reacting carbon particle-laden oxygen gas behind a shock wave

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REACTING CARBON PARTICLE-LADENOXYGEN GAS BEHIND A SHOCK WAVEJun Sung Park a & Seung Wook Baek aa Division of Aerospace Engineering, Department of MechanicalEngineering , Korea Advanced Institute of Science and Technology ,Yuseong-gu, Daejeon, Republic of KoreaPublished online: 01 Sep 2006.

To cite this article: Jun Sung Park & Seung Wook Baek (2005) REACTING CARBON PARTICLE-LADENOXYGEN GAS BEHIND A SHOCK WAVE, Numerical Heat Transfer, Part A: Applications: An InternationalJournal of Computation and Methodology, 47:3, 269-289, DOI: 10.1080/10407780590886403

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REACTING CARBON PARTICLE-LADEN OXYGEN GASBEHIND A SHOCK WAVE

Jun Sung Park and Seung Wook BaekDivision of Aerospace Engineering, Department of Mechanical Engineering,Korea Advanced Institute of Science and Technology, Yuseong-gu,Daejeon, Republic of Korea

An analysis of the flow field, which develops when a shock wave hits a two-phase medium

comprising carbon particles and oxygen gas, has a practical application to industrial acci-

dents such as explosions at coal mine and in grain elevator. Therefore, its successful predic-

tion of thermo-fluid mechanical characteristics would be very crucial and imperative. This

paper describes and inherent interaction phenomenon behind a shock wave for a two-phase

medium of gas and particles with chemical reaction. A carbon particle-laden oxygen gas is

considered to be located along a ramp so that numerical integration is accomplished from

the tip of ramp for a finite period. For numerical solution, a fully conservative unsteady

implicit 2nd order time-accurate sub-iteration method and the 2nd order Total Variation

Diminishing (TVD) scheme are used with the finite volume method (FVM) for gas phase.

For particle phase, the Monotonic Upstream Schemes for Conservation Laws (MUSCL)

as well as the solution of the Riemann problem for the particle motion equations is used.

The transient physical development is discussed in comparison with the cases of the pure

gas and the reacting particle-laden gas. The results are then extended to changing the initial

gas temperature as well as the particle diameter and particle mass fraction. Major results

reveal that for the reacting particle-laden gas flow, the adverse pressure gradient is so high

that there exists some region in which the particle velocity exceeds the gas velocity. When

the particle diameter is smaller and the particle mass fraction is higher, the thermo-fluid

dynamic behavior is significantly affected due to stronger interaction of momentum and

thermal energy in two-phase mixture.

INTRODUCTION

Particle-laden gas flow is a general category of two-phase flow which showsdifferent behavior according to initial composition, phase (solid, liquid, or gas),chemical reaction, and other parameters (temperature, pressure, etc.). It is foundin a lot of fundamental as well as practical engineering applications such as cycloneseparators and classifiers, pneumatic transport of powder, and so on. When chemicalreaction is involved, it has a close practical relation to other industrial applications(e.g., solid rocket engines in which aluminum particles are used to reduce combus-tion instability) as well as industrial accidents such as explosions in coal minesand grain elevators.

Received 17 July 2004; accepted 4 September 2004.This research was financially supported by the Combustion Engineering Research center at the

Korea Advanced Institute of Science and Technology.Address correspondence to Seung Wook Baek, Division of Aerospace Engineering, Department

of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong,

Yuseong-gu, Daejeon 305-701, Republic of Korea. E-mail: [email protected]

269

Numerical Heat Transfer, Part A, 47: 269–289, 2005

Copyright # Taylor & Francis Inc.

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407780590886403

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There are a few experimental as well as numerous analytical or numerical stu-dies that examine the effects of two-phase medium on the gas flow field withoutconsidering chemical reaction. Carrier [1] and Rudinger [2] proposed the generalequations governing a shock-wave interaction with a two-phase medium of airand glass beads, and studied their thermofluid mechanical behavior, based on a per-fect gas assumption. A comprehensive review of gas-particle nozzle flows waspresented by Hoglund [3]. In this article, two-phase theoretical and experimental stu-dies and discussion about the one-dimensional approximation were given. Chang [4]used the MacCormack scheme to study the gas and particle flow patterns in axisym-metric convergent-divergent nozzles by changing such parameters as particle diam-eter and particle mass fraction. Bendor et