Article pubs.acs.org/IECR
Model for Degradation of Galactoglucomannan in Hot Water Extraction Conditions Juha A. Visuri,† Tao Song,‡ Susanna Kuitunen,*,† and Ville Alopaeus† †
Chemical Engineering Research Group, School of Chemical Technology, Aalto University, P.O. Box 16100, FI-00076 Aalto, Finland Laboratory of Wood and Paper Chemistry, Process Chemistry Centre, Åbo Akademi University, Porthansgatan 3, FI-20500 Turku/Åbo, Finland
‡
ABSTRACT: A kinetic analysis is presented for the homogeneous degradation of galactoglucomannan (GGM) in hot water extraction conditions. In the experiments used for constructing the kinetic model, GGM extracted from spruce wood meal was treated at temperatures from 150 to 170 °C and in buffered pH 3.8−4.2. The hydrolysis of glycosidic bonds in GGM was modeled as a random process which is catalyzed by hydrogen ions. The real hydrogen ion concentration at the reaction temperatures was estimated. The Arrhenius equation was used to describe the temperature dependence of the hydrolysis rate and an activation energy of 102 kJ/mol was obtained for the reaction. The presented model predicts accurately the evolution of the molecular weight distribution with lower computational load than other similar models.
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recent years.3−6 Most of the hemicellulose in spuce wood is galactoglucomannan (GGM). GGM is a linear polysaccharide with a backbone consisting of β-(1,4)-linked mannose and glucose units. The backbone contains α-(1,6)-linked galactose units as side branches and the galactose:glucose:mannose ratio is 1:1:3.7 The degradation of linear polysaccharides follows the same mechanism as the hydrolysis of simple glycosides in acidic conditions.8 According to Timell,9 the reaction starts with a rapid equilibrium-controlled protonation of the glycosidic oxygen, and as a result, conjugate acid is formed. The acid decomposes in a rate controlling slow reaction into carboniumoxonium ion which most likely exists in a half-chair conformation. Finally, water is added and mono-, oligo-, or polysaccharides are formed and hydrogen ions are released. If all glycosidic bonds linking the monomer units are similar in a linear polysaccharide, it can be assumed that every bond has the same hydrolysis rate, and the hydrolysis of the bonds is a pure random process. The mathematical modeling of polysaccharide degradation was started already in the 1930s when Kuhn10 and Freudenberg et al.11 used a statistical treatment for estimating the degradation rate for cellulose. Their work was later extended by Ekenstam12 while studying homogeneous degradation of cellulose in phosphoric acid. The Ekenstam model describes the evolution of an average degree of polymerization, and it has been used for describing aging of paper in a wide range of conditions.13 While the Ekenstam model gives information only regarding the average degree of polymerization, some models have been developed to predict the evolution of the entire molecular weight distribution (MWD).14−17 Basedow et al.14 presented a set of equations to describe the kinetics of acid
INTRODUCTION In the past decade, there have been large structural changes in the pulp and paper markets. The prices of paper products have lowered due to increased competition and overproduction in the paper industry. New pulp mills are built in areas of cheap labor and wood. Areas of traditional pulp and paper industry need new competitive advantages to stay in the business. For these reasons, a large effort has been directed toward developing new ways of utilizing wood. Forest products biorefineries have become the center of interest. The concept aims at the fractionation of wood to its components, which are cellulose, hemicellulose, and lignin, and at upgrading them further into chemicals and new materials. In the paper pulp production, the hemicelluloses are desired in the product pulp to increase the pulp yield and paper making properties. Unfortunately, a major fraction of hemicelluloses ends up in the black liquor and is burned with a low heating value in traditional kraft process. Thus, one great economical opportunity is to convert existing kraft pulp mills into integrated forest product biorefineries and extract the hemicelluloses from wood chips prior to pulping and turn them into value-added chemicals and polymers.1 Mastering the extraction technology is one key element