1050
Ind. Eng. Chem. Res. 1988,27, 1050-1058
A Study of Zinc Sulfate Decomposition at Low Heating Rates Ravi Narayan, Ali Tabatabaie-Raissi,t and Michael Jerry Antal, Jr.* Renewable Resources Research Laboratory, Department of Mechanical Engineering, University of Hawaii at Manoa, Honolulu, Hawaii 96822
Studies of the thermal decomposition of zinc sulfate in a conventional thermogravimetric analyzer reveal the critical influence of various experimental parameters on the course of decomposition. These parameters include the sample size, boat material, water of hydration, and carrier gas composition. Competing reactions consume the ZnS04 substrate, forming either the high-temperature ZnS04 p phase, or an intermediate oxysulfate, whose probable composition is Zn0.2ZnS04. The oxysulfate decomposition is a pseudo-zero-order reaction, while its formation can be modeled as a first-order reaction. Kinetic parameters for the reaction network are derived by using a nonlinear least-squares optimization program. Values of these kinetic parameters for various experimental conditions all evidence the compensation effect. Thermochemical water-splitting cycles which produce hydrogen are a prospective application of solar energy (Bowman, 1984). The decomposition of zinc sulfate is the high-temperature reaction step in many such thermochemical cycles (Hosmer and Krikorian, 1979; Krikorian and Shell, 1982; Shell et al., 1983). In various sulfuric acid based cycles, the decomposition of zinc sulfate can also be used to replace reactions involving the decomposition of sulfuric acid vapor (Shell et al., 19831, thereby reducing materials and corrosion problems and improving the cycle efficiency. Zinc sulfate decomposition is also an important reaction in the production of zinc from sphalerite (ZnS), which accounts for 90% of the zinc produced in the world today (Kirk-Othmer, Encyclopedia of Chemical Technology, 1984). The kinetics of zinc sulfate decomposition have been studied by many researchers using both isothermal and nonisothermal thermogravimetric techniques. The decomposition temperature, kinetics, and (possibly) reaction mechanism, as well as the formation of intermediate compounds, depend on many parameters. These include sample properties (such as particle size, density, and moisture content); shape, size, and material of the sample holder (“boat”);heating rate; flow rate; and type of carrier gas used. Often these parameters were not well-defined in earlier studies. Consequently, much disagreement exists in the literature concerning the values of the kinetic parameters for the decomposition process. Because of the difficulties encountered by earlier workers, this study gave special emphasis to details which are often presumed to be unimportant. As described in the following sections, our first step was to identify the experimental variables which affect the decomposition process. To assess the validity of our results, the accuracy and precision of the thermogravimetric temperature and weight measurements were determined. Friedman signatures (Antal, 1983) were employed to analyze the experimental data and identify suitable reaction pathways. When these pathways were incorporated into a model for the decomposition process, a nonlinear least-squares optimization algorithm was used to determine values of the kinetic parameters which best fit the data. The error bars on these parameters were also evaluated. The low heating rate results presented here form the basis for future research at high heating rates, which is of considerable interest to the solar community since it would involve the use of concentrated solar energy at temperatures to 1700 K and radiant flux densities above 100 Present address: Florida Solar Energy Center, 300 State Road 401, Cape Canaveral, FL 32920. 0888-5885/88/ 2627-1050$01.50/0
W/cm2. Researchers at the Lawrence Livermore Laboratory proposed that rapid heating of zinc sulfate could accelerate its decomposition by avoiding the formation of a refractory intermediate. Since radiant heating is an effective way of rapidly heating solid particles, this method may also increase the overall efficiency of the process. The results of this low heating rate study will be used to simulate the decomposition behavior of zinc sulfate at high heating rates. Future papers will compare these simulations with experimental results obtained in the radiative environment of an arc image furnace, which simulates a solar furnace.
Prior Work Recent studies on the decomposition of inorganic sulfates, including zinc sulfate, have been carried out at the Lawrence Livermore Laboratory (Shell et al., 1983), The University of Pennsylvania (Mu and Perlmutter, 1981), the University of Houston (Ibanez et al., 1984), CNRS in France (Ducarroir et al., 1982), and the Yokohama National University in Japan (Tagawa, 1984). Zinc sulfate exists at room temperature as a stable heptahydrate (ZnSo4.7H20). Mu and Perlmutter (1981) established the presence of two other stable hydrates at higher temperatures, ZnSO4.4Hz0 and ZnS04.H20. The decomposition of zinc sulfate ultimately results in zinc oxide (ZnO) production and involves the formation of one or more intermediates for which various compositions have been reported in the literature. Ingraham and Kellogg (1963), Krikorian and Shell (19821, and Ibanez et al. (1984) all concluded that the most likely composition for the intermediate oxysulfate was Zn0.2ZnS04. On the basis of a thermodynamic study using phase diagrams, Ingraham and Kellogg (1963) suggested that the possibility of two different oxysulfates of zinc existing was unlikely and other reported compositions were probably erroneous. In addition to the formation of an intermediate, zinc sulfate undergoes a solid-state phase transition from the low-temperature cy phase to the high-temperature P phase (Krikorian and Shell, 1982; Hosmer and Krikorian, 1980) at about 1015 K. This transformation from a close-packed orthorhombic structure to a cubic high-cristobalite-type structure is accompanied by a 28% volume expansion. Earlier workers have characterized the decomposition behavior by measuring the “initial decomposition temperature”. Values of this temperature vary widely. Apart from inconsistencies in experimental procedures, the initial decomposition temperature is not a well-defined point because it is a function of the heating rate as well as the extent of decomposition used in its definition; consequently, it should not be used as a rigorous measure 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 1051 Table I. Comparison of Kinetic Parameters Reported by Earlier Investigators kinetic parameters (E in kJ/mol) for researcher ZnS04 Zn0.2ZnS04 Ibanez et al., 1984 E = 306 i 25 E = 320 i 34 1nA = 20 2 1nA = 21 i 4 zero order zero order Ducarroir et al., 1982 E = 150 E = 197 Mu and Perlmutter, 1981 E = 353 In A = 37.7 order = 2/3 Kolta and Askar, 1975 E = 96 ~~
