High-Pressure Shock Tube Studies on Carbon Oxidation Reactions

The heterogeneous reactions of solid carbon reacting with gaseous carbon dioxide and steam have been studied under high-pressure and high-temperature ...
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Energy Fuels 2009, 23, 5806–5812 Published on Web 11/16/2009

: DOI:10.1021/ef900641r

High-Pressure Shock Tube Studies on Carbon Oxidation Reactions with Carbon Dioxide and Water Brad Culbertson and Kenneth Brezinsky* Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 2039 Engineering Research Facility, 842 West Taylor Street, Chicago, Illinois 60607 Received June 24, 2009. Revised Manuscript Received October 29, 2009

The heterogeneous reactions of solid carbon reacting with gaseous carbon dioxide and steam have been studied under high-pressure and high-temperature conditions. The high-pressure shock tube facility at the University of Illinois at Chicago was used to perform experiments with post-shock pressures ranging from 40 to 412 atm, temperatures ranging between 1404 and 2534 K, and nominal reaction times of 1 ms. Amorphous carbon particles were injected into the shock tube by means of a particle injector specifically designed for these experiments. The carbon particles were injected into the shock tube containing mixtures of varying concentrations of H2O or CO2 in Ar as balance. The overall reaction rates have been calculated and compared to rates of previous theoretical and experimental values. The wide temperature range of the experimental conditions allowed for the observation of both reaction- and transport-limiting regimes. Using theoretical techniques for computing the species transport properties, the transition temperatures where transport dominates the reaction were determined and match the experimental results.

The present study has been initiated under such extreme conditions to determine if the rate coefficient for the reactions of solid carbon with carbon dioxide and steam change at these high-pressure conditions. To study these reactions at high pressures, the carbon particles were introduced into the shock tube by means of a particle injector. The high-pressure, hightemperature conditions behind the reflected shock wave enabled reactions between the gas-phase molecules and the particles to occur at isothermal and stagnant conditions. The change in concentrations of H2O and CO2 as well as the formation of CO and H2 as functions of the temperature, pressure, and reaction time provide the information necessary to calculate the rate coefficients.

Introduction Post-combustion gases, primarily CO2 and H2O, from the combustion chamber of a rocket are hypothesized to react with the inner surface of the rocket nozzle. The high-temperature, high-pressure environment that exists in the nozzle can facilitate these reactions, thereby leading to erosion of the nozzle surface and potentially affecting the rocket performance. Designs for future rockets call for increased range and other capabilities by means of increased operating pressures. A previous theoretical analysis1 of the nozzle erosion, which takes into account many processes, including chemical erosion, found that increasing the rocket operating pressure further increases the amount of nozzle erosion. A major assumption used for this analysis1 was that the two heterogeneous reactions of solid carbon with CO2 and H2O, reactions I and II, can be described by the same rate coefficient determined by Libby and Blake.2 The extreme high-pressure environment of the rocket nozzle exceeds any existing experimental data;3-7 consequently, an experimental study of these key heterogeneous reactions under these extreme conditions is essential for optimum rocket design. k1

ðIÞ

k2

ðIIÞ

s CO þ CO CðsÞ þ CO2 f CðsÞ þ H2 O f s CO þ H2

Experimental Section The high-pressure shock tube (HPST) facility at the University of Illinois at Chicago (UIC) was used to conduct these experiments. A detailed description of the shock tube and its operation can be obtained from prior studies.8,9 The shock tube is a wellcharacterized experimental setup9,10 and has been used recently in related work on the water-gas shift reaction at similar highpressure, high-temperature conditions.11 The experiments were performed behind the reflected shock wave using a driver section (length of 6000 with a 200 bore) separated from the driven section (length of 10100 with a 100 bore) by means of pre-scored brass and aluminum diaphragms. Reaction pressures ranged from 40 to 412 atm, while reaction temperatures ranged from 1404 to 2534 K, with reaction times lasting 1 ms. The reaction temperatures are calculated using a chemical thermometer technique that is outlined in a previous work.9