for the success of the integrated forest products biorefinery and hot water extraction is one potential environmentally friendly technology for the extraction task. Hot water extraction (also known as autohydrolysis, hydrothermal treatment, and water prehydrolysis) is a process in which wood material is treated in water under elevated temperature and pressure. Hemicelluloses degrade in the process and form polymers with varying degree of polymerization (DP), monomers, and degradation products of monomers. The pH lowers to a range of 3−4 in the process due to hydrogen ions generated by autoionization of water and dissociation of acetic acid released from hemicellulose.2 Much effort has been aimed toward understanding the phenomena related to hot water extraction of spruce wood in © 2012 American Chemical Society
Received: Revised: Accepted: Published: 10338
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formed from the breakage with the probability of formation of an L sized polymer:
hydrolysis of dextran, and an analytical solution was given for the case of random hydrolysis. Nattorp et al.17 developed a model to describe the thermal hydrolysis of mannan at temperatures from 160 to 220 °C in buffered pH of 3.8−4.0. They showed that the formation of oligosaccharides during thermal hydrolysis of mannan can be modeled independent of the initial distribution. Tayal and Khan18 used the model of Basedow et al. to study the enzymatic hydrolysis and sonification of guar galactomannan. The evolution of the MWD during the degradation polysaccharides has also been studied by means of Monte Carlo method.15,16 In this work, the acid catalyzed hydrolysis kinetics of the glycosidic bond of GGM is studied at temperatures from 150 to 170 °C and in buffered pH of 3.8−4.2. A mathematical model based on population balances is developed to describe the evolution of MWD of GGM. A high accuracy numerical method of Alopaeus et al.19 is applied for solving the population balance equations. This numerical method makes our model capable for predicting accurately the evolution of MWD with low computational load. With other similar type of degradation models either the computational speed or the accuracy suffers when polysaccharides with wide MWD are considered. The kinetic parameters for the hydrolysis of glycosidic bonds in GGM were optimized by using the number and weight average molecular weight data analyzed with HPSEC-MALLS. Simulated MWDs were compared against the measured MWDs. Two mechanisms were tested for describing the degradation of GGM. First, it was assumed that the degradation is a random process and all the glycosidic bonds in GGM hydrolyze at the same rate. Then, it was tested, if the process could be modeled more accurately if some of the bonds hydrolyze with faster rate. The hydrolysis rates depend on the hydrogen ion concentration which is estimated at reaction temperatures. The kinetic parameters and the overall approach for modeling the changes in MWD will be used in the model for hot water extraction of wood. The model will take into account reaction kinetics, mass transfer, and thermodynamic aspects using general physicochemical laws. To enhance the identification of the reaction kinetic parameters, data from simple, one liquid phase, experimental setup was utilized. Thus, mass transfer and difficulties in characterization of MWD of GGM in wood, for instance, can be ruled out.
β(L , Lj) =
g (L) = h[H+](L − 1)
(3)
where h is the hydrolysis rate parameter. In the first case of parameter optimization, it is assumed that the hydrolysis rate is the same for all bonds in the polymer, thus the hydrolysis rate parameter is constant (h = k). In the second case, it is assumed that some of the randomly located bonds in the polymer hydrolyze at different rate and h is obtained using the following equation: h = αk fast + (1 − α)kslow
(4)
where α is the fraction of fast reacting bonds in the polymer and kfast and kslow are the reaction rate constants for the hydrolysis of fast and slow reacting bonds. The rate of change for the fraction of fast reacting bonds can be derived as follows. The total number of bonds in a polymer is the sum of fast and slow reacting bonds: ntot = n fast + nslow (5) The fraction of fast reacting bonds is n α = fast ntot
(6)
The rate of change for the number of fast and slow reacting bonds is dn fast = −k fastn fast , dt
dnslow = −kslownslow dt