*
of solid-phase decomposition behavior. As shown in Table I, values of the kinetic parameters reported by earlier investigators differ markedly. These differences may be attributed to the range of experimental parameters employed by them. Later in this paper, we show that sample size exerts a critical influence on kinetic data. Mu and Perlmutter (1981) reported studies of the decomposition of 10 different inorganic sulfates in nitrogen. Using samples between 5 and 20 mg, they found that sample size did not affect the decomposition behavior. Larger samples were used by other investigators. Ibanez et al. (1984) used samples between 800 and 1600 mg, while Tagawa (1984) and Ducarroir et al. (1982) used sample masses of 100-300 mg. Another important experimental consideration is the type and flow rate of the carrier gas. Ducarroir et al. (1982) found an empirical relationship between the nitrogen gas flow rate and time of completion of reaction for their isothermal experiments with sulfates of magnesium, zinc, and nickel. Kolta and Askar (1975) reported "decomposition temperatures" in air of six metal sulfates (including zinc sulfate). Their values differed greatly from earlier literature values. Mu and Perlmutter (1981) carried out experiments using air as the carrier gas and compared them to their earlier experiments in nitrogen. They found that air had no effect on the decomposition of zinc sulfate. On the other hand, Tagawa (1984) carried out experiments on 16 metal sulfate hydrates and observed that the decomposition of these sulfates was delayed in air as compared to nitrogen. The material of the sample boat is an important experimental parameter which is often overlooked. Platinum boats were used by Ducarroir et al. (1982) and Tagawa (1984), whereas Ibanez et al. (1984) used a quartz vial. Hildenbrand (1978, 1979, 1980) showed that the decomposition of zinc sulfate and zinc oxysulfate could be catalyzed by the metals Pt, Ru, and Rh, as well as small amounts of certain transition metal oxides (such as Fe,O,). Platinum was found to catalyze the decomposition at temperatures as low as 530 "C.
Table I1 summarizes the experimental conditions employed by earlier investigators. Differences displayed in Table I1 could easily account for the lack of agreement of kinetic data displayed in Table I. Clearly, a careful study of the decomposition behavior of zinc sulfate requires well-defined experimental conditions. Instrument Characterization Isothermal studies of solid-phase decomposition reactions are never truly isothermal because it is not possible to establish an isothermal condition before a substantial degree of reaction has occurred in the solid. Hence, dynamic kinetic studies are usually executed, where the solid reactants are assumed to follow the programmed temperature closely. The thermal analysis instrument used in this work was a Setaram simultaneous TGA/DTA system. This unit is capable of operating at temperatures to 1000 "C and heating rates to 20 "C/min. The Setaram unit employs specially designed and fabricated thermocouples in direct contact with the sample and reference boats. This design minimizes errors in the sample temperature measurement, which are often present in commercial TGA instruments. The boats and sample were weighed on a Mettler H51AR balance prior to the experiment. The system was purged with about 5 times its volume of carrier gas prior to an experiment, and a steady downward flow of gas (55 mL/min) was maintained during the experiment. The samples used were zinc sulfate heptahydrate (Baker Analyzed Reagent grade) and an anhydrous zinc sulfate prepared by a technique described by Shell et al. (1983). A. Accuracy of the Temperature Measurement. The calibration of the type S sample thermocouple was verified by using several ICTA certified Curie point reference materials issued by the National Bureau of Standards (SRM GM-761). These reference materials (including nickel and the alloys Permanom 3, Mumetal, and Trafoperm) were placed in alumina boats and covered with high-purity aluminum oxide powder to simulate sample conditions. Helium was used as the carrier gas at a flow rate of approximately 60 mL/min. Two different heating rates (1 and 5 "C/min) were used. In the presence of a magnetic field, these materials provided detectable change in the apparent weight at the temperature of demagnetization. Table I11 compares this reported temperature with values provided by NBS. These values give a measure of the instrumental error under dynamic temperature conditions. The above experiments do not account for heat- and mass-transfer effects on sample temperature measurements under actual reaction conditions. Several experiments were conducted using samples of anhydrous zinc sulfate wherein the actual sample temperature was mea-
Table 11. ComDarison of the ExDerimental Conditions Used by Earlier Investigators Tagawa Ducarroir et al. Ibanez et al. Mu and Perlmutter researcher 1982 1983 1981 1984 year Mettler H54.-AR ATS3110 Du Pont TGA Cahn TGA Kanthal instrument furnace furnace ZnS0,.7H20 dehydrated ZnS04.7H20 dehydrated sample type none 12 h/450 "C at 400 OC preparation none powder ground to 300 mesh powder compacts powder sample form 80-160 mg 10-20 mg 50-300 mg sample mass 120 mg p1atinum quartz boat material platinum nitrogen N2/argon none N2/air carrier gas isothermal 1, 1.5, 5 6 heating rate, 2 and 5 OC/min a
-: indicates no data were available.