*To whom correspondence should be addressed. E-mail: kenbrez@ uic.edu. (1) Kuo, K. K.; Keswani, S. T. Combust. Sci. Technol. 1985, 42, 145– 164. (2) Libby, P. A.; Blake, T. R. Combust. Flame 1981, 41, 123–147. (3) Golovina, E. S. Carbon 1980, 18, 197–201. (4) Bradley, D.; Dixon-Lewis, G.; El-Din Habik, S.; Mushi, E. M. J. Symp. Int. Combust. Proc. 1984, 20, 931–940. (5) Long, F. J.; Sykes, K. W. J. Chim. Phys. 1950, 47, 361–378. (6) Turkdogan, E. T.; Koump, V.; Vinters, J. V.; Perzak, T. F. Carbon 1968, 6, 467–484. (7) Turkdogan, E. T.; Vinters, J. V. Carbon 1970, 8, 39–53. r 2009 American Chemical Society

(8) Tranter, R. S.; Brezinsky, K.; Fulle, D. Rev. Sci. Instrum. 2001, 72, 3046–3054. (9) Tranter, R. S.; Sivaramakrishnan, R.; Srinivasan, N.; Brezinsky, K. Int. J. Chem. Kinet. 2001, 33, 722–731. (10) Tang, W.; Brezinsky, K. Int. J. Chem. Kinet. 2006, 38, 75–97. (11) Culbertson, B.; Sivaramakrishnan, R.; Brezinsky, K. J. Propul. Power 2008, 24, 1085–1092.

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: DOI:10.1021/ef900641r

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Figure 1. Sequence of carbon injection: (a) prior to injection, (b) during injection, and (c) post-injection.

Figure 2. SEM photos of carbon black. Table 1. C(s) þ CO2 Experimental Conditions

To introduce the solid carbon into the shock tube, a particle injector was designed and constructed on the basis of Professor Nick Glumac’s12 particle injector at the University of Illinois at Urbana-Champaign. Examining the injection characteristics of the particles is essential; therefore, in addition to the injector, a plastic model of the shock tube was fabricated to view the injection process. A high-speed camera captured hundreds of images during each carbon injection. The images in Figure 1 are characteristic of the three phases of the injection. As the particle jet forms and hits the opposite wall of the tube, a portion of the particles injected appear to slide down the sides until they come to rest, never going into suspension. To properly calculate the rate constants for these reactions, any particles injected into the shock tube must be available to react with the gaseous mixture. A literature review revealed that an entire area of research is devoted to a phenomena related to coal dust lifting. A review by Fedorov13 outlines these studies and discusses the Saffman lift force, a force perpendicular to and resulting from a velocity gradient. As a shockwave travels across a dusty layer, the particles are entrained in the flow following the shock wave because of the Saffman lift force, which increases in intensity with a decreasing boundary-layer thickness and decreasing particle size. This literature review gave concrete evidence that the particles that may settle to the tube surface upon injection are entrained in the flow following the incident shock wave, thus making all of the carbon injected into the shock tube available for reaction because (1) as our visualization tests (Figure 1) have shown, most of the particles are already in suspension, (2) the UIC HPST boundary layer is relatively small (because of the high pressures), and (3) the nominal particle size is 2.5 μm. In addition to the injector, a new end section of the shock tube was designed, to which the injector is mounted. The new section was manufactured using the same 17-4PH alloy stainless steel as the existing shock tube. The new end section joins the end wall of the tube to the existing driven section while accommodating pressure transducers similar to the standard end section. The new end section contains a passage for the injected carbon to enter the shock tube. Each joint is sealed by means of o-rings that have been proven to seal the tube to less than 5 mtorr. Each piece has a 100 bore down the center, which was honed to 32 rms to match the finish in the existing driven section. Carbon black, supplied by CABOT, was injected into the shock tube close to the end wall by means of the injector just prior to the bursting of the diaphragm. Characterization using scanning electron microscopy (SEM) (Figure 2) revealed that the individual particles were on the order of 50 nm in size but formed agglomerations that averaged 2.5 μm in size. To minimize the agglomerations as well as possible surface oxides, the particles were placed in an oven at 150 C until needed for experimentation. Information provided by the manufacturer listed the specific gravity at 1.85 and the specific surface area to be 210 m2/g. The Brunauer-Emmett-Teller (BET) surface area analysis performed at UIC confirmed the value with a measured 215.99 m2/ g of total surface area. An average pore diameter of 24.79 nm was also calculated from the surface area analysis. The pore diameter

set

temperature (K)

pressure (atm)

reaction time (ms)

xCO2 (ppm)

mc (mg)