(7)
From eqs 5−7 it follows that the rate of change for the total number of bonds is dntot = −k fastn fast − kslownslow dt
MODEL DEVELOPMENT A continuous population balance equation describing the evolution of MWD of polymers can be adopted from ref 20, where model for breakage in dispersed phase systems has been developed:
∫L
(2)
The degradation of GGM is assumed to be random process and the mechanism for the hydrolysis of glycosidic bonds in GGM is in this work assumed to be the same as in more acidic conditions although some controversial evidence can be found in the literature.21 The hydrolysis rate depends on the hydrogen ion concentration and number of bonds:
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dn(L) = −g (L) n(L) + dt
2 Lj
= −(k fastα + kslow(1 − α))ntot
(8)
Now the rate of change for the fraction of fast reacting bonds is obtained from eqs 7 and 8: dα d ⎛n ⎞ = ⎜ fast ⎟ dt dt ⎝ ntot ⎠
∞
β(L , Lj) g (Lj) n(Lj) dLj (1)
=
where n(L) is the number of polymers with a size L, β(L,Lj) is the daughter size distribution for the breakage of larger polymers Lj forming smaller L sized polymers, and g(Lj) is the chain breakage (hydrolysis) rate of polymers of size Lj. The model is formulated using polymer chain lengths and the degree of polymerization (DP) is used as a size variable L. The model gives the chain length distribution, but the term MWD is used instead for simplicity. The daughter size distribution β(L,Lj) is the product of multiplying the number of polymers
ntot
dn fast dt
− n fast
dn tot dt
ntot 2
= k fast(α 2 − α) + kslow( −α 2 + α)
(9)
Arrhenius temperature dependence is assumed for reaction rate constants. To reduce the correlation between activation energy and frequency factor, the following form of Arhenius equation is used in the parameter optimization:22 k = A e−Ea /(RTtr)e−Ea /(RTave) = kavee−Ea /(RTtr) 10339
(10)
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where Ttr is the transposed temperature, Tave is the average temperature of the experiments, and kave is the reaction rate constant in average temperature. The values for kave and Ea are obtained in the parameter optimization. Ttr is calculated from the following equation: 1 1 1 = − Ttr T Tave
μβi = [A](βi (vi, Lj)ΔLi)
where [A] is a matrix and vi is a vector that describes in which categories moments are distributed from breakage into category i. The vector vi is obtained from equations below: ⎧ ⎛ NM ⎞ ⎟, ⎪ vi(n) = i − floor⎝⎜ 2 ⎠ ⎪ ⎪ ⎨ vi(n) = vi(1) + (n − 1), ⎪ ⎪ n = 2, ..., (NM − 1) ⎪ v (n) = v (1) + NM − 1, ⎩ i i
(11)
When the numerical method of Alopaeus et al.19 is used for solving the continuous population balance equation, the MWD is divided into chosen number of categories. Representation of the continuous MWD can then be recovered using categories so that the eq 1 can be written as dYi = − g (Li)Yi + dt
∑ (β(vi , Lj) ΔLig(Lj))Yj (12)
where NC is the number of categories, and Yi is the concentration (mol/dm3) of polymers belonging to category i. In the model, 14 size categories were chosen to represent the MWD of GGM. Increasing the number of categories increased the accuracy of the representation of the measured MWD, but did not change the values of kinetic parameters. Lowering the number of categories had a slight effect on the values of kinetic parameters and the accuracy of the representation of the measured MWD was also reduced. Thus, the 14 size categories were found to be suitable for describing the degradation kinetics and the MWD in the investigated case. All polymers belonging to a certain category Li have the same DP representing the characteristic value of the category i. The following discretization was used to emphasize computational accuracy for concentration of polymers with low DPs:
ai , j = Lvmi((nn)) ,
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RESULTS AND DISCUSSION The unpublished experimental results in this work are weight average molecular weight results in 150 and 160 °C for all the three pH levels and number average molecular weight results for all the three temperatures and pH levels. In addition, the data of the entire MWD of the untreated GGM and MWD analyzed after treating GGM at 150 °C and pH of 3.8 for 30 and 90 min are reported. The weight average molecular weight results from treating GGM in 170 °C and in pH of 3.8, 4.0, and 4.2 have been published earlier.3 The experimental setup described by Song et al.3 was simulated using a batch reactor model. The numerical solution of the model equations was obtained using an initial value ordinary differential equation solver DASLL.23 The ODEs for the concentrations of polymers are shown in eq 12. The acid− base equilibrium of the phthalate buffer and water was computed to obtain the hydrogen ion concentration at the reaction temperatures. Table 1 summarizes the pKa values for the acid−base equilibrium reactions at 25 °C and at the reaction temperatures. The hydrogen ion concentrations at 25
