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Energy & Fuels 2000, 14, 1156-1160
New Simplified Rate Equation for Gas-Phase CO Oxidation at Combustion Jaakko J. Saastamoinen* VTT Energy, Box 1603, FIN-40101 Jyva¨ skyla¨ , Finland
Pia T. Kilpinen and Tommy N. Norstro¨m Åbo Akademi Process Chemistry Group, c/o Combustion Research, Lemminka¨ isenkatu 14-18 B, FIN-20520 Turku, Finland Received February 10, 2000. Revised Manuscript Received September 1, 2000
The available global kinetic expressions for gas-phase CO oxidation have been shown to be of insufficient accuracy in the wide temperature range that exists in many modern combustion systems. A simplified formula for the kinetics of CO oxidation in the presence of water vapor is here developed to cover the whole temperature range of 973-1673 K. Comprehensive kinetic calculations with a validated, elementary reaction mechanism serve at ideal plug flow conditions as a basis for the development work of the simplified mechanism. It is shown, in contrast to the simplified rate equations in the literature, that this new simplified model can predict the oxidation rate fairly well both at low and high temperatures.
Introduction The calculation of the rate of CO oxidation with the comprehensive chemistry in connection with computational fluid dynamics codes developed for furnaces is time-consuming. The applicability of this comprehensive model has been verified by comparing to experimental data.1-7 Simplified CO oxidation schemes have been presented in the literature8-10 and used frequently in practice. However, their accuracy is poor outside the range of validity. At high temperatures, the oxidation * Corresponding author. VTT Energy, Box 1603, FIN 40101 Jyva¨skyla¨, Finland. E-mail:
[email protected]. Fax: + 358 14 672 596 (1) Miller, J.; Bowman, T. Prog. Energy Combust. Sci. 1989, 15, 287. (2) Glarborg, P.; Dam-Johansen, K.; Kristensen, P. Reburning Rich Lean Kinetics, Final Report, Gas Research Institute Contract No. 5091260-2126, Nordic Gas Technology Centre Contract No. 89-03-11, Dec 1993. Nordic Gas Technology Centre: Lyngby, Denmark. (3) Glarborg, P.; Kristensen, P. G.; Jensen, S. H.; Dam-Johansen, K. Combust. Flame 1994, 98, 241. (4) Kilpinen, P.; Hupa, M.; Aho, M.; Ha¨ma¨la¨inen, J. Selective NonCatalytic NOx Reduction at Elevated Pressures-Studies on the Potential for Increased N2O Emissions. In Proceedings 7th International Workshop on Nitrous Oxide, Cologne, 1997; K. H. Becker, K. H., and P. Wiesen, P., Eds.; Bergische Universita¨t Gesamthochschule Wuppertal. (5) Kilpinen, P.; Hupa, M. Nitrogen Chemistry at Combustion and Gasification-Mechanisms and Modelling. In LIEKKI Technical Review 1993-1998; Hupa, M., Matinlinna, Eds.; 1998; Vol. 1, pp 327-359, ISBN 952-12-0271-8, ISSN 1235-6858. (6) Kilpinen, P.; Norstro¨m, T.; Mueller, C.; Kallio, S.; Hupa, M. Homogeneous NO and N2O Chemistry in FBC. Presented at the Workshop on NO/N2O Formation and Destruction in Fluidised Bed Combustors, 38th IEA Fluidised Bed Conversion Meeting, May 1516, 1999, Savannah. (7) Kilpinen, P.; Kallio, S.; Hupa, M. Advanced Modeling of Nitrogen Oxide Emissions in Circulating Fluidized Bed Combustors: Parametric Study of Coal Combustion and Nitrogen Compounds Chemistries. Presented at the 15th International Conference on Fluidized Bed Combustion (ASME), May 9-13, Savannah, 1999 (Paper FBC-990155). (8) Fristrom, R. M.; Westenberg, A. A. Flame Structure; McGrawHill: New York, 1965.