Kolta and Askar 1975 Gebruder-Netzch-Selb G.F.R dehydrated at 400 OC -
-
-
1052 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 Table 111. Accuracy of the Temperature Measurement" source ICTA SET.TGAb SET.TGA ICTA SET.TGA SET.TGA
rate, OC/min 1-2, heating 1, heating 5, heating 1-2, cooling 1, cooling 5, cooling
Permanorm 3 253.3 f 5.3 246 242 266.4 f 6.2 256 257
demannetization temm "C nickel Mumetal 351.4 356 354.4 356 358
* 4.8
5.4
377.4 f 6.3 384 382 385.9 f 7.2 391 393
Trafoperm 749.5 f 10.9 746 746 754.3 f 11.0 756 758
" Standard deviations (OC) . , for eiven mean values from ICTA Certified Reference Materials for Thermogravimetry, Certificate. * SET.TGA I
= SETARAM TGA.
sured by using a type K thermocouple embedded within the sample. The sample was heated to 970 "C under two widely differing conditions. In the first case, a 52.0-mg sample was heated at a nominal heating rate of 17 "C/min, whereas in the second experiment the sample size and its nominal heating rate were reduced to 12.2 mg and 5 "C/min, respectively. The results of these experiments are summarized in Table IV. The effect of heat transfer is pronounced for larger sample sizes (52 mg) and higher heating rates (17 "C/min). The decrease in the difference between actual and reported sample temperatures at higher temperatures can be explained by enhanced heat transfer due to radiation. These results lead us to conclude that limiting values of sample sizes and heating rates should be determined at the outset of any thermogravimetric study to ensure that heat- and mass-transfer intrusions are minimized. All experiments following these were performed at heating rates less than 5 OC/min and sample sizes smaller than 15 mg, resulting in intraparticle sample temperature gradients below 3 "C (Table IV). B. Accuracy and Calibration of the Weight Measurement. Since weight measurements on a TGA are made relative to the initial sample mass, it is not meaningful to define the accuracy of the weight measurement on an absolute scale. Nevertheless the instrument was calibrated by using several NBS traceable weights (Rice Lake Bearings Inc., class 1 (1 g) and 1.1 (1, 10100 mg)) prior to each experiment. In all cases, the error in the recorded value of the weight (under static conditions) was within the manufacturer's stated accuracy of 0.01 mg. C. Precision and Reproducibility of the Instrument. A series of five blank runs was made on the Setaram TGA/DTA at a heating rate of 5 "C/min using alumina boats and helium as the carrier gas (flow rate of 55 mL/min) to study the reproducibility of its temperature and weight measurements. The boats were heated from room temperature to about 950 "C. The experiments were carried out in groups of three and two runs. Experiments within each group were performed consecutively without altering any of the instrument settings. The instrument recorded an apparent weight gain of the empty sample boats due to the decreasing density of the carrier gas with increasing temperature. The measurements of temperature vs time and apparent weight vs temperature of the empty boats showed no regular pattern and the fluctuations appeared to be random. The sample standard deviation (corresponding to a 60% confidence interval based on Student's t distribution) for the weight measurement increases with increasing temperature from 0.05 to 0.15 mg. Presumably this behavior reflects fluctuations in carrier gas flow at higher temperatures. It is noted that, in spite of our use of a back-pressure flow regulator, it was a very difficult task to reduce flow fluctuations. This may be due to the use of larger sample pans in our TGA required for simultaneous DTA measurement. The temperature range emphasized in this study was
Table IV. Accuracy of the Temperature Measurement under Actual Reaction Conditions sample temp ("C) recorded by the temp of TGA thermocouple embedded TC, 12.15 mg a t "C 5 OC/min 52 mg at 17.5 "C/min 198 479 762 916 981
195.5 (-2.5)" 477.5 (-1.5) 759 (-3) 919 (+3) 981
184 (+14) 472 (-7) 759 (-3) 917.5 (+1.7) 983 (+2)
'T~emam- Tembedded* 500-900 "C. Within this range, the 95% confidence interval in the weight measurement was about 0.4 mg. To have a maximum relative error of 5%, the minimum weight of the sample, which is at the end of the experiment, should exceed 7 mg. Thus, the required weight of anhydrous zinc sulfate at the beginning of the experiment must exceed 14 mg.
Results and Discussion
A. Effects of Thermogravimetric Parameters. As mentioned earlier, inconsistent results of earlier investigators probably resulted from differing experimental conditions and sample properties. Hence, our first goal was to identify the effects of the various parameters on the decomposition behavior. The sensitivity of the results to sample size, water of hydration, boat material, and carrier gas was explored. 1. Sample Size. Experiments with anhydrous zinc sulfate in helium using ceramic boats revealed that a decrease in sample mass from 24 to 16 mg shifted the TG curve toward lower temperatures by about 20 "C. To reduce heat- and mass-transfer intrusions, it is necessary to have a small sample size, whereas to have a small relative error in weight measurement it is advantageous to have a large sample mass. About 12-14 mg of anhydrous zinc sulfate or 20-24 mg of zinc sulfate heptahydrate, forming a thin layer at the bottom of the 8-mm boats, was judged to be an optimum sample size. Because earlier investigators (Ducarroir et al., 1982; Ibanez et al., 1984) used much larger samples, their results may have been compromised by heat- and mass-transfer intrusions. 2. Water of Hydration. Weight loss curves for the decomposition of zinc sulfate (heptahydrate as well as anhydrous powder) in helium using both ceramic and platinum boats are shown in Figure 1. The anhydrate appears to decompose at a higher temperature than the heptahydrate. Since the initial mass of the anhydrate was somewhat higher than the heptahydrate, this delay may have been due to heat/mass-transfer intrusions. On the other hand, it is likely that pretreatment of the sample (dehydration in air at 380 "C)resulted in a rearrangement of the lattice imperfections, thereby providing more stable ions which necessitate higher decomposition temperatures (Brown et al., 1983).
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 1053
:
2
d 0
100
ZOO
Y)O
400
530
600
700
800
900
1000
Figure 1. Effect of dehydration pretreatment on the decomposition of zinc sulfate in helium a t 2.5 OC/min.
----"$
d 100
200
YIO
400
500
600
100
200
300
400
500
600
700
800
'-900
IO00
TEMPERATURE I C 1
TEMPERRTURE I C )
0
*" 0
700
800
900
1000
TEVPERATURE I C 1
Figure 2. Effect of boat-material on zinc sulfate heptahydrate decomposition in helium a t 2.5 OC/min.
