Melt-phase nylon 612 polycondensation kinetics: effects of sodium hypophosphite catalyst.
During the manufacture of nylons, a variety of catalysts are used to reduce the reaction time required for high molecular weight products to be produced. Reducing the length of time that molten nylon is held at high temperatures increases reactor throughput and decreases the amount of undesirable thermal degradation that occurs in commercial polymerization processes (Schaffer et al., 2000). Some catalysts also serve as additives to improve product properties. Catalytic additives are used to reduce the colour of products (Cottle and Yong, 1955; Burrows and Hepworth, 1976), to make nylon more thermally stable (Stamatoff, 1955; Furukawa and Isukamoto, 1975; Burrows and Hepworth, 1976) and to inhibit branching reactions, keeping more of the molecules linear for better fibre spinning (Flory, 1941; Walker, 1951; Cocci, 1960; Sum, 1965; Aharoni et al., 1983; Coffey et al., 1984; Hofmann et al., 1984; Buzinkai et al., 1992; Cramer et al., 1995). Most catalysts for polyamidation reactions are phosphorus compounds (Genas, 1951; Sum, 1965; Wujciak, 1968; Burrows and Hepworth, 1976; Tomek, 1976; Aharoni et al., 1983; Coffey et al., 1984; Hofmann et al., 1984; Curatolo et al., 1985; Wheland, 1990; Wheland and Miller, 1990; Buzinkai et al., 1992; Wheland, 1992; You et al., 1994). This study examines the catalytic effect of sodium hypophosphite (SHP). SHP is added to polyamides because, in addition to its catalytic properties, SHP acts as a whitening agent (Lysek and Ables, 2001), as well as a thermal stabilizer and antioxidant (Agouri and Muller, 1968; Kelmchuk, 1972; Kazuhiko et al., 1995), which gives desirable end-use properties to nylon fi bres and moulded products.
SHP catalyzes a variety of esterification and amidation reactions. For example, Fourier Transform Infrared (FTIR) and High-Performance Liquid Chromatography (HPLC) studies have confirmed that: SHP catalyzes the formation of anhydrides and water from polycarboxylic acids (Yang et al., 1996; Schramm and Rinderer, 2004), an important intermediate step in the formation of ester linkages between polycarboxylic acids and cellulose; SHP catalyzes esterification reactions between anhydrides and hydroxyl groups on cellulose (Kazayawoko et al., 1997; Ibrahim and ElZawawy, 2004); and SHP catalyzes amidation reactions between carboxyl groups on citric acid cross-linker and amine groups on polyacrylamide (Save et al., 2002).
Solid-Phase Polymerization (SPP) rate experiments have shown that SHP is an effective catalyst for polyamidation, but only when the water concentration in the polymer particles is low (Dujari et al., 1998). For example, when nylon particles were contacted with moist nitrogen (a nitrogen/water vapour mixture with a dew point temperature of 50 [degrees]C, as is commonly used in SPP reactors) for 2 h at 200 [degrees]C, polymer particles containing 20 ppm SHP had similar polyamidation rates as uncatalyzed particles. However, when the same particles were contacted with dry nitrogen (a nitrogen/water vapour mixture with a dew point temperature of only -40 [degrees]C), polyamidation rates in the particles containing SHP catalyst were approximately three times higher than polyamidation rates in the uncatalyzed nylon particles. Unfortunately, there is no information in the literature concerning how SHP catalyst and water interact to influence the rate of polyamidation.
In the present work, experiments are performed using nylon 612 to develop quantitative knowledge of the joint effects of SHP and water on the melt-phase kinetics of the polyamidation reaction at high temperature and low water content. The experimental results are used to build a semi-empirical dynamic mathematical model that predicts the effects of catalyst concentration, temperature, moisture level and time on the concentrations of unreacted carboxyl and amine end groups. This type of information is important for reactor design and for optimization of operating conditions in commercial reactors. The current kinetic study and modelling work build on previous studies of uncatalyzed nylon 612 polyamidation kinetics (Schaffer et al., 2003a; Zheng et al., 2005).
Figure 1 shows a schematic diagram of the batch reactor system. The reactor itself is based on the design of high-viscosity finisher reactors described in several patents (Pinney, 1973; Iwasyk, 1978; Kendall et al., 1982; Livingston, 1983). Detailed information about the reactor system and operating procedures (including agitation, gas sparging, temperature control, and sample removal) has been presented by Schaffer et al. (2001), Zheng (2004) and Zheng et al. (2005).