1 2 3 4 5 6

1580-2336 1553-2204 1617-2199 1596-2176 1567-2212 1404-2183

412 406 248 256 268 40

0.99 0.963 1.015 1.032 1.009 1.104

2392 1108 2107 1018 501 2226

1.3 1.3 1.3 1.3 1.3 1.3

Table 2. C(s) þ H2O Experimental Conditions set

temperature (K)

pressure (atm)

reaction time (ms)

xH2O (ppm)

mc (mg)

1 2 3 4

1503-2454 1619-2534 1412-2419 1508-2506

386 347 205 213

0.975 0.946 1.023 0.987

471 239 994 471

1.3 1.3 1.3 1.3

and porosity in general have an influence on the species diffusion but should not change the intrinsic reaction kinetics. Before an experiment was performed with the carbon particles, care was taken to ensure that no carbon particles were left behind from a previous experiment to possibly alter the results. After a shock in which carbon was injected, the tube was disassembled and cleaned via lint-less towels. After the length of the tube was physically cleaned and reassembled, two experiments using O2 as the driven gas in place of the experimental mixture were performed. The O2 shocks were performed to burn any excess carbon that was possibly missed by the physical cleaning. This was confirmed from gas chromatography (GC) analysis of the postshock samples. For the cases studying the reaction between solid carbon and CO2, a shock with no carbon injection was performed with the CO2 mixture to determine if any pyrolysis was occurring and possibly influencing the heterogeneous reaction with solid carbon. It was concluded that CO2 pyrolysis was not a factor under these conditions. Finally, the carbon injection experiments were performed last in this sequence of four experiments. Consequently, to obtain one experimental point for the heterogeneous reaction, a total of four shock tube experiments were performed. In total, 10 sets of experiments were conducted, with the experimental conditions given in Tables 1 and 2. The nozzle of the particle injector was loaded with a constant amount of 1.3 mg of solid carbon for each heterogeneous experiment. Zero-grade purity (99.998%) helium supplied by Airgas was used as the driver gas to burst the diaphragms and generate the shock wave. The test gas mixtures were prepared in a heated 50 L stand-alone mixture vessel and allowed to stand overnight to homogenize before use. The reaction mixtures with CO2 contained varying quantities of CO2 and Ne, used as an internal standard, with Ar as balance. The experimental details for the carbon-carbon dioxide sets are listed in Table 1. Because of different analysis techniques, the carbon-steam reaction mixtures contained Xe as an internal standard instead of Ne, and the details for those experiments are shown in Table 2. After each shock experiment, post-shock gas-phase samples were withdrawn from a port in the end wall of the shock tube and subsequently analyzed using gas chromatographic techniques.

(12) Glumac, N.; Krier, H.; Bazyn, T.; Eyer, R. Combust. Sci. Technol. 2005, 177, 485–511. (13) Fedorov, A. V. Combust. Explos. Shock Waves 2004, 40, 17–31.

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Figure 4. Species profiles from the reaction of C(s) þ CO2 at varying initial CO2 mole fractions with reaction pressures of approximately 260 atm: 2107 ppm, (9) CO2 and (0) CO; 1018 ppm, (b) CO2 and (O) CO; and 501 ppm, (2) CO2 and (4) CO. Lines are fits to experimental data.

Figure 3. Species profiles from the reaction of C(s) þ CO2 at varying pressures with an initial CO2 mole fraction of approximately 2000 ppm: 410 atm, (9) CO2 and (0) CO; 260 atm, (b) CO2 and (O) CO; and 40 atm, (2) CO2 and (4) CO. Lines are fits to experimental data.