kinetics has a minor effect compared to mixing when considering the CO oxidation. In many combustion devices, such as fluidized bed furnaces, grate combustion furnaces, and fireplaces, the temperatures can be as low as 973-1073 K, and the oxidation kinetics of CO becomes an important factor. At such low temperatures, however, the accuracy of the existing rate equations is poor. The aim of the present work is to develop a rate equation that is valid for the whole temperature range of 973-1673 K, and is sufficiently simple to be used with CFD codes. Development of the Simplified Rate Equation The rate of the formal reaction CO + 1/2O2 ) CO2 is usually presented in the literature8-10 by the equation of type
-
dXCO b ) kXaCO XO Xc 2 H 2O dt
(1)
where X is mole fraction, t is time and k is the temperature dependent reaction rate coefficient, which is usually presented in the Arrhenius form k ) Aexp[E/(RT)], and the exponents a, b, and c are constants. For example, Dryer and Glassman10 give (in units K, cm, mole, s) A ) 1014.6, a ) 1, b ) 1/4, c ) 1/2, E/R ) 20140 K. The effect of water vapor is included in eq 1, since the radicals formed from it, especially the hydroxyl (9) Howard, J. B.; Williams, G. C.; Fine, D. H. Proceedings of the 14th Symposium (International) on Combustion; The Combustion Institute: Pittsburgh, PA, 1973; p 975 (10) Dryer, F. L.; Glassman, I. Proceedings of the 14th Symposium (International) on Combustion; The Combustion Institute: Pittsburgh, PA, 1972; pp 987-1003.
10.1021/ef000021e CCC: $19.00 © 2000 American Chemical Society Published on Web 10/21/2000
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Energy & Fuels, Vol. 14, No. 6, 2000 1157
radical (OH), play a major role in the underlying actual chemical mechanism for CO oxidation. Correspondingly the oxygen is consumed in the reaction according to
dXO2 dt
)
1 dXCO 2 dt
(2)
The CO oxidation was calculated at plug flow mode by applying a detailed elementary reaction mechanism4 that has been validated elsewhere11 with initial oxygen concentration of 1-10 vol % and that of carbon monoxide 0.5-7 vol %. Two concentrations of water vapor at 0.01 and 15%, respectively, were applied. The oxidation of CO to CO2 was calculated at the temperatures 973, 1073, 1173, 1273, 1473, and 1673 K. When the concentrations in two different transient situations A and B are known from the calculations with the comprehensive chemistry, it is possible to derive from eq 1 the following expressions to evaluate the exponents a, b, and c
[( ) ( ) (
ln a)
c)
b
XO2,B
XH2O,A
c
)(
)]
(3)
)(
)]
(4)
)(
)]
dXCO,B dXCO,A / dt dt
XH2O,B
ln(XCO,B/XCO,A)
ln b)
XO2,A
[( ) ( ) ( XCO,A XCO,B
a
XH2O,A
c
dXCO,B dXCO,A / dt dt
XH2O,B
ln(XO2,B/XO2,A)
[( ) ( ) (
XCO,A ln XCO,B
XO2,A
a
XO2,B
b
Figure 1. Calculated effect of temperature and average ratio h CO ) 0.5(XCO,A + XCO,B), XO2,0 ) 0.10. X h CO/XO2 on exponent b. X Case X h CO/XO2 ) 0.044 (O, XH2O ) 0.0001; 2, XH2O ) 0.15) was calculated by using XCO,A,0 ) 0.005, XCO,B,0 ) 0.01; XCO,A ) 0.00134, XCO,B ) 0.00684. Case X h CO/XO2 ) 0.447 (∆, XH2O ) 0.0001) was calculated by using XCO,A,0 ) 0.05, XCO,B,0 ) 0.07; XCO,A ) 0.0303, XCO,B ) 0.0503.