3. Boat Material. Weight loss curves for zinc sulfate heptahydrate decomposition using platinum, quartz, ceramic, and gold-plated platinum boats are shown in Figure 2. Platinum boats which have the same dimensions as the ceramic boats appear to have a catalytic effect on the decomposition relative to the ceramic boats. Ingraham and Kellogg (1963) and Hildenbrand (1978,1979,1980) found that the decomposition of various metal sulfates (including ZnSOJ yields sulfur trioxide as the sole gaseous product, although sulfur dioxide should be the dominant gaseous species under equilibrium conditions. They postulated that the influence of these metals on zinc sulfate decomposition resulted from the catalytic conversion of the primary reaction product, SO3, to SO2 and 02. From Figure 1, it is seen that platinum catalyzes the decomposition of both anhydrate and heptahydrate by as much as 60 "C. This difference cannot be ascribed to error in the temperature measurement. This error was found to be less than 3 "C even for ceramic boats, which have a lower thermal conductivity than platinum. The curves for quartz and ceramic do not coincide. This may be explained by the different physical dimensions of
Figure 3. Effect of carrier gas on the decomposition of zinc sulfate heptahydrate at 2.5 OC/min.
the quartz vial. Its larger cross-sectional area (16-mm diameter compared to the 8-mm diameter of the platinum boats) would provide an easier exit for the product gases and thereby influence the reaction. In what follows, we presume that quartz or ceramic do not catalyze the decomposition of the zinc sulfate. In order to verify this assumption, the platinum boats were coated with a layer of gold (which is inert), and the results were compared with the platinum and quartz runs. Unfortunately, the goldcoated platinum boat results proved inconclusive because all of the gold (-2 mg) was lost during the course of the experiment, probably as a result of diffusion into the platinum. 4. Carrier Gas. Figure 3 compares the weight loss curves for the decomposition of zinc heptahydrate in air and helium. By the use of either platinum or ceramic boats, decomposition in air was delayed by 30-60 "C. This difference increased with increasing heating rate. These results are in agreement with the findings of Tagawa (1984) and may be the result of the higher partial pressure of oxygen in air. Since oxygen is one of the products of the decomposition (SO3dissociates to SO2and OJ, its higher partial pressure reduces the rate of dissociation of SO3 which in turn delays or shifts the decomposition of the sulfate to higher temperatures. It must be noted that platinum has a catalytic effect on the decomposition even in air. B. a-to-@Phase Transition. Figure 4 presents DTA results from experiments using anhydrous zinc sulfate at a nominal heating rate of 12.5 "C/min. The DTA trace evidences three endotherms associated with the decomposition reaction, two of which correspond to the formation and decomposition of the oxysulfate. The a-to-0 transition is accompanied by a sharp endotherm peaking at 1025 K or 752 "C. This occurs on top of a much broader endotherm associated with the formation of the oxysulfate. Similar results were obtained for various heating rates between 5 and 17.5 "C/min. In all cases, the a-to-(3 transition temperature was found to lie between 750 and 755 O C . Below 5 "C/min, the DTA did not reveal any transition. This is probably due to the fact that, at the lower heating rates, by the time the sample reached the transition temperature, very little a-phase zinc sulfate remained for conversion to the p phase. I t is important to note that this study emphasizes the decomposition of
1054 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 -.. m i
A
oxysulfite decomposition.
3 0
0
100
mo
300
5 00
730
600
80C
TEPPERAVRf
905.
1000
(C:
Figure 4. DTA curve for the decomposition of anhydrous zinc sulfate in helium a t a nominal heating rate of 12.5 "C/min. , 3
",,
\
"I
" I!
ff!z 11.94 Iq
1 1 . 7 7 nq
1 0
IOC
200
4
j
1 1 . 2 3 ng
300
400
500
600
700
800
9CO
400
y70
TENPERATURE
\
I
:2C;
TENPERATbRE : C 1
Figure 5. Effect of heating rate on the decomposition of zinc sulfate heptahydrate in helium using ceramic boats.
the a phase, which is the dominant phase at the low heating rates employed here. Krikorian and Shell (1982) speculated that the phase transformation would result in a shattering of the ZnS04crystallites, thereby influencing the decomposition kinetics. C. Formation of the Intermediate. The weight loss curves for all the experiments showed a "plateau" corresponding to the formation of an intermediate. This plateau was less pronounced when platinum boats were used, but the DTG curves confirmed the presence of two distinct peaks. In all cases, the plateau laid between 82% and 87% weight fraction. A plateau a t a weight fraction of 83.5% would be consistent with the widely accepted oxysulfate composition Zn0.2ZnS04. In order to verify the composition of the oxysulfate, eight experiments (using different boat materials and carrier gases and at different heating rates) were carried out where the decomposition was stopped at the plateau. The weight of the oxysulfate was measured and compared to the initial
600
700
mo
900
ico:
[C!
Figure 6. Effect of heating rate on the decomposition of zinc sulfate heptahydrate in helium using ceramic boats.
