequence of the construction of the model.

It should be noted that this type of model assumes that the three types of lignin react in parallel and not in series, resulting in compensation for the positive/negative values of the exponents in the other lignin subspecies L2 and L3.

Thus, the overall influence of chemical concentrations is low at low degrees of delignification . This gives a faulty chemical description of the reactions in the initial phase caused by the assumption of parallel reacting subspecies. The fact that other models presented in the literature, using the same assumption, have not arrived at the same conclusion can be explained by the fact that very few have considered the entire kraft cook, and instead have used data starting at a degree of delignification of approximately 75%11 or have used a limited amount of data in the early part of the cook4.

Furthermore, the amount of lignin reacting according to third phase kinetics (L3), the so called residual phase lignin, is higher than can be found in the literature6. This is primarily a result of the model and the ensemble effects that arise when a large number of parameters are simultaneously estimated in the model. When the continuous distribution of reactivity model is used, a more likely and straightforward description of the kinetics is obtained, and does not give rise to any obviously incorrect conclusions regarding reactions with chemicals

kraf_46
Figure 4.6. Distribution of the reactivity calculated from the model of continuous distribution of reactivity at different temperatures: OH-=0.26 mol/kg solvent, HS- =0.26 mol/kg solvent.

kraf_47
Figure 4.7. Reactivity of the three subspecies according to the three-phase model at different temperatures: OH-=0.26 mol/kg solvent, HS-=0.26 mol/kg solvent.


When the model using a time-dependent rate constant is used, the distribution of reactivity can be calculated10. The results at five different temperatures are shown in figure 4.6. At low temperatures, the reactivity is very low; nevertheless a small part reacts very fast resulting in a very broad distribution. At higher temperatures the distribution curve is narrower and a fairly high amount of lignin has high reactivity. The corresponding plot for the three-phase model, where the reactivities are equal to the rate constants, is shown in figure 4.7. In this case, the amount of subspecies reacting in each phase at the corresponding reactivity is shown. When the two figures are compared, it can be concluded that both show a narrower distribution of reactivity at higher temperatures. The levels of the reactivity are also comparable in the two figures. This means that if additional phases, i.e. lignin subspecies, are added to the threephase model (i.e. a four-phase model, a five-phase model, etc.), the plot would eventually be similar to the one in figure 4.6. However, this is impossible since the number of parameters needed for such a model would be far too many. Therefore, the continuous distribution of reactivity model is the most suitable of these two models for describing the reaction kinetics of delignification.

The two models perform very well when fitted to well-defined experimental data. However, the physical and chemical explanation of the model using three phases becomes dubious, and is therefore not suitable for use. The number of parameters used in the models differs greatly, making it more convenient to use the model utilising continuous distribution of reactivity. Another advantage of the continuous distribution of reactivity model is the easier mathematics, since only one differential equation is used, instead of three as in the three-phase model.

Consequently, the model using a continuous distribution of reactivity is recommended for modelling the reaction kinetics of delignification.

5. ACKNOWLEDGEMENTS
The authors would like to thank everyone involved at Avancell - Centre for Fibre Engineering, for fruitful cooperation. Financial support from the foundation, Södra Skogsägarnas stiftelse för forskning, utveckling och utbildning, is gratefully acknowledged.

6. REFERENCES
1. Gierer, J. Chemical aspects of kraft pulping, Wood Sci. Technol., 1980,
14(4), pp. 241-266.

2. Obst, J.R. Kinetics of alkaline cleavage of ß-aryl ether bonds in lignin models: Significance to delignification, Holzforschung, 1983, 37(1), pp. 23-28.

3. Vroom, K.E. The H-factor: A means of expressing cooking times and temperatures as a single variable, Pulp Pap. Mag. Can., 1957, 38(2), pp. 228-231.

4. Chiang, V.L., Yu, J. and Eckert, R.C. Isothermal reaction kinetics of kraft delignification of Douglas fir, J. Wood Chem. Technol., 1990, 10(3), pp. 293-310.

5. Lindgren, C. and Lindström, M. The kinetics of residual delignification and factors affecting the amount of residual lignin, J. Pulp Pap. Sci., 1996, 22(8), pp. J290-J295.

6. Gustavsson, C. A., Lindgren, C. T. and Lindström, M. E. Residual phase lignin in kraft cooking related to the conditions in the cook, Nord. Pulp Pap. Res. J., 1997, 12(4), pp. 225-229.

7. Blixt, J. and Gustavsson, C. A. Temperature dependence of residual phase delignification during kraft pulping of softwood, Nord. Pulp Pap. Res. J., 2000, 15(1), pp. 12-17.

8. Montané, D., Salvadó, J., Farriol, X., Jollez, P. and Chornet, E. Phenomenological kinetics of wood delignification: Application of a timedependent rate constant and a
generalized severity parameter to pulping and correlation of pulp properties. Wood Sci. Technol., 1994, 28(6), pp. 387-402.

9. Bogren J, Brelid, H and Theliander H. Reaction kinetics of kraft delignification – General considerations and experimental data, Nord. Pulp Pap. Res. J., 2007, (In press).

10. Plonka, A. Lecture Notes in Chemistry- Time-dependent reactivity of species in condensed media, Springer-Verlag, 1986.
 
11. Andersson, N.; Wilson, D. I. and Germgård, U. An improved kinetic model structure for softwood kraft cooking. Nord. Pulp Pap. Res. J., 2003, 18(2), pp. 200-209.

12. Kondo, R. and Sarkanen, K. V. Kinetics of lignin and hemicellulose dissolution during the initial stage of alkaline pulping, Holzforschung, 1984, 38(1) , pp. 31-36.

13. Olm, L. and Tistad, G. Kinetics of the initial stage of kraft pulping, Svensk Papperstidning, 1979, 82(15), pp. 458-464.

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