Figure 2. Calculated dependence of exponent a on the ratio XCO/XO2 at T ) 1273 K.
dXCO,B dXCO,A / dt dt
ln(XH2O,B/XH2O,A)
(5)
However, the exponents evaluated from these equations depend from each other. This difficulty can be avoided. For example, when applying eq 3 to evaluate a, we choose a situation at which XO2,A ) XO2,B and XH2O,A ) XH2O,B. Then (XO2,A/XO2,B)b ) 1 and (XH2O,A /XH2O,B)c ) 1. In this case eq 3 is reduced to
(
)
dXCO,B dXCO,A / dt dt a) ln(XCO,B/XCO,A) ln
(6)
which no longer depends on b and c. In the same way, eqs 4 and 5 can be applied directly to evaluate the coefficients. When evaluating b, we choose XCO,A ) XCO,B and XH2O,A ) XH2O,B, and when evaluating c, we choose XO2,A ) XO2,B and XCO,A ) XCO,B. Conversion of CO is calculated with the comprehensive chemistry, but the transient concentration of O2 is estimated from the relation XO2 ) XO2,0 - 0.5(XCO,0 - XCO), which is obtained by integrating eq 2. Subscript 0 denotes the initial value. The calculated exponent a is shown in Figure 1. It is seen that it depends both on T and XCO/XO2. The calculated dependence on XCO/XO2 at 1273 K is presented in Figure 2. At low temperatures the behavior is not very clear. (11) Glarborg, P.; Kubel, D.; Kristenssen, P. G.; Hansen, J.; DamJohansen, K. Combust. Sci. Technol. 1995, 110-111, 461.
Figure 3. Calculated effect of temperature on exponent b at two initial levels of CO concentration: XCO,0 ) 0.01 (b, XO2,A ) 0.0168, XO2,B ) 0.0968, XO2,A,0 ) 0.02, XO2,B,0 ) 0.10) and XCO,0 ) 0.05 (2, XO2,A ) 0.0542, XO2,B ) 0.0842, XO2,A,0 ) 0.07, XO2,B,0 ) 0.10). Filled symbols are for XH2O ) 0.0001, open symbols are for XH2O ) 0.15.
The calculated dependence of exponent b on temperature is presented in Figure 3. Also here some dependence on the ratio XCO/XO2 can be seen. Here we simply assume a ) 1 and approximate b by an average linear correlation (the unit of T is K),
b ) 7.89 × 10-4T - 0.90
(7)
which does not depend on the ratio XCO/XO2. Then later the dependence on the ratio XCO/XO2 will be found in the evaluation on the reaction rate coefficient k. The calculated dependence of the exponent c0 is shown in Figure 4, where also line calculated by the following
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Saastamoinen et al.
Figure 4. Calculated approximate exponent c0, when XCO,0 ) 0.005 (2), 0.01 (b) 0.05 (l). The points have been evaluated at conversion XCO,A/XCO,0 ) XCO,B/XCO,0 ) e-1.
Figure 6. Effect of ratio XCO/XO2 on the reaction rate coefficient k0 at different temperatures (0 973, ∆ 1073, b 1173, × 1273, O 1473 and 2 1673 K), when XH20 ) 0.0001. Points have been evaluated from data calculated using oxidation rate with comprehensive chemistry; trendlines have been calculated by using eq 11.
Figure 5. Dependence of k0 on T at different conditions. The reaction rate coefficient was evaluated approximately at conversion XCO/XCO,0 ) e-1. Points indicate different initial conditions for ratio CO/O2 (%/%) (0.5/10 ×, 0.5/1 , 7/10 2, 1/2 ), 5/ b, 1/10 ∆, and 5/10 0, when X 7 H2O ) 0.0001 and 0.5/10 9, when XH2O ) 0.15).