At the start of each experimental run, approximately 2270 g of pre-weighed additive-free polymer pellets, supplied by DuPont Canada, were charged to the reactor. The vessel was sealed and thoroughly purged with nitrogen to remove residual oxygen. After the temperature reached the desired value (290 [degrees]C), a solution of SHP in water was added to the reactor to obtain the desired level of catalyst in the molten nylon, resulting in catalyst concentrations of 33, 112 and 249 ppm, respectively, for the three different runs. During the catalyst addition procedure, the melt temperature dropped as low as 240 [degrees]C due to evaporation of water from the added solution.
After the melt temperature returned to 290 [degrees]C, the sparge gas composition was switched to pure steam. Figure 2 shows step changes in the water content of the sparge gas during a typical dynamic experimental run. After the first sample was taken (time zero on the plots), one or two hours were allowed to elapse so that the polymer could reach equilibrium under steam. Several polymer samples were then taken to permit verification of equilibrium conditions by subsequent amine and carboxyl end-group concentration measurements. The concentration of water in the sparge gas was then reduced to 20 mol.% by introducing nitrogen into the sparge stream and lowering the water pumping rate. The polymer was then allowed to re-equilibrate while more polymer samples were taken. Pure steam conditions were then re-established and the polymer was allowed to equilibrate a third time. The concentration of water in the sparge gas was then reduced again by the addition of nitrogen (5 mol.% steam/95 mol.% [N.sub.2]) and the polymer was allowed to re-equilibrate once more. As a result, each experimental run contained four equilibrium conditions where end-group concentrations and moisture concentration remained constant, and three dynamic segments where concentrations were changing. Polymer samples were taken more frequently immediately after each change in sparge-gas composition because concentration variables changed rapidly during these time periods.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
Three experimental runs were performed at 290 [degrees]C with different SHP catalyst concentrations (33, 113 and 249 ppm by mass). To determine the SHP content of the molten polymer in each run, polymer samples were hydrolyzed, by heating in a sealed tube with concentrated hydrochloric acid for one hour at 175 [degrees]C. The phosphorous content of the resulting hydrolysate was determined using Inductive Coupled Plasma Analysis (ICP).
Experimental end-group data are plotted versus reaction time in Figures 3 to 5 for runs conducted using the three different catalyst concentrations. In these figures, lower precision in the carboxyl end-group concentration measurements is reflected by the greater degree of scatter in these data than in the amine end-group concentration data (Schaffer et al., 2003a; Zheng et l., 2005). In the initial stage of each experimental run, equilibrium was established under a steam atmosphere. When the water partial pressure, [P.sub.w], decreased, the water concentration in the melt decreased rapidly, causing the polycondensation reaction rate to become faster than the hydrolysis reaction rate, and the equilibrium shown below to shift toward the right.
---COOH + ---[H.sub.]2N [??] ---CONH--- + [H.sub.2]O
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
Hence, the concentrations of both the carboxyl and amine decreased as amide linkages were formed. Equilibrium was re-established at these conditions of lower water concentration. Next [P.sub.w] was increased and the water concentration in the melt increased back to its original level. Under these higher-water conditions, the hydrolysis reaction rate became faster than the polycondensation reaction rate, leading to a net hydrolysis of amide linkages, and an increase in the end-group concentrations until equilibrium was re-established under conditions similar to those at the beginning of the experimental run. Finally, [P.sub.w] was decreased and the water concentration decreased again to a lower level, causing the end-group concentrations to decrease as amide linkages were formed. Equilibrium was re-established at these conditions.
Data from the above experiments were fitted using the following second-order kinetic model developed by Schaffer et al. (2003a):
d[A]/dt = d[C]/dt = [-k.sub.p] ([C][A] - [L][W]/[k.sub.a] (1)
d[W]/dt = [k.sub.p] ([C][A] - [L][W]/[K.sub.a]) - [K.sub.m] ([W] - [W.sub.leq]) (2)
Equations (1) and (2) are dynamic material balances on amine end groups (A), carboxyl end groups (C) and water (W) in the molten polymer phase. For nylon 612, [L], the concentration of amide links, can be computed from:
[L] = [10.sup.6] - 115.15[C] 58.10[A] 18.02[W]/ 155.23 (3)
Note that the typical concentration units used in the nylon polymerization industry are mole equivalents per 106 g of polymer. Equation (3) uses the molar masses of carboxyl ends, amine ends, water, and amide links (which are 115.15, 58.10, 18.02 and 155.23 g mol-1, respectively) to compute [L], which is the number of moles of amide links in 106 g of polymer.