(14)
ΔLi ΔLi , Li + = Li + (15) 2 2 19 The numerical method of Alopaeus et al. uses moments of MWD for solving the eq 12. The moments for breakage from category j into category i are calculated from the following equation:
β(Li , Lj)Lim(k) dLi
(20)
EXPERIMENTAL SECTION High molecular weight GGM, starting material for kinetic experiments, was obtained from spruce ground wood with particle size smaller than 1 mm. Ground wood was treated in accelerated solvent extractor (ASE-300) with water at 170 °C for 20 min. Polysaccharides were precipitated out of solution by the addition of ethanol into water. The precipitate was separated, washed with ethanol, acetone, and MTBE, and vacuum-dried. The isolated water-soluble high molecular weight GGM was treated in temperatures of 150, 160, and 170 °C using three pH levels of 3.8, 4.0, and 4.2. The pH levels were kept constant using potassium hydrogen phthalate buffer. The experimental setup, procedures, and materials used in the experiments have been described earlier by Song et al.3 The MWD was analyzed from each sample once using HPSECMALLS.
Li − = Li −
i−
(19)
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The category width ΔL and minimum Li‑ and maximum Li+ limits of the categories were calculated using the following equations:
Li +
(18)
where ΔLi is the size of category i.
(13)
∫L
n = 1, 2, .., NM
(βi (vi , Lj)ΔLi) = [A]−1 μβi
= 1, 2, 3, 5, 8, 12, 18, 26, 38, 55, 80, 116, 168, 242
μβki =
n = NM
With the following matrix inversion the contribution of breakage from category j into category i can be solved:
L1...14 = DP1...14
⎧ ΔL1 = 1 ⎪ ⎪ ⎛ ⎞ ⎨ ΔL = 2 × ⎜L − ⎛⎜L + ΔLi − 1 ⎞⎟⎟ , i i i−1 ⎪ ⎝ 2 ⎠⎠ ⎝ ⎪ ⎩ i = 2...14
n=1
where NM is the number of moments set to be conserved and the function floor() rounds toward the smaller integer number. The matrix elements ai,j for matrix [A] can be calculated from following equation:
NC j=1
(17)
(16)
where m(k) is the kth element of vector which describes order of the moments set to be conserved and μβki is the moment of category i which corresponds to the order of moment m(k). The term β(vi,Lj)ΔLI in eq 12 refers to the contribution of breakage from category j into category i. This term is solved using the following linear transformation between the moments and table elements: 10340
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Table 1. pKa Values of Equilibrium Reactions at 25 °C and at the Reaction Temperatures
between molecular weights log M and log M + d(log M) and molecular weight. The initial differential mass fractions of the first five size categories (DP < 8) are zero, and they are not shown in Figure 1. The Arrhenius equation (eq 10) parameters Ea and kave were first optimized for the reaction rate constant k. In the second case, Ea and kave were optimized for the reaction rate constants kfast and kslow, and the initial fraction of fast reacting bonds in GGM α0 was also optimized. The relative differences between measured and simulated values of number and weight average molecular weights were squared, and the squares were summed. The sum of squared errors was the target function used in the parameter optimization. The target function was minimized with KINFIT26 software using the Levenberg−Marguardt algorithm.27 The confidence intervals and the cross-correlations for the optimized parameters were also obtained. Parameter Optimization with a Constant Hydrolysis Rate Parameter (h = k). Figure 2 shows the evolution of number and weight average molecular weight (Mn and Mw) for the pH levels 3.8, 4.0, and 4.2 for the case were the hydrolysis rate is assumed to be the same for all the glycosidic bonds in GGM. The optimized parameters are reported in Table 3, and their values are discussed later. Figure 2 shows that the developed model describes the degradation of GGM quite well with the single reaction rate constant k. The largest