correlations are shown. When T e 1173,
c0 ) 0.67 - 0.76θ1 + 1.53exp(-1.026/θ1)
(8)
where θ1 ) (T - 973K)/200 K. When T g 1173 K
c0 ) 0.46 + 0.04θ2 - 0.06exp(-0.278/θ2)
(9)
where θ2 ) (T - 1173 K)/600 K. Index 0 is used here to note that this is an initial approximation that is valid only for low concentrations of CO, which will later be improved. Now a, b, and c0 or their approximations are known and the apparent reaction rate coefficient k0 can be solved from eq 1
k0 ) -
1 b XaCOXO Xc0 2 H 2O
dX dt
(10)
The calculated values of ln(k0) as a function of 1/T are shown in Figure 5. It is seen that the points do not fall on a single line, but there is some scatter. In addition to temperature, k0 depends on the ratio XCO/XO2 and on XH2O as shown in Figures6 and 7. The dependence of k0 on T and XCO/XO2 can be correlated as
ln(k0) ) f0(T) + f1(T) ln XH2O + [f2(T) + f3(T) ln XH2O]XCO/XO2 (11)
Figure 7. Effect of ratio XCO/XO2 on the reaction rate coefficient k0 at different temperatures (0 973, ∆ 1073, b 1173, × 1273, O 1473, and 2 1673 K), when XH20 ) 0.15. Points have been evaluated from data calculated using oxidation rate with comprehensive chemistry; trendlines have been calculated by using eq 11.
The calculations by this correlation are shown in Figures 6 and 7 by the lines. The scatter of points from the correlation is rather high in the range XCO/XO2 ) 0.15...0.45 at 1073 K, when the moisture content is low (Figure 6). The functions are, when T e 1173 K,
f0(T) ) -0.37 - 4.17θ1 + 17.44exp(-0.427/θ1)
(12)
f1(T) ) -0.103 + 0.179θ1 - 1.056exp(-2.063/θ1) (13)
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Energy & Fuels, Vol. 14, No. 6, 2000 1159
f2(T) ) 4.44 + 40.64θ1 - 97.14exp(-0.798/θ1) (14) f3(T) ) 0.548 + 4.685θ1 - 14.212exp(-1.016/θ1) (15) and when T g 1173 K,
f0(T) ) 6.85 + 6.65θ2 - 9.98exp(-1.265/θ2)
(16)
f1(T) ) -0.0585 + 0.094θ2 - 0.117exp(-0.904/θ2) (17) f2(T) ) 0.96 + 5.52θ2 - 26.98exp(-1.075/θ2) (18) f3(T) ) 0.086 + 0.359θ2 - 2.395exp(-1.102/θ2) (19) Now, using eq 11, an expression for c that is valid for higher CO concentrations can be developed as
c ) c0 + f1(T) + f3(T)XCO/XO2 ) f4(T) + f3(T)XCO/XO2
(20)
where
Figure 8. Effect of ratio XCO/XO2 on the reaction rate coefficient k at different temperatures (0 973, ∆ 1073, b 1173, × 1273, 1473, and 2 1673 K). Points have been evaluated from data calculated using comprehensive chemistry; trendlines have been calculated by using eq 23.
f4(T) ) 0.567 - 0.581θ1 + 1.53exp(-1.026/θ1) 1.056exp(-2.063/θ1) (21) when T e 1173 K, and
f4(T) ) 0.402 + 0.134θ2 - 0.06exp(-0.278/θ2) 0.117exp(-0.904/θ2) (22) when T g 1173 K. The final expression for k is
k ) exp[f0(T) + f2(T)XCO/XO2]
(23)
which is shown in Figure 8 by the straight lines. The points have been evaluated by using eq 1 with the oxidation rate from calculations with comprehensive chemistry. The correlations are not valid beyond the range 9731673 K. In practical combustion cases it can usually be assumed that k ) 0 when T < 973 K and ln(k) ≈ 10, when T > 1673 K. At high temperatures the reverse, dissociation reaction CO2 f CO + 1/2O2 becomes influential. Discussion It is seen that a (Figure 1) is close to 1 at high temperatures and relatively high concentrations, but a steep decrease in it is noticed when the temperature decreases. At lower concentrations a can be almost as high as 1.5 at 1200 K. Recently Li et al.12 found by fundamental analysis that the order for CO is 3/2. Both Howard et al.9 and Dryer and Glassman,10 and recently Li et al.,12 give the constant value 1/2 for the exponent of the water vapor. This is in close agreement with our prediction presented in Figure 3 showing a somewhat higher value at low temperatures and lower value at high temperatures. At higher concentrations (12) Li, S. C.; Williams, F. A.; Gebert, K. Combust. Flame 1999, 119, 367-373.
Figure 9. Oxidation rate of CO under various conditions calculated by using the comprehensive and simplified chemistry.
of CO, c depends also on the ratio XCO/XO2 as shown by eq 20. Values of 1/4 (refs 8 and 10) and 1/2 (ref 9) for the oxygen order have been presented. Our calculations indicate that the oxygen order cannot be presented by a single constant but is approximately linear function of the temperature (Figure 3). Zero order for oxygen was found by Li et al.12 The difficulty to describe the CO oxidation by a single activation energy in the whole temperature range is seen in Figure 5, where points do not fall on a single straight line. In the new model, accuracy is improved by the linear dependence of ln(k) on both temperature and XCO/XO2 as shown in Figure 8. The oxidation rates calculated with the comprehensive chemistry and the new correlation are compared in Figure 9. As an example, comparison between different methods at low concentration of CO is presented in Figures10 and 11. The method presented here is valid at both temperatures. Only one (ref 8) of the three models presented in the literature gives the same oxidation rate
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Energy & Fuels, Vol. 14, No. 6, 2000
Figure 10. CO oxidation with different models at 973 K, 10% O2, and 0.01% H2O. Results ref 4, this work and ref 8 are practically overlapping.
Figure 11. CO oxidation with different models at 1073 K, 10% O2, and 0.01% H2O.
Saastamoinen et al.
Figure 13. Oxidation of CO at 1273 K. Initial composition is 7% CO, 10% O2, 0.01% H2O (thin lines, lower time scale) and 15% H2O (thick lines, upper time scale). Solid lines are calculated with full chemistry; dashed lines are calculated with the simplified formula.
counted for. The ignition delay is due to the initial development of the radicals. It does not exist in furnaces, where the radicals are already present in the case of steady combustion. At lower temperature 1073 K, The accuracy is not so good (Figure 13). Two different stages are clearly seen at low concentration of H2O in Figure 13. The CO oxidation with exact chemistry is approximately linear indicating a zero reaction order (a ≈ 0) after a critical CO concentration of about 1.5% is reached. The simplified formula, which includes the effect of high ratio of XCO/XO2 not in exponent a, but in k, can, to some extent, describe this behavior, but the oxidation rate is somewhat too high. This low value of a at low temperature and high concentration of CO is also shown in Figure 1. The reason for this is probably that at low temperatures, the rate of generation of radicals is so low in comparison to the amount of CO that the process becomes dominated by the amount of radicals and does not depend on the CO concentration. The situation is different at higher concentrations of H2O. Conclusions The following new rate equation
Figure 12. Oxidation of CO at 1273 K. Initial composition is 7% CO, 10% O2, and 0.01% H2O. Thick line is calculated with full chemistry; thin line is calculated with the simplified formula.
as the comprehensive calculations at 973 K (Figure 10), but at higher temperature 1073 K (Figure 11) this models predicts too low oxidation rate whereas the accuracy of the other models (ref 9 and ref 10) is improved. The model presented in this work gives a good agreement at both temperatures as shown in Figures 10 and 11. The two curves (this work) in Figure 11 are identical in shape, but the other is shifted to account for the ignition delay. The comparison for CO oxidation at 1273 K with a higher CO concentration is shown in Figure 12. It is seen that the oxidation rates with the full chemistry and the correlation are in a quite close agreement, if the shift due to the ignition in the comprehensive case is ac-
dXCO ) dt
7.89×10 ef0(T)+f2(T)XCO/XO2XCOXO 2
-4T-0.90
f4(T)+f3(T)XCO/XO2 XH (24) 2O
for rapid calculation of kinetics limited CO oxidation in the presence of H2O is presented. In contrary to existing rate equations, the accuracy of the new formula is good in the whole temperature range 973-1673 K at low concentrations of CO. Relatively good accuracy is obtained in a wider range XCO/XO2 < 0.65. Acknowledgment. The support from the Finnish National Research Program TULISIJA, National Technology Agency TEKES, and Academy of Finland is gratefully acknowledged. The authors wish to thank Mr Edcardo Coda Zabetta for his assistance in performing part of the calculations. EF000021E