weight of the heptahydrate. In all the experiments, the ratio of the weight of the heptahydrate to the oxysulfate corresponded to the theoretical ratio for the composition Zn0.2ZnS04. The oxysulfate samples were collected and sent to a private laboratory for analysis. The results of this analysis by atomic absorption and combustion analysis, within their experimental accuracy, also agree with the above composition. D. Reaction Network Analysis. Figures 5 and 6 display TG and DTG curves for the decomposition of zinc sulfate heptahydrate at three different heating rates. The reproducibility of the experiments at 2.5 OC/min is noteworthy. For each heating rate, the first derivative of the degree of conversion (Figure 6) shows two separate peaks, indicating the presence of at least two single-step reactions. For the first reaction, the derivative peak heights increase with increasing heating rates, suggesting competitive reactions (Flynn, 1980). The decreasing peak heights for the second reaction are indicative of a simple, single reaction step. The model suggested by the derivative curves is that of competitive reactions followed by a single reaction step. Solid-phase thermal decompositions can often be described by the following rate law (Sestak et al., 1973; Wendlandt, 1964): rate of conversion = dc/dt = k(T)f(c) (1) where c=
(Win
- wJ
/Win
(2)
and
k ( T ) = A exp(-E/RT) (3) where E is the apparent activation energy and A is the preexponential factor. The most commonly used function, f(c), which accounts for the effect of the conversion on the rate of decomposition, is f ( c ) = (1- c)" (4) where n is the apparent order of the reaction. Combining eq 1,3, and 4 and taking the natural logarithm of both sides, Friedman (1964) showed that the resulting equation In (dc/dt) = In A - E/RT + n In (1 - c) (5) could be used to evaluate E at fixed c by plotting In (dcldt) against 1 / T for various heating rates. The Friedman
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 1055 / (I)
I
/
(11)
1'..
27 ;
Zf
\
,_,,.._.......... .........__..,,_ "......
.." z 9 1"...j._.... .,. j
:I,, 0.5
,
I:, /x"
......,..,,..-..,............*~-+:.:"
,,,,,__, ,_, ,,,,,,__, ,,
C %
,
ZnSO4(B )
, , ,, ,
,,
0.6
,
,
, 0.7
,, , , , , ,
,
,
,
,
, ,
,
,
, . , , , ,. , , , , , , , , , , , . , I
0.0
0.s
1.0
WEIGHT FRACTION
Figure 7. Friedman signatures for the decomposition of zinc sulfate a t low heating rates (1-5 OC/min).
method of kinetic analysis, which provides a graph of E vs c, is commonly employed to estimate an average value of the apparent activation energy associated with solidphase decompositions (Antalet al., 1980; Urban and Antal, 1982). More recently, Antal(1983) observed that the Friedman algorithm defines a mathematical method for mapping a set of TGA weight loss vs temperature curves into a single curve which acts as the signature of the reaction network active during the solid-phase decomposition. In his work, Antal emphasized the shape of the E vs c curves, rather than the absolute values of E calculated by the algorithm. By employing the Friedman algorithm to analyze synthetic weight loss curves generated numerically for a variety of networks, Antal was able to compile a catalog of signatures. Each reaction network studied was found to have a unique signature. This catalog of signatures can be used to identify tentative reaction networks involved in a solidphase decomposition. These signatures have been used in studies of sewage sludge (Urban and Antal, 1982), levoglucosan (Mok and Antal, 19831, Avicel cellulose (Antal et al., 1985),and Kraft lignin and ground corn cob material (Antal, 1985). Friedman signatures have also been employed to detect the onset of heat-transfer intrusions in TG studies of Avicel cellulose pyrolysis at moderate heating rates (Antal et al., 1985). Figure 7 shows the Friedman signatures for the decomposition of zinc sulfate. The similarity between the signatures for the various thermogravimetric conditions indicates that the reaction network governing the decomposition process is not altered by changing the boat material or the carrier gas. However, the reaction kinetics are greatly affected by the experimental parameters. The signatures show two distinct regions: an increasing activation energy region to a weight fraction of about 83% and a constant activation energy region between 80% and 50% weight fraction. This signature suggests competitive reactions for the formation of the oxysulfate followed by a single reaction step. This conclusion is corroborated by the first derivative data presented earlier. Based on these findings, and recalling that the formation of two different zinc oxysulfates is unlikely (Ingraham and Kellog, 1963), the following two models can be used to describe the decomposition of zinc sulfate at low heating rates:
ZnS0,
-
1 ZnO.PZnSO4 Zn0.2ZnS04
ZnS04(a)
\
/
-
ZnO
ZnO
In the case of model i, an overall oxysulfate weight fraction of 82-87% can be explained if the pathway leading to the formation of the oxysulfate is dominant at the lower heating rates employed here. E. Evaluation of Kinetic Parameters. Kinetic parameters for a simple single-step mechanism are often estimated by a routine least-squares fit of the data on a plot of In k vs 1/T. However, the transformation of the measured k and T to In k and 1/T, respectively, results in a transformation of the error bars. If the least-squares procedure does not account for these transformed error bars using a complicated weighting mechanism, values of E and In A deduced from these plots may be compromised (Cvetanovic et al., 1979). Moreover this approach is not suitable for multiple reaction networks. Meaningful values of all the kinetic parameters in a complex reaction network can be obtained by using a sophisticated nonlinear leastsquares (NLS) algorithm. Following the mathematical approach of Nowak and Deuflhard (1982), a NLS algorithm based on the MINPACK subroutine LMDER was developed to evaluate apparent kinetic parameters involved in solid-phase reaction networks. Using mass action kinetics to describe the trial reaction network, the governing ordinary differential equations (ODE) for each individual reaction in the network can be written by using eq 1 and 3 as dy/dt = f h p )