deviation between the measured and predicted values is during the first 10 mins of the reaction. The measured values of Mn and Mw decrease rapidly in the beginning after which the degradation rate slows down significantly. It seems that there are two reaction stages in the degradation of GGM: a fast start and a slower second stage and it is evident that more parameters are needed in the model to describe both of the stages accurately. An explanation for the two reaction stages could be that there are two types of glycosidic bonds present in the GGM that are cleaved with different reaction rates. Timell9 studied the acid hydrolysis of disaccharides that have similar bonds as GGM and xylan. He discovered that the glycosidic bond between two mannose units (mannobiose) breaks more easily than the bond between mannose and glucose units (glucosyl mannose). The behavior can be assumed to be analogous for linear polysaccharides so that the mannose−mannose bond in the GGM hydrolysizes more rapidly than the mannose−glucose bond. The xylans and pectins that the isolated GGM contained as impurities could have also affected the degradation patterns. The bond between two xylose units is known to break more easily than the bond between two mannose units.9 With the same analogy as used above, it can be assumed that xylan hydrolysizes more rapidly than GGM. The developed model predicts also the evolution of the MWD. Figure 3 shows the measured and predicted MWD for a pH of 3.8 and a temperature of 150 °C. Figure 3 shows that there is some deviation between the measured and predicted MWD. After 30 min of reaction time, the model underpredicts the amount of degraded polymers and after 90 min reaction time, the model seems to overpredict the amount of polymers that have molecular weights between 3000 g/mol and 7000 g/mol. The same observations apply for all the temperatures and pH levels. The reason for this systematic deviation could be the two reaction stages discussed earlier. Parameter Optimization with Hydrolysis Rate Parameter h = αkfast + (1 − α)kslow. The observed two reaction
pKa values at the temperatures reaction stoichiometry 1a
+
−
H 2O ↔ H + HO
25 °C
150 °C
160 °C
170 °C
13.99
11.64
11.56
11.48
b
H 2Ph ↔ H+ + HPh−
2.95
3.04
3.05
3.06
b
HPh− ↔ H+ + Ph2 −
5.40
5.47
5.48
5.49
2 3 a
(1) The values were obtained based on the work of Shock and Helgeson.24 b(2 and 3) The values were calculated using equations: Ka = exp{(ΔH0 − TΔS0)/(RT)} and pKa = −log10(Ka). The ΔH0 and ΔS0 values used in the calculations were reported by Gagliardi et al.25 for the temperature range of 15−50 °C.
°C and at the reaction temperatures were computed using the pKa values and they are shown in Table 2. Table 2. The Estimated Hydrogen Ion Concentrations at 25 °C and at Reaction Temperatures estimated [H+] (mol/dm3) at the temperatures pH at 25 °C
25 °C
150 °C
160 °C
170 °C
3.8 4.0 4.2
1.58 × 10−4 1.00 × 10−4 6.31 × 10−5
1.25 × 10−4 7.83 × 10−5 4.91 × 10−5
1.23 × 10−4 7.72 × 10−5 4.85 × 10−5
1.21 × 10−4 7.62 × 10−5 4.78 × 10−5
The initial MWD for the parameter optimization was obtained through discretization of the measured initial MWD of the GGM into 14 size categories (eq 13) so that zeroeth, first, and second moment of the measured distribution were conserved. The same three moments were conserved in the parameter optimization. The moments μk of the discretized distribution can be calculated using the following equation: NC
μk =
∑ niMik i=1
(21)
where k refers to the order of the moment and M is the molecular weight of polymers in category i. The molecular weight of each monomer unit in a polymer chain was assumed to have the same molecular weight as mannose (180.16 g/mol). The measured and discretized initial MWDs are presented in Figure 1 as a plot of the differential mass fraction eluting
Figure 1. The discretization of the measured initial MWD into size categories. Solid line presents the measured initial distribution. The dots present the categories and dashed line present the MWD of size categories. 10341
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Figure 2. The evolution of number and weight average molecular weight of GGM when h = k. The lines represent the predicted values and the dots represent the measured values. Relative errors are shown for the measured values. The symbols refer to (○) initial material and temperature of the experiments: (■) 150, (□) 160, and (Δ) 170 °C.