[FIGURE 5 OMITTED]
The amine and carboxyl end-group concentrations on the polymer molecules are very important, because they influence the rate of polyamidation, and because they can be used to determine the average molecular weight of the polymer that is produced using this step-growth polymerization process. Since each linear polymer chain has two ends, the number-average molar mass of the polymer can be computed from:
[M.sub.n] = mass of polymer/moles of polymer molecules = 115.15[C] + 58.10[C] + 155.23[L]/0.5([A] + [C]) (4)
[k.sub.[rho]] in Equations (1) and (2) is the forward rate constant for the polyamidation reaction. Large values of [k.sub.[rho]], which can be obtained using a good catalyst, result in fast polyamidation, and hence in the production of high-molecular-weight nylon in a short period of time.
[K.sub.a] in Equations (1) and (2) is the apparent equilibrium constant, which is equal to [k.sub.[rho]] divided by the rate constant for the reverse reaction. This apparent equilibrium constant is related to the true thermodynamic equilibrium constant, [K.sub.t], via the activity coefficients for the various species present in the molten nylon:
[K.sub.a] = [[L].sub.eq][[W].sub.eq]/[[C].sub.eq][[A].sub.eq] = [gamma]C[gamma]A/[gamma]L[gamma]W (5)
Giori and Hayes (1970) showed that the moisture content of molten nylons has a significant influence on the apparent equilibrium constant for polyamidation. Schaffer et al. (2003a) and Zheng et al. (2005) developed the following semi-empirical expression that accounts for the influence of moisture level and temperature on the ratio of the activity coefficients:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)
[K.sub.a0] is the value of the apparent equilibrium rate constant at a reference temperature, [T.sub.o], under low-moisture conditions where [[W].sub.eq] [right arrow] 0, and [[gamma].sub.W0] is the activity coefficient for water in the molten nylon under these same low-moisture conditions. For nylon 612, [[W].sub.eq], the polymer-phase water concentration at equilibrium, can be estimated from the temperature and water content of the gas phase using a correlation based on Flory-Huggins theory (Schaffer et al., 2003b):
[[W].sub.eq] = 5.55 x [10.sup.4] ([P.sub.w]/[P.sup.sat].sub.w]) exp (-9.624 + 3613/T) (7)
in which [[P.sup.sat].sub.w], the vapour pressure of pure water at the temperature of molten polymer, can be calculated from the Wagner equation (Poling et al., 2001):
ln [P.sup.sat].sub.w]/[P.sub.c] = (8)
-7.77224 (1 - T/[T.sub.c]) + 1.45684 [(T/[T.sub.c]).sup.1.5] -2.71492 [(1 - T/[T.sub.c]).sup.3] - -1 41336 [(T/[T.sub.c]).sup.6]/T[T.sub.c]
Values for the mass-transfer coefficient, [k.sub.m], in Equation (2) and equilibrium-constant parameters [alpha], [beta], [[gamma].sub.0,] [K.sub.a0] and [DELTA]H in Equation (6) are listed in Table 1. The large value of [k.sub.m], which results from vigorous mixing and sparging within the vessel, is consistent with the mass transfer of water between the gas and liquid phases being fast compared with the reaction kinetics. A reference temperature of [T.sub.0] = 549.15 K was used because it is in the middle of the temperature range in Schaffer's et al. (2003a) study of the effects of temperature on polyamidation kinetics.
In an exploratory study of the influence of reactor operating conditions on [k.sub.p], weighted non-linear least-squares estimation was used to estimate different values of [k.sub.p] (using amine and carboxyl end-group data from Figures 3 to 5 and Equation (1)) corresponding to each catalyst level and equilibrium water concentration studied in the experiments. Weighting factors used in the least-squares objective function for parameter estimation (Zheng et al., 2005) reflected the different levels of precision of the carboxyl and amine-end group measurements:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)
The resulting estimates of [k.sub.p], which are reported in Table 2 along with their 95% confidence intervals, provide information about the influence of catalyst concentration and moisture level on the polyamidation rate. During this exploratory study, we assumed that the moisture level in the molten polymer attained its new equilibrium concentration instantaneously after each change in sparge gas composition. Because of this assumption, [W] = [[W].sub.eq] could be calculated using Equation (7), and Equation (2) did not need to be solved to determine [W]. Note that the assumption of instantaneous vapour-liquid equilibrium was later relaxed when the fi nal parameter estimates (shown in Tables 3 and 4) were estimated using the full model (using km in Equation (2) to account for the rate of moisture transport to and from the polymer phase).