(6)
where the vector y specifies the weight fraction (or degree of conversion) of the n, species involved in the reaction network. The vector p denotes the various parameters to be evaluated (such as E, In A, and order and stoichiometry for each path of the reaction network). Assuming nkin parameters per path and npathpathways in the network, a total of np = nkinnpathparameters must be evaluated. The total weight fraction, w, which is experimentally measured by using the TGA, is the sum of the individual weight fractions of the n, reacting species:
na
dw/dt = C dyi/dt = g b , p ) i=l
(7b)
There are nl experimental measurements of the total weight fraction, wj at different times ti for each of the nb experiments performed at different heating rates. Hence, a total of m = nb nl measurements are available. The weighted least-squares norm, S(p), of the residuals is given by m
S b ) = Cs(tj)TDjs(tj)
(8)
d t j ) = w(Y(tj);P) - wj
(9)
j=1
where and Dj is a diagonal (m,m) weight matrix. The NLS problem is most efficiently solved by using the Jacobian of the objective function, S b ) . To evaluate the Jacobian, values of dw/dpi are required at each of the m data points. Values of d/api(dw/dt), which is equal to d/dt(aw/dpi), can be obtained from eq 7b. This gives rise
1056 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988
to a set of ODE'S in awlap, which have to be solved simultaneously: a/ap,(dw/dt) = d/dt(aw/ap,) =
w a p , + ]=1 ?(ag/aY,)(aY,/aP,)
(10)
Values of ayi/ap, appearing in eq 10 are unknown. These are also determined in a similar way by using eq 6 and solving another set of ODE'S: a/ap,(dy,/dt) = d/dt(ay,/ap,) =
af, / ap, + k? cafl/aYk) (aYk /ap,) =l
(11)
The above differential equations, together with those governing the time evolution of the weight fractions, y,, must be provided as input to the NLS algorithm and are specific to each reaction network. These equations were integrated by using the DGEAR algorithm in the International Mathematical and Statistical Library (IMSL). LMDER, a modification of the Levenberg-Marquardt algorithm (More, 1977), determines values of p which minimize the weighted least-squares norm S(p). An initial estimate of the kinetic parameters is provided using values taken from the Friedman signatures and results of other investigators. The weights for the least-squares norm were based on the precision of the weight measurement at the various temperatures. To estimate the confidence interval associated with the values of the parameters p, the experimental error variance 2, was set equal to S(p)/(m- np). The covariance matrix V,whose elements are the variance of the kinetic parameters, is estimated by the following equation (Come, 1983; Bevington, 1969):
v = o.~[a2~/ap,ap,1-~~2,,,
(12)
Values of the second derivative of the objective function, d2S/dpiap, are evaluated from values of the Jacobian of S(p) by using a forward difference formula. The data for oxysulfate formation were modeled by using a single reaction step as well as competitive reactions. A first-order reaction was obtained by using the single step model, and the fit was within our experimental error. The standard deviation based on an internal error estimate (Cvetanovic et al., 1979), qnt,is 0.01-0.02 compared to the value of U~ of 0.02-0.08. When competitive reactions were used to model the data, the values of the kinetic parameters for one of the steps were the same as those for the single-step model. The fit did not appreciably improve. These results can be explained by model i where the amount of @-phasezinc sulfate is probably too small to have a significant effect on the overall kinetics. This view is reinforced by the fact that the Friedman signature for the competitive reaction was more pronounced for the experiments performed using ceramic boats. Decomposition temperatures were typically higher when ceramic boats were used instead of the platinum ones. Higher decomposition temperatures would result in a higher proportion of the @ phase, thereby emphasizing the competitive nature of the reaction network. Values of the kinetic parameters for the other reaction step were inconsistent, possibly due to the small amounts of the phase present. The zinc oxysulfate decomposition was similiarly modeled by a single-reaction-step model. In this case, a zero-order reaction was obtained with a value of uextbetween 0.02 and 0.03. Values of the activation energy ( E ) ,the preexponential factor (In A ) , and the order of the reaction obtained for
Table V. Values of Kinetic Parameters from T h i s S t u d y kinetic parameters ( E in kJ/mol) for exptl conditions first step second step 1 anhydrous powder E = 264 f 21" E = 260 f 21 In A = 29 f 2.5 In A = 24 f 2.5 ceramic boats order = 1.1 f 0.2 order = 0 f 0.05 helium gas 2 hydrated crystals E = 248 f 8 E = 290 4 I n A = 27 f 1 In A = 28 f 0.5 ceramic boats order = 0.9 f 0.1 order = 0 f 0.1 helium gas 3 hydrated crystals E = 298 f 13 E = 252 f 0.5 ceramic boats In A = 32 f 1.5 In A = 23 f 0.05 air order = 0.9 f 0.1 order = 0 f 0.01 4 hydrated crystals E = 227 f 6 E = 290 f 1 In A = 24 f 1 In A = 28 f 0.1 platinum boats air order = 0.6 f 0.06 order = 0 f 0.01 5 hydrated crystals E = 243 f 4 E = 315 f 25 platinum boats In A = 27 f 1 In A = 32 3 order = 0.85 f 0.06 order = 0 f 0.1 helium gas
*
*
All deviations refer to lu limits (eq 12). 40
Bi
4 Heptahydrate/plalinum/alr
32
X
I
I
I
Oxysulfate formation Oxysulfate decomposition
I 200
240
280 320 E,Activation energy (kJimol)
360
Figure 8. Compensation effect for zinc sulfate decomposition with la limits (from eq 12).