Table 3. The Optimized Activation Energies and Frequency Factors and Reaction Rate Constants at 165 °C and the Initial Fraction of Fast Reacting Bonds and Residual Sums of Squares for Parameter Optimization reaction rate constant
value at 165 °C (dm3/(s·mol))
Ea (kJ/mol)
A (dm3/(s·mol))
α0
RSS
k kfast kslow
0.27 ± 0.03 45 ± 63 0.20 ± 0.04
102 ± 20 100 ± 135 118 ± 27
3.74 × 1011 4.09 × 1013 2.30 × 1013
0.01 ± 0.008
2.6 1.7
Figure 3. Measured and predicted MWD for pH 3.8 and temperature 150 °C at time steps 30 and 90 min when the hydrolysis rate parameter is constant (h = k). The solid line represents the measured MWD. The dots represent the categories, and dashed line represents the predicted MWD of size categories.
Figure 4. The evolution of weight average molecular weight of GGM when h = αkfast + (1 − α)kslow. The lines represent the predicted values and the dots represent the measured values. Relative errors are shown for the measured values. The symbols refer to (○) initial material and temperature of the experiments: (■) 150, (□) 160, and (Δ) 170 °C.
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stages in the degradation of GGM were taken into account in further parameter optimization by using eq 4, and the results are shown in Figure 4. Figure 4 shows that the model predicts the degradation of GGM with enhanced accuracy when the two separate reaction rates for the hydrolysis of bonds are used. The optimized activation energies Ea and reaction rate constants at 165 °C are listed in Table 3 with their 95% confidence intervals. The temperature 165 °C was the average temperature of the experiments used in the parameter optimization. The frequency factors A calculated from the Arrhenius equation (eq 10) and the initial fraction of fast reacting bonds α0 with its 95% confidence interval and the residual sums of squares (RSS) for the optimization of the kinetic parameters are also shown in Table 3. Even though the model with hydrolysis rate parameter h = αkfast + (1 − α)kslow gives better fit for the experimental data, it cannot be considered as an improvement to the simpler model with a constant hydrolysis rate parameter. The confidence intervals of α0, kfast, and its activation energy are so wide that it is clear that they were not truly identified in the optimization process. The reason for the wide confidence intervals can be that there are too many parameters that are to be optimized with too low amount of measurement data, or that the model itself is inadequate to describe the phenomenon. The crosscorrelation matrix shown in Table 4 was checked to see
constants instead of the values reported at 25 °C. The model presented by Nattorp et al. aims at modeling the change in oligosaccharide concentrations and extracting the degradation kinetics regardless of the initial MWD. Our model requires the initial MWD, but has the advantage that it is capable of describing the evolution of MWD of polysaccharides that have wide MWD in addition to oligosaccharides. In the modeling approach used by Nattorp et al., the evolution of concentrations of every single chain length is computed and this makes their model too slow for describing the evolution of the MWD of polysaccharides like cellulose. A comparison was also made between the optimized activation energies and literature values reported for acid hydrolysis of disaccharides. An activation energy of 137 kJ/mol has been reported for the hydrolysis of mannobiose,9 and for cellobiose the reported values are in the range of 132−135 kJ/ mol.9,15,21 Thus, the activation energy obtained for k does not compare with these values. The model that assumes same hydrolysis rate for all bonds in hemicellulose chain will be used as a part of the comprehensive process model for hot water extraction of wood. In that model, the thermodynamics and mass transfer and all chemical reactions related to the process (hydrolysis of glycosidic bonds, deacetylation, and degradation of monosaccharides) are considered. The model will be applied to describe the evolution of MWD of other hemicelluloses and cellulose.