The [k.sub.p] estimates and confidence intervals in Table 2 indicate that [W] and [SHP] jointly influence the rate of polyamidation. For example, [k.sub.p] = 0.0389 Mg [mol.sup.-1] [h.sup.-1], which was obtained using 249 ppm of SHP and the low-moisture-content sparge gas, is 1.5 times higher than [k.sub.p] = 0.0260 Mg [mol.sup.-1] [h.sup.-1] obtained using the same catalyst concentration with steam as the sparge gas. This result, which is consistent with the findings of Dujari et al. (1998) for solid-phase polymerization of nylon 66, is important because it confirms that SHP is a more effective catalyst when the melt-phase moisture concentration is low. Therefore, removal of water from melt-phase and solid-phase reactors that use SHP catalyst benefits the polyamidation rate in two ways: low water concentrations reduce the rate of the reverse (hydrolysis) reaction, which would be the case even if [k.sub.p] did not depend on [W]; low water concentrations increase the effectiveness of SHP catalyst, resulting in a significant increase in the forward (polyamidation) reaction rate. Mathematical models that account for the joint effects of SHP and water on [k.sub.p] can be used to select economically desirable catalyst concentrations and reactor operating conditions that can achieve the desired polymer molecular weight using a reduced reactor residence time.
To further illustrate and explore the influence of SHP and water on [k.sub.p], plots of [k.sub.p] vs. water mole fraction in the gas phase are shown in Figure 6 for the three different levels of SHP investigated. These plots confirm that [k.sub.p] decreases with increasing water content, so that as the water content increases, SHP catalyst becomes less effective at increasing the polyamidation rate. The relationship between [k.sub.p] and [W] is not linear. At low water content, [k.sub.p] increases substantially with increasing catalyst concentration.
From Figure 7, it appears that [k.sub.p] increases linearly with increasing [SHP], over the water content and SHP concentration ranges studied. It also appears that [k.sub.p] vs. [SHP] curves for the three different water content levels may have converge to the same value when [SHP] = 0, indicating that [k.sub.p] is not influenced by water concentration when no catalyst is present. The validity of this assumption was tested by estimating parameters in an empirical expression of the form:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)
[[k.sup.*].sub.p0] is the rate constant at the reference temperature, [T.sub.0], when [SHP] = 0 and [W] [right arrow] 0, E is the activation energy and a, b, c and d are empirical parameters to account for the effects of catalyst and water on the rate of polyamidation. The semi-empirical expression in Equation (9) was selected after considerable thought. Note that Equation (10) collapses to a regular Arrhenius expression when [SHP] = [W] = 0. Equation (6) results in a linear effect of [SHP] on [k.sub.p] when [W] [right arrow] 0. If b is negative, the influence of SHP on [k.sub.p] becomes negligible as [W] becomes large. The c exp(d[W]) term is only important if [W], by itself with no catalyst present, has a significant influence on [k.sub.p]. If c = 0, then water content does not influence [k.sub.p], except by interaction with SHP.