the various experimental conditions and reaction steps are given in Table V. The values of E and In A vary considerably, which may explain the disparity in the results of earlier investigators. Figure 8 depicts a graph of E versus In A for the various experimental conditions listed in Table V. It is interesting to note that these values lie on a straight line, indicating the role of a compensation effect (Garn, 1975; Zsako, 1976) in the decomposition measurements. Figure 9 compares our experimental data to theoretical curves generated using the evaluated parameters at three different heating rates. The model used for the theoretical curves was a first-order reaction for the oxysulfate formation followed by a zero-order reaction for its decomposition. The deviations lie within the 95% confidence interval, and this excellent agreement leads us to conclude that the above model (using eq 4) can be used to satisfactorily describe zinc sulfate decomposition at low heating rates. It must be noted that the evaluation of kinetic parameters employing data at more than one heating rate is a complicated task. When the experiment involves data at only one heating rate or isothermal data at only one temperature, it is relatively easy to arrive at kinetic parameters which give an excellent fit to the data, and often there is no countercheck on the validity of these values. Using data at several heating rates serves to provide this check. As a final check for heat- and mass-transfer intrusions, values of the characteristic times for chemical kinetics ( T ~ = K-'), thermal diffusion through the particle ( ~ =dL 2 / a P ) , and heat transfer due to radiation between the crucible and the particle (7ht = mpAH/u,b(T; - T,4)Ap) were eveluated. A value of 7ck of 5-100 s compared with values
~
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 1057 position of zinc sulfate. Our future goal is to attempt to influence the decomposition process, using the high heat flux environments provided by a solar furnace, and to accelerate the formation of the zinc oxysulfate and possibly the formation of zinc oxide. The difficulties encountered in this research are universal to the field of solid-phasedecompositions and are due to the inherent limitations of thermogravimetry and the accuracy of the data obtained from it. Often, complex reactions can be modeled as single-step reactions within the experimental accuracies of the data obtained. Therefore, the evaluation of kinetic data using mathematical methods necessitates a careful study of the inaccuracies in the input data in order to obtain comparable and meaningful results. Acknowledgment 0
0.0
100.0
200.0
330.0
400.0
90.0
800.0
700.0
800.0
900.0 1M O . O
TEMPERATURE ( C 1
Figure 9. Comparison of experimental and predicted TG curves for the decomposition of zinc sulfate heptahydrate in helium using ceramic boats.
of 5 X 10-4 for +ht and 0.4 s for +d reinforced our conclusion that chemical reaction was the rate-determining step for our experiments. Conclusions and Future Research The following conclusions were drawn from this study: 1. Thermogravimetric parameters, including sample size, boat material, water of hydration, and carrier gas composition, all influence the decomposition of zinc sulfate. Consequently, the effect of these factors on the experimental results must be carefully considered. 2. Platinum acts as a catalyst for the decomposition process, possibly by influencing the vapor-phase decomposition of sulfur trioxide. The carrier gas also influences the decomposition kinetics by changing the partial pressure of the gaseous products above the sample. 3. Pretreatment of the zinc sulfate in air delays the decomposition process, probably by creating a more stable starting material. This view is supported by the fact that decomposition temperatures are higher for the experiments which used either dehydrated samples or air as the carrier gas. 4. The solid-state transition from the CY phase to the P phase takes place a t 752 "C and is highly reproducible. 5. The decomposition of zinc sulfate proceeds in two steps, with the formation of an intermediate. The formation of the intermediate takes place via a competitive reaction, whereas its decomposition is a simple, singlereaction step. This model is not affected by the experimental conditions, but the values of the kinetic parameters obtained depend on the thermogravimetric parameters. 6. Our results agree with the formula Zn0.2ZnS04 for the intermediate oxysulfate. 7. The formation of the oxysulfate can be modeled by a single-step first-order reaction at low heating rates, while ita decomposition is a pseudo-zero-order reaction. 8. The most likely model for the decomposition appears to be model i. It is noted that our analysis accounted for only the random errors in the measurements (assuming a Gaussian distribution) and did not consider any systematic errors present due to the influence of the chemical reaction. An important outcome of this research is the possible existence of alternative pathways during the early decom-
The authors thank Dr. M. Bowman, Prof. D. G. M. Anderson (Hamud), Dr. G. Varhegyi (Hungarian Academy of Sciences),Dr. H. Friedman (Textile Research Institute), Dr. A. W. Czanderna (SERI), and W. S. L. Mok and S. Cooley (University of Hawaii) for their assistance in the research. This research was supported by the US.Department of Energy under Contract No. DE-AC0384SF12200. Nomenclature A = preexponential constant A , = exposed area of the particle c = degree of conversion
D = diagonal weight matrix E = activation energy k ( T ) = rate of the reaction L = characteristic length of the particle m = number of experimental data points m, = mass of the particle n = reaction order np = number of parameters n, = number of data points for each heating rate nb = number of heating rates nkin= number of kinetic parameters for each pathway npath = number of reaction pathways node= number of independent y's n, = number of species in the reacting system p = parameter vector R = universal gas constant s = vector of residuals S = objective function ti = time T = temperature Ti = initial decomposition temperature T f = final decomposition temperature T , = temperature of the crucible T, = temperature of the sample V = covariance martix w = weight fraction of reacting species w, = weight of the reacting species win = initial weight of the reacting species wi = experimentally measured weight fraction y i = weight fraction of an individual reacting species Greek Symbols = thermal diffusivity of the particle AH = specific heat for phase transition
(Y
K
= reaction rate constant
uext= external standard deviation uint = internal standard deviation
= Stephan-Boltzmann constant = characteristic time for chemical kinetics T~~ = characteristic time for heat transfer (radiation) Tck
Ind. Eng. Chem. Res. 1988,27, 1058-1065
1058 rtd =
characteristic time for thermal diffusion Registry No. ZnSO,, 7733-02-0; ZnS04.7H20, 7446-20-0.