Table 4. the Cross-Correlation Matrix
CONCLUSIONS A model was developed for describing the degradation of linear polysaccharides. The main benefit of the model compared to the earlier work is the low computational load when wide MWD is modeled. The performance of the model was demonstrated by modeling the degradation of water-soluble high molecular weight GGM in hot water extraction conditions. The model fits quite well the evolution of the number and weight average molecular weight of the GGM in the hot water extraction conditions when all the glycosidic bonds break at the same rate. The obtained kinetic parameters were in good agreement with the earlier results obtained for the degradation of linear mannan during homogeneous thermal hydrolysis.
kfast Ea,fast kslow Ea,slow α0
kfast
Ea,fast
kslow
Ea,slow
α0
1 0.62 0.58 −0.40 −0.76
1 0.14 −0.27 −0.22
1 −0.34 −0.84
1 0.48
1
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whether the more complex model is poorly formulated and there is too large a correlation between the parameters disturbing the optimization. There is large correlation between α0 and the reaction rate constants kfast and kslow, but not large enough to disturb the optimization. A correlation coefficient higher than ±0.95 would indicate too much correlation.22 It can be concluded that the hypotheses presented earlier to explain the two reaction stages in the degradation of GGM could not be proven to be correct because the parameters α0, kfast, and Ea,fast were not identified in the optimization. Our work is closely related to the earlier work of Nattorp et al.17 They studied the thermal hydrolysis of mannan in temperatures of 160−200 °C in the pH range of 3.8−4.0 measured at 25 °C. The conditions used in their work are similar to ours and the structures of mannan and GGM differ only slightly. Their model assumes the same hydrolysis rate for all the glycosidic bonds in mannan, and they obtained an activation energy of 113 kJ/mol for the hydrolysis of the bonds. This value is close to the value of 102 kJ/mol obtained in this work. They also reported an apparent rate constant of 3.0 × 10−5 s−1 at temperature 160 °C. When our reaction rate constant k is calculated from eq 10 for the temperature 160 °C and multiplied with the [H+] in the of pH 3.8 and 4, the obtained apparent rate constants are 2.4 × 10−5 s−1 and 1.5 × 10−5 s−1. The reported and calculated apparent rate constants at 160 °C are very close to each other and might be even closer if the true pH at 160 °C was used for calculating the apparent rate
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AUTHOR INFORMATION
Corresponding Author
*E-mail: susanna.kuitunen@aalto.fi. Tel.: +358 9 470 22642. Notes
The authors declare no competing financial interest.
■ ■
ACKNOWLEDGMENTS Forestcluster, Ltd. is thanked for the funding.
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NOMENCLATURE A = frequency factor (dm3/(s·mol)) [A] = matrix A ai,j = element for matrix [A] Ea = activation energy (J/mol) g(L) = hydrolysis rate of polymers with a size L (s−1) h = hydrolysis rate parameter (dm3/(s·mol)) [H+] = hydrogen ion concentration (mol/dm3) k = reaction rate constant (dm3/(s·mol)) L = size variable; degree of polymerization is used as a size variable in the model L − 1 = number of the bonds in a polymer dx.doi.org/10.1021/ie3000826 | Ind. Eng. Chem. Res. 2012, 51, 10338−10344
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m() = vector which describes order of the moments set to be conserved M = molecular weight (g/mol) n = molar amount of component (mol) n(L) = number of polymers with a size L (mol) NM = number of moments set to be conserved R = universal gas constant, 8.314 J/(mol K) Y = concentration of polymers in a category (mol/dm3) Greek Symbols
α = fraction of fast reacting bonds in the polymer β(L,Lj) = daughter size distribution for the breakage of larger polymers Lj forming smaller L sized polymers β(vi,Lj) ΔLi = contribution of breakage from category j into category i ΔL = size of a category μ = moment of the distribution Subscripts
ave = average temperature of the experiments fast = fast reacting fraction i = category i i−, i+ = minimum and maximum of a category j = category j k = order of the moment slow = slow reacting fraction of hemicellulose tr = transposed value 0 = initial value Abbreviations
DP = degree of polymerization GGM = galactoglucomannan MWD = molecular weight distribution NC = total number of categories RSS = residual sum of squares
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dx.doi.org/10.1021/ie3000826 | Ind. Eng. Chem. Res. 2012, 51, 10338−10344