Since the effects of [[k.sup.*].sub.p0], a and c in Equation (10) are highly correlated, they cannot easily be estimated at the same time. To address this problem, parameters were separated into two groups ([[k.sup.*].sub.p0], c, d, E) and (a, b). The first group of parameters was estimated using data from six uncatalyzed runs reported by Schaffer et al. (2003a) and Zheng at al. (2005) using the objective function in Equation (9) and the complete mathematical model (Equations (1) to (3), (5) to (7), (9)). Then the remaining parameters were estimated using the new data shown in Figures 3 to 5. To estimate the first parameter group, Equation (10) was simplified to Equation (11) by letting [a.sub.1] = [[k.sup.*].sub.p0] c and [a.sub.2] = d. The parameter estimation results are in Table 3.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)
[FIGURE 6 OMITTED]
Based on the parameter estimation results in Table 3, the term involving [a.sub.1] and [a.sub.2] can be removed from the model because of the large confidence intervals that contain zero. Therefore, water content does not influence the apparent rate constant significantly when there is no SHP catalyst in the reaction system, and Equation (10) can be simplified to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)
When no SHP catalyst is present, the model equations simplify to the uncatalyzed polyamidation model of Zheng et al. (2005), which is able to simulate a wide variety of dynamic polymerization experiments. Using data from uncatalyzed experiments, Zheng et al. (2005) estimated [[k.sup.*].sub.p0] = 1.91 E-02 Mg [mol.sup.-1] [h.sup.-1] and E = 45.09 kJ [mol.sup.-1], along with some of the parameters listed in Table 1. The new parameters, [[theta].sub.1] and [[theta].sub.2], which account for the effect of the SHP catalyst, were estimated using the complete set of model equations (Equations (1) to (3), (5) to (7), (9)) and the data in Figures 3 to 5, with all of the other kinetic and equilibrium parameters were held at their previous estimates (Schaffer et al., 2003a; Zheng et al., 2005). The resulting estimates of [[theta].sub.1] and [[theta].sub.2] and their 95% confidence intervals are listed in Table 4. Note that the estimates for both parameters are significantly different from zero. Simulation results using the full model and these parameter estimates, which are shown in Figures 3 to 5, indicate that the model fits the experimental data very well.
[FIGURE 7 OMITTED]
The joint effects of sodium hypophosphite (SHP) catalyst and water content on the reaction kinetics of nylon 612 polyamidation were investigated in the high-temperature and low-water-concentration regime that is encountered in the fi nal stages of industrial melt-phase polyamidation processes. Within the range of concentrations studied, sodium SHP catalyst caused a significant increase in the polyamidation rate constant, [k.sub.p]. For each steady-state moisture level studied, [k.sub.p], increased linearly with the SHP concentration. The effectiveness of the SHP catalyst decreased with increasing water concentration, with the catalyst having no significant effect at the highest water concentration studied (~0.055 mass % in the molten polymer). At the lowest water concentration studied (~0.0025 mass %) and highest SHP concentration (249 ppm by weight), the catalyst resulted in approximately a 50% increase in the polyamidation rate constant. A semi-empirical mathematical model was developed to describe the joint effects of SHP and water. This model, which simplifies to the model of Zheng et al. (2005) when no catalyst is present, can simulate the experimental data very well.
NOMENCLATURE a, b, c, d empirical parameters in Equation (10) accounting for the effects of water and SHP on the polyamidation rate constant [a.sub.1], [a.sub.2] empirical parameters in Equation (11) accounting for the effects of water on the polyamidation rate constant [A] concentration of amine end groups, mol Mg-1 [C] concentration of carboxylic acid end groups, mol Mg-1 E activation energy, kJ mol-1 [DELTA]H apparent enthalpy of polyamidation, kJ mol-1 J objective function for weighted least-squares parameter estimation [K.sub.a] apparent polyamidation equilibrium constant [K.sub.a0] apparent polyamidation equilibrium constant at T0 (276 [degrees]C) and [W].0 [K.sub.m] volumetric liquid-phase mass-transfer coefficient for nylon/water system, h-1 [K.sub.p] polyamidation rate constant, Mg mol-1 h-1 [k.sub.p0] polyamidation rate constant at reference temperature T0 (276 [degrees]C) [[k.sup.