For the CO2 reactions, ultra-high purity helium was used as a carrier gas through a HP-MOLSIEVE column eluting to a thermal conductivity detector (TCD), which was used to detect and quantify the CO, CO2, and Ne mole fractions. For the reactions with H2O, a different analytical setup was used to test a hypothesis based on the conclusions of Bews et al.14 that suggest that a significant amount of O2 is formed when a surface oxide reacts with another surface oxide on a carbon surface. To measure if O2 was in fact formed during this reaction, ultra-high purity argon was used as the carrier gas through the HP-MOLSIEVE column in line with the TCD to measure O2, CO, and Xe mole fractions, another HP-MOLSIEVE column eluting to a pulse discharge detector (PDD) measured the H2 mole fractions, an HP-PLOT-Q column in line with a mass spectrometer measured the CO2 and H2O mole fractions, and another PLOT-Q column eluting to a flame ionization detector (FID) measured the C2H2 mole fraction. When conducting the carbon-steam experiments, the shock tube was heated uniformly to 50 C to prevent condensation of the water vapor. Although the water mole fraction in the mixture was sufficiently low to assume no condensation, the shock tube and other lines were heated as a precaution. The lines connecting the mixing tank to the shock tube were heated with standard 1/200 wide HTS/Amptek heating tape controlled by a 12 A variac. The shock tube was heated with more precise control. The heating tape used for the shock tube was custom-made, with an adhesive backing for direct contact between the tape and the shock tube purchased from Clayborn Lab. The heat tape was controlled by three 4-Zone Omega controllers that were able to keep the inner temperature of the shock tube within (1 C to ensure that shock tube conditions did not vary and possibly affect the chemistry.

Figure 5. Species profile from the reaction of C(s) þ H2O at 994 ppm of H2O and 205 atm: (O) CO, (9) H2, () CO2, and (2) C2H2.

reaction begins increases from 1500 K for the 40 atm data set to 1600 K for the higher pressure sets. Another characteristic of these heterogeneous experiments is that, as the reaction pressure increases, the amount of reaction that occurs before the plateau is reached decreases, with an extent of reaction of 50% for the 40 atm data set to a 40% extent of reaction for the 410 atm data. Correspondingly, the temperature at which the reaction profiles level off increase from 1900 K for the 40 atm data to 2200 K for the 410 atm data set. Figure 4 displays the three data sets performed at average pressures of 260 atm. This figure shows that, as the initial CO2 concentration increases, the absolute amount of reaction that occurs when the plateau is reached also increases. However, the extent of reaction averages roughly 50% for all three sets. It should be noted that the temperature at which the reaction begins does not change substantially; only the “rate” and amount of reaction change. The experiments carried out using H2O had very similar results to the CO2 experiments. Figure 5 shows a representative species profile from experimental set 3 with 205 atm and 994 ppm of initial water mole fraction. The figure shows that, as temperature is increased, the amount of reaction increases and the primary products, CO and H2, are formed until a leveling off occurs, as was seen in the carbon-carbon dioxide reactions. In these reactions with H2O, there was some production of CO2 throughout the temperature regime, with

Results For the carbon-carbon dioxide experiments, the CO2 decay and subsequent CO formation from the three experimental sets with an initial CO2 mole fraction of approximately 2000 ppm are shown in Figure 3. The CO2 and CO profiles from the three experimental sets with reaction pressures of approximately 260 atm are shown in Figure 4. The two plots show the overall trends that were seen in all of the experimental sets. In general, as reaction temperatures increase, the amount of reaction increases until a leveling off occurs. Figure 3 shows that, as the post-shock pressure (or reactant concentration) increases, the temperature at which the (14) Bews, I. M.; Hayhurst, A. N.; Richardson, S. M.; Taylor, S. G. Combust. Flame 2001, 124, 231–245.

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disassembly and cleaning. Visual confirmation also provided confidence that the particles were not inhibited by the presence of the end wall. Some particles were found on the end wall during shock tube cleaning but in relatively small amounts. Once the above quantities are measured or calculated, the rate constant can be determined for each experiment. For the carbon-carbon dioxide experiments, the change in carbon dioxide was used to calculate the rate constant, while for the steam reactions, the production of H2 was used. It has been assumed that, when the carbon particles are injected into the shock tube, they are instantaneously heated to the reaction temperature upon exposure to the reflected shock wave. This assumption is mainly due to the miniscule size of the carbon particles. The assumption can be validated using the lumped capacitance model15 for transient heat transfer. The shock wave heating of the gas is a stepwise increase in the gas temperature. As the gas surrounding the particles heats up, the particles require a finite amount of time to acquire the energy from the ambient gas. The lumped capacitance equation, shown in eq 3, is an ideal and simple method to calculate the length of time required by the particle heating process. The expression can be rearranged to solve for the time to heat up the particle as shown in eq 4. Using lookup values for the specific heat, density, and heat-transfer coefficient of carbon black (509 J kg-1 K-1, 1.85 g/cm3, and 1.6 W m-1 K-1, respectively), the time necessary to heat the particle to temperature T can be calculated. dT ¼ -hAs ðT -T¥ Þ ð3Þ F C cp V dt   F cp D Ti -T¥ t ¼ C ln ð4Þ 6h T -T¥

Figure 6. H2 profiles from the reaction of C(s) þ H2O at varying pressures and initial H2O mole fractions: (0) 386 atm and 471 ppm, (O) 347 atm and 239 ppm, (9) 205 atm and 994 ppm, and (b) 213 atm and 471 ppm. Lines are fits to experimental data.

C2H2 formation at the extreme high temperatures; however, no O2 production was ever observed. The H2 production from the four carbon-steam experimental results are plotted and compared in Figure 6. As was seen in the carbon-carbon dioxide experiments, the amount of reaction increases with the temperature until leveling off occurs. The H2 production appears to be dependent upon the initial H2O concentration, as expected. The H2 production was greatest for set 3, having an average of 205 atm and 994 ppm of H2O initially, followed by set 1 at 386 atm and 470 ppm, etc. Discussion To reduce the data from the heterogeneous experiments into a rate constant, eq 1 was derived and, subsequently, the rate constant expression in eq 2, using measurable quantities or known properties. Here, r_c,i is the graphite rocket nozzle regression rate from Kuo and Keswani,1 ki is the reaction rate coefficient, pi is the initial pressure of CO2 or H2O in the reactant mixture, n is the order of oxidizer pressure dependence, Fc is the density of the carbon, and MWc is the molecular weight of the carbon. The properties as and mC are used as a means to determine the total surface area available to participate in the reaction. The volume VC is used to describe the reaction volume of the experiment; therefore, the quantity asmC/VC is the carbon particle surface area concentration in the reaction volume (cm2/cm3). r_c, i ¼

ki p i n d½CO2  VC 1 ¼ MWC dt as mc Fc Fc

ki ¼ MWC

d½CO2  VC 1 dt as mc pi n

The total reaction time of the heterogeneous experiments was nominally 1 ms, and if the time to heat the particles is more than a few percent of the reaction time, the rate constants calculated would not be representative of the actual reaction rate constant. First, the heat-transfer coefficient, h, is determined from the Nusselt number, the conductive heat-transfer coefficient, k, and a characteristic particle length or diameter. The Nusselt number was determined from the Ranz and Marshall correlation,15 as seen in eq 5. Nu ¼ 2 þ 0:6Re1=2 Pr1=3

ð5Þ

Because the velocity behind the reflected shock is negligible, the Nusselt number then becomes a value of 2, making the heat-transfer coefficient

ð1Þ

h ¼ ð2Þ

2k dp

ð6Þ

Using the lumped capacitance model with the appropriate heat-transfer properties reveals a particle heating time of less than 1 μs. This time is sufficiently small to permit the assumption that the particles are at the reaction temperature for the entirety of the reaction time. The Biot number under these conditions is 0.005, indicating little error from using the lumped capacitance assumption. With the particle temperature analysis concluded, the experimental data can now be reduced using eq 2. The Arrhenius plot shown in Figure 7 shows the reaction rate

The volume of the reaction is an important parameter in the data processing. The volume of the reaction was calculated for each individual shock tube experiment. The volume behind the reflected shock wave, where the reaction takes place, was determined using the reflected shock wave velocity multiplied by the reaction time and the cross-sectional area of the shock tube. The axial distance of the reaction volume, calculated from the reflected shock velocity and reaction time, averaged 58 cm but ranged from 52 to 62 cm and varied for each experiment. The calculated axial length of the reaction volume correlated very well to the particle dispersion found upon shock tube

(15) Incropera, F. P.; DeWitt, D. P. Fundamentals of Heat and Mass Transfer; John Wiley and Sons: New York, 1996.

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dependence), experimental results of Golovina3 (first-order dependence), and the suggested carbon-carbon dioxide reaction rate of Bradley et al.4 (1/2-order dependence). The lower temperature region of the plot shows that the rate constant calculated for all of these shock tube experiments, both the CO2 and H2O, are fit with a value of   41000 k1 ¼ k2 ¼ 6:11  102 exp ð8Þ RT where the units of k are kg m-2 s-1 atm-1/2 and the activation energy is in units of cal/mol. All of the experimental sets are fit by this value with the exception of the 213 atm carbon-steam experiments, which appear slightly lower than the other experimental sets. This is attributed to the sensitivity of H2O measurements using our GC techniques. These shock tube data are consistent with the prior investigators, but of the existing rates from the literature, the rate constant calculated from Golovina3 qualitatively matches closest to the shock tube data. Golovina3 observed a change in the slope of the rate constant, as was seen in the shock tube data, and attributed the variation to site coverage changing with temperature. However, a diffusion analysis was able to explain the leveling off in these shock tube experiments. The diffusion rate of the oxidizing species can be calculated from Fick’s law, as seen in eq 9. The oxidizing species concentration in the bulk, Cox,b, is taken as the oxidizer concentration within the mixture. When the oxidizer comes in contact with the surface, it is assumed that the species loses an O atom to the surface and is chemically altered. Therefore, the oxidizer surface species concentration, Cox,s, is taken to be zero. ð9Þ J ¼ kc ðCox, b -Cox, s Þ

Figure 7. Reaction rate coefficients: (9) 412 atm and 2392 ppm CO2, (b) 406 atm and 1109 ppm CO2, (0) 248 atm and 2107 ppm CO2, (O) 256 atm and 1018 ppm CO2, (open rectangle) 268 atm and 501 ppm CO2, () 40 atm and 2226 ppm CO2, (1) 386 atm and 471 ppm H2O, (tilted solid triangle) 347 atm and 239 ppm H2O, (3) 205 atm and 994 ppm H2O, (tilted open triangle) 213 atm and 471 ppm H2O, (;) Arrhenius fit to data, ( 3 3 3 ) Libby and Blake,2 where k has units of kg m-2 s-1 atm-1, (- - -) Golovina,3 where k has units of kg m-2 s-1 atm-1, and (- 3 -) Bradley,4 where k has units of kg m-2 s-1 atm-1/2.

coefficients calculated from the experimental results for both the carbon-carbon dioxide and carbon-steam reactions. The data were best fit when the order of oxidizer dependence, n, was assumed to be 1/2. An assumption of first-order dependence did not best fit the data and led to separate data trends that were separated on the basis of the initial oxidizer concentration. Previous investigators14,16,17 have attempted to explain the source of the 1/2-order dependence with creative reaction mechanisms. Koenig et al.16,17 proposed that the 1 /2-order dependence was due to carbon dioxide dissociatively chemisorbing onto the carbon surface, forming both an adsorbed oxygen atom as well as an adsorbed carbon monoxide molecule. However, recently, Xu et al.18 concluded in a theoretical study of carbon dioxide dissociative adsorption onto graphite surfaces that this is not the case and only an oxygen atom is adsorbed, producing a gaseous CO molecule. The reaction mechanism achieving a 1/2-order dependence by Bews et al.14 requires the formation of O2 from the reaction of multiple surface oxides, which was not observed in these shock tube experiments. These details point to the conclusion of Hurt and Haynes19 that surface heterogeneity is possibly to blame for the power-law behavior and not a creative Langmuir-Hinshelwood reaction mechanism. Figure 7 shows that the rate constant increases with temperature until reaching a maximum value, at which the rate constant appears to level off. For the lower pressure of 40 atm carbon-carbon dioxide experiments, the leveling off occurs at 2000 K, but for the higher pressure experiments, the leveling off does not occur until 2100 K. Also plotted with these experimental shock tube data are the experimental fit to the data, the theoretical results of Libby and Blake2 (first-order

To calculate the diffusion rate, only the mass-transfer coefficient, kc, is needed. The flux can be calculated from the Sherwood number via the Fr€ ossling correlation20 in eq 10. As was performed previously for the particle heating analysis, the Reynolds number in the volume behind the reflected shock wave can be neglected because the velocity is negligible, resulting in a Sherwood number of 2. Sh ¼

kc dp ¼ 2 þ 0:6Re1=2 Sc1=3 DAB

ð10Þ

With the oxidizer concentration and the carbon black particle size (2.5 μm) known, only the species diffusivity, DAB, is needed to calculate the diffusion rate. However, the gas diffusivity can be determined by the physical properties of the diffusing gases from Chapman and Enskog’s method,21 shown in eq 11, which exhibits the relationship between the diffusivity and the gas properties. DAB ¼

0:00266T 3=2 PMwAB 1=2 σ AB 2 ΩD

ð11Þ

The gas diffusivity is dependent upon the temperature and pressure as well as the Leonard-Jones energies, characteristic lengths, and collision integrals. The Leonard-Jones energies and the collision integrals are intrinsic properties of the diffusing gas, while the temperature and pressure are properties of the mixture. Although Chapman and Enskog’s expression takes

(16) Koenig, P. C.; Squires, R. G.; Laurendeau, N. M. Carbon 1985, 23, 531–536. (17) Koenig, P. C.; Squires, R. G.; Laurendeau, N. M. Fuel 1986, 65, 412–416. (18) Xu, S. C.; Irle, S.; Musaev, D. G.; Lin, M. C. J. Phys. Chem. B 2006, 110, 21135–21144. (19) Hurt, R. H.; Haynes, B. S. Symp. Int. Combust. Proc. 2005, 30, 2161–2168.

(20) Fogler, H. S. Elements of Chemical Reaction Engineering; Prentice Hall PTR: Upper Saddle River, NJ, 1999. (21) Kuo, K. K. Principles of Combustion; John Wiley and Sons, Inc.: New York, 2005.

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Figure 9. Experimental rate of the reaction and theoretical diffusion limit for H2O reactions: (b) 386 atm and 470 ppm experimental rate, (O) 386 atm and 470 ppm diffusion rate limit, (2) 347 atm and 239 ppm experimental rate, and (4) 347 atm and 239 ppm diffusion rate limit. r is in units of kg m-2 s-1.

Figure 8. Experimental rate of the reaction and theoretical diffusion limit for 260 atm CO2 experiments: (9) 2067 ppm experimental rate, (0) 2067 ppm diffusion rate limit, (b) 990 ppm experimental rate, (O) 990 ppm diffusion rate limit, (2) 491 ppm experimental rate, and (4) 491 ppm diffusion rate limit. r is in units of kg m-2 s-1.

pressure into account, real mixture diffusivities vary from ideal at high pressures and need to be corrected using a high-pressure diffusivity formula taken from Takahashi.22 Takahashi’s22 expression relates a high-pressure correction factor to the reduced pressure and reduced temperature of the system, which were empirically determined within Takahashi’s work.22 In addition to the adjustment for high pressure, Knudsen pore diffusion also needs to be accounted for because of the porosity present in the carbon black particles. Knudsen pore diffusivity can be calculated,23 as in eq 12, from the pore radius, particle temperature, and molecular weight of the diffusing gas. pffiffiffiffiffiffiffiffiffiffiffiffi ð12Þ DK ¼ 9700re T=M When the system pressure is sufficiently low, the process of diffusion can be exclusively limited by pore diffusion, and at intermediate pressures, the two diffusion processes can combine in transport resistance. The combined transport resistances form an effective diffusivity governed by eq 13.23   1 1 -1 Deff ¼ þ ð13Þ DK DAB

Figure 10. Experimental rate constant data below Mears’ criterion: (9) C(s) þ CO2 data and () C(s) þ H2O data. k has units of kg m-2 s-1 atm-1/2. (;) Arrhenius fit to the data.

is in fact a result of the presence of a diffusion limitation and not a change in the chemistry, the data are now revisited to ensure that the original fit to the rate constant data was not calculated from a region affected by diffusion. If the experimental rate data falls below Mears’ criterion,20 eq 14, it can be assumed that the data are not affected by diffusion. Mears’ criterion is calculated from the experimentally measured reaction rate (rA0 ), the order of dependence of the oxidizing gas (n), and the diffusion rate (kcCox,b). The experimental rate constant data meeting Mears’ criteria are plotted in Figure 10 for both the carbon-carbon dioxide and carbon-steam reactions and coincide with that used for the original fit to the experimental rate constant data. rA 0 n < 0:15 ð14Þ kc Cox, b =Fb R

With the effective diffusivity calculated, the diffusion rate can now be determined. The theoretical diffusion rate was calculated for each shock tube experiment. The experimental rate data and the theoretical diffusion rate calculated from the experimental conditions for the experiments performed at approximately 260 atm for the carbon-carbon dioxide reactions are shown in Figure 8. The figure shows that, as the temperature is increased, the experimental rate reaches the diffusion limit. The data from all three experimental sets at approximately 260 atm matched the diffusion limit. Figure 9 shows the rate and diffusion rate limit for the two carbonsteam experimental sets performed at approximately 365 atm. The experimental rates for the carbon-steam reactions also approach the diffusion limit as the temperature is increased. With the diffusion analysis performed resulting in the conclusion that the leveling off of the high-temperature data

Conclusion The heterogeneous reactions between solid carbon with carbon dioxide and steam were experimentally investigated at the elevated temperatures and pressures that reflect the

(22) Takahashi, S. J. Chem. Eng. Jpn. 1974, 7, 417–420. (23) Satterfield, C. N.; Sherwood, T. K. The Role of Diffusion in Catalysis; Addison-Wesley Publishing Company, Inc.: Reading, MA, 1963.

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: DOI:10.1021/ef900641r

Culbertson and Brezinsky

conditions existing in a rocket nozzle environment. A particle injector was designed, fabricated, and characterized specifically for these high-pressure heterogeneous experiments. Reaction- and diffusion-limited regimes were observed in the experimental data. At the lower temperatures where chemistry dominated the reaction, it was determined that the order of oxidizer dependence was 1/2 for both the carbon-carbon dioxide and carbon-steam reactions. The rate constant calculated from these highpressure shock tube experiments was close to values suggested by Libby and Blake,2 who studied the reactions at lower pressures. The two rate constants determined in the current work for the carbon-carbon dioxide and carbon-steam reactions were found to be equal and best represented by   41000 kg m-2 s-1 atm-1=2 k ¼ 6:11  102 exp RT

Nomenclature As =particle surface area as =particle specific surface area cp =specific heat capacity of carbon particles D=gas diffusivity Dk =Knudsen diffusivity dp =carbon particle diameter h=heat-transfer coefficient k=reaction rate coefficient k=conductive heat-transfer coefficient kc =mass-transfer coefficient Lc =characteristic particle length mc =mass of carbon injected into the shock tube n=order of dependence of reacting gas Ω=gas particle collision integral P=total pressure pi =partial pressure of reacting gas R=universal gas constant re =particle pore radius rA0 =reaction rate r_c,i =rocket nozzle regression rate F=density of carbon particles Fb =bulk density of carbon particles σAB =characteristic bond length T=particle temperature T¥ =ambient temperature V=volume of the carbon particle VC =volume of the reaction

The rate constant is applicable over the temperature range of 1400-2000 K. Acknowledgment. We extend our appreciation to Prof. Nick Glumac for his generosity in sharing his advice and particle injector design. We also thank Dr. Robert Tranter for his advice and overall guidance on this work. This research is supported by subcontract 2799-UI-ONR-0683 from Penn State University through an Office of Naval Research Multi-university Research Initiative (MURI) (prime award N0014-04-1-0683) under the direction of Prof. Ken Kuo.

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