Literature Cited Antal, M. J., Jr. “Thermogravimetric Signatures of Complex Solid Phase Pyrolysis Mechanisms and Kinetics”. In Thermal Analysis; Miller, B., Ed.; Wiley: New York, 1983; Vol. 11, p 1490. Antal, M. J., Jr. “Biomass Pyrolysis: A Review of the Literature. Part 11. Lignocellulose Pyrolysis”. In Advances in Solar Energy; Boer, K. W., Duffie, J. A., Eds.; American Solar Energy Society: New York, 1985; Vol. 2. Antal, M. J., Jr.; Friedman, H. L.; Rogers, F. E. Combust. Sci. Technol. 1980,21, 141-152. Antal, M. J., Jr.; Mok, W. S.-L.; Roy, J. C.; T-Raissi, A. J . Anal. Appl. Pyrol. 1985, 8, 291. Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1969; Chapter 11, p 242. Bowman, M. G. “Solar Thermal Hydrogen”. Presented at the Proceedings of the International Symposium on Hydrogen Production from Renewable Energy, Honolulu, 1984. Brown, M. E.; Dollimore, D.; Galwey, A. K. Comprehensive Chemical Kinetics; Bamford, C. H., Tipper, C. H. F., Eds.; Elsevier: New York, 1983; Vol. 22, p 41. Come, G. M. comprehensive Chemical Kinetics; Bamford, C. H., Tipper, C. H. F., Eds.; Elsevier: New York, 1983; Vol. 24, p 249. Cvetanovic, R. J.; Singleton, D. L.; Paraskevpoulos, E. J. Phys. Chem. 1979, 83(1),50-60. Ducarroir, M.;Romero-Paredes, H.; Steinmetz, D.; Sibieude, F.; Tmar, M. ”On the Kinetics of the Thermal Decomposition of Sulfates Related with Hydrogen Water Splitting Cycles”. Proc. 4th World Hydrogen Energy Conf. 1982,2, 451-463. Flynn, J. Thermochim. Acta 1980,37, 225-238. Friedman, H. L. J. Polym. Sci. 1964, C6, 183-195. Garn, P. D. J. Thermal Anal. 1975, 7, 475. Hildenbrand, D. L. ”High Temperature Chemistry of Hydrogen Production Cycles”. Technical Status Report, September 1978; SRI International, PYU-6788. Hildenbrand, D. L. ”High Temperature Chemistry of Hydrogen Production Cycles”. Technical Status Report, October 1979; SRI International, PYU-6788.
Hildenbrand, D. L. ”High Temperature Chemistry of Hydrogen Production Cycles”. Technical Status Report, October 1980; SRI International, PYU-6788. Hosmer, P. K.; Krikorian, 0. H. “Solar Furnace Decomposition Studies of Zinc Sulfate” Lawrence Livermore Laboratory Report, Preprint UCRL 83634, 1979. Hosmer, P. K.; Krikorian, 0. H. High Temp.-High Pressures 1980, 12, 281. Ibanez, J. G.; Wentworth, W. E.; Batten, C. F.; Chen, E. C. M. “Kinetics of the Thermal Decomposition of Zinc Sulfate”. Rev. Int. Hautes Temp. Refract. 1984, 21, 113-124. Ingraham, T. R.; Kellogg, H. H. Trans. Metall. SOC.M M E 1963,227, 1419-1426. Kirk-Othmer, Encyclopedia of Chemical Technology, 3rd ed.; Wiley-Interscience; New York, 1984; Vol. 24, p 807. Kolta, G. A.; Askar, M. H. Thermochim. Acta 1975, 11, 65-72. Krikorian, 0. H.; Shell, P. K. Int. J. Hydrogen Energy 1982, 7(6), 463-469. Mok, W. S.-L.; Antal, M. J., Jr. Thermochim. Acta 1983, 68, 165. More, J. J. “The Levenberg-Marquardt Algorithm. Implementation and Theory”. In Numerical Analysis; Lecture Notes in Mathematics 630; Springer-Verlag; New York, 1977; pp 105-116. Mu, J.; Perlmutter, D. D. Znd. Eng. Chem. Process. Des. Dev. 1981, 20, 640. Nowak, U.; Deuflhard, P. “Towards Parameter Identification for Large Chemical Reaction Systems”. International Workshop on Numerical Treatment of Inverse Problems for Differential and Integral Equations, Heidelberg, FRG, 1982. Ostroff, A. G.; Sanderson, R. T. J . Inorg Nucl. Chem. 1959,9,45-50. Sestak, J.; Satava, V.; Wendlandt, W. W. Thermochim. Acta 1973, 7, 333. Shell, P. K.; Ruiz, R.; Yu, C. M. “Solar Thermal Decomposition of Zinc Sulfate”. Lawrence Livermore National laboratory Report UCRL 53370, 1983. Tagawa, H. Thermochim. Acta 1984, 80, 23-33. Urban, D. L.; Antal, M. J., Jr. Fuel 1982, 61, 799-806. Wendlandt, W. W. Thermal Methods of Analysis; Interscience: New York, 1964. Zsako, J. J . Thermal Anal. 1976,9, 101. Received for review April 28, 1987 Accepted January 12, 1988
Parameters for the Perturbed-Hard-ChainTheory from Characterization Data for Heavy Fossil Fuel Fluids Shao-Hwa Wangt and Wallace
B. Whiting*
Department of Chemical Engineering, West Virginia University, Morgantown, West Virginia 26506-6101
Correlations for pure-component parameters have been developed for the Perturbed-Hard-Chain equation of state in terms of conventional characterization data (molecular weight, normal boiling point, and specific gravity). The correlations are comparable to and often better than correlations using molecular-structure information (which require elaborate measurements) in predicting saturated liquid densities and vapor pressures for heavy fossil fuel fractions. A universal temperature dependence of the hard-core volume is introduced. The Perturbed-Hard-Chain equation of state with the proposed parameter correlations and hard-core-volume temperature dependence gives better VLE calculation results than typical cubic equations of state for heavy hydrocarbons. This thermodynamic model is extended for continuous mixtures. The economic value of recovering more liquid from heavy fossil oil is increasing. It is, therefore, important to be able to calculate the properties of these heavy hydrocarbons which have not been studied as extensively as have lighter hydrocarbons.
* Author t o whom correspondence should be addressed. Presently at Morgantown Energy Technology Center, Morgantown, WV 26505.
0888-5885/88/2627-lO58$01.50/0
Conventionally, the Redlich-Kwong equation of state and its modifications have been used to describe the thermodynamic properties of low molecular weight hydrocarbon mixtures. However, theoretically this type of equation of state is limited to simple spherical molecules. As indicated by Sim and Daubert (1980) and Alexander et al. (1985), such equations of state are inadequate in predicting the phase equilibria of ill-defined heavy hydrocarbons. The Perturbed-Hard-Chain (PHC) equation 0 1988 American Chemical Society