*].sub.p0] polyamidation rate constant at reference temperature T0 (276[degrees]C) and [W].0 [K.sub.t] true thermodynamic equilibrium constant for polyamidation, which is independent of the end-group and water concentrations [L] concentration of amide links, mol Mg-1 [n.sub.A], [n.sub.c] number of amine end group and carboxyl end-group measurements used to fit the model parameters [P.sub.c] critical pressure of water, 22 050 kPa [P.sub.w] partial pressure of water in the gas phase, kPa [[P.sup.sat].sub.w] vapour pressure of pure liquid water, kPa R ideal gas constant, kJ mol-1- K-1 [SHP] concentration of SHP catalyst, ppm by weight t time (h) T temperature, K [T.sub.0] reference temperature, 549.15 K [T.sub.c] critical temperature of water, 647.3 K [W] concentration of water in the molten polymer, mol [Mg.sup.-1] [[W].sub.eq] concentration of water in the molten polymer in equilibrium with the gas phase, mol [Mg.sup.-1] Greek Symbols [alpha] parameter in Equation (6) accounting for the effect of T on the apparent equilibrium constant, exp([alpha]) in Mg [mol.sup.-1] [beta] parameter in Equation (6) accounting for the effect of T and [W]eq on the apparent equilibrium constant, exp(T/T) in Mg mol-1 [[gamma].sub.i] activity coefficient of species i in the molten polymer phase [[gamma].sub.w] activity coefficient for water dissolved in molten polymer at reactor temperature [[gamma].sub.w0] activity coefficient for water dissolved in molten polymer at reference temperature [[theta].sub.1] empirical parameter in Equation (12) accounting for the effect of SHP and water on the polyamidation rate constant, ppm-1 [[theta].sub.2] empirical parameter in Equation (12) accounting for the effect of water and SHP on polyamidation rate constant, Mg [mol.sup.-1] [[[sigma].sup.2] variances of amine end and carboxyl end .sub.A], measurements , respectively, [mol.sup.2] [[[sigma].sup.2] [Mg.sup.-2] .sub.C] Subscript eq equilibrium concentrations (mol [Mg.sup.-1])
The authors are grateful for the financial support provided by NSERC, DuPont Canada and Queen's University, and for the end-group analyses that were conducted by DuPont Canada.
Manuscript received December 23, 2005; revised manuscript received June 27, 2006; accepted for publication September 13, 2006.
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Wei Zheng (4), Kim B. McAuley (1) *, E. Keith Marchildon (2) and K. Zhen Yao (3)
(1.) Department of Chemical Engineering, Queen's University, Kingston, ON, Canada K7L 3N6
(2.) Current address: E.I. DuPont Canada Company, Research, Engineering and Business Development, Kingston, On, Canada K7L 5A5
(3.) Current address: Institute of Polymerization Reaction Engineering, Zhejiang University, Hangzhou, China 310027
(4.) Iogen Corporation, 310 Hunt Club Road East, Ottawa, ON, Canada K1V 1C1
* Author to whom correspondence may be addressed. E-mail address: firstname.lastname@example.org
Table 1. Reaction equilibrium and mass-transfer parameters used in the model Parameter Units Value Source A exp([alpha]) in Mg/mol 22.6039 Zheng et al., 2005 B K -1.45E+04 Zheng et al., 2005 [K.sub.a0] dimensionless 53.7841 Zheng et al., 2005 [DELTA]H kJ/mol -85.752 Zheng et al., 2005 [GAMMA]. 20.97 Schaffer et al., sub.w0] 2003b [k.sub.m] [h.sup.-1] 24.3 Schaffer et al., 2003a Note: [K.sub.a0], [DELTA]H and [[GAMMA].sub.w0] are shown for the reference temperature, [T.sub.0]=549 K. Table 2. Apparent amidation rate constant (with SHP) [SHP], T, [N.sub.2]/Steam (mol %) ppm [degrees] 100% Steam 80% [N.sub.2], 20% Steam [k.sub.p] 95% C.I. [k.sub.p] 95% C.I. +/- +/- 33 290 0.0248 0.0032 0.0261 0.0040 113 290 0.0253 0.51% 2.0282 0.0085 249 290 2.60% 0.39% 0.0324 0.0079 [SHP], [N.sub.2]/Steam (mol %) ppm 95% [N.sub.2], 5% Steam [k.sub.p] 95% C.I. +/- 33 0.0298 0.0042 113 0.0341 0.0077 249 0.0389 0.0072 Table 3. Parameter estimation results for Equation (11) Parameter Estimate results 95% C.I. +/- [k*.sub.p0] 1.84E-02 2.40E-03 [a.sub.1] 1.36 6.26E+02 [a.sub.2] -1.41 118.1 E 53.2 37.2 Table 4. Final parameters estimation results Parameter Units Estimate results 95% C.I. +/- [[theta].sub.1] 1/ppm 3.59E-03 1.84E-03 [[theta].sub.2] Mg/mol -8.17E-02 2.75E-02
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|Author:||Zheng, Wei; McAuley, Kim B.; Marchildon, E. Keith; Yao, K. Zhen|
|Publication:||Canadian Journal of Chemical Engineering|
|Date:||Apr 1, 2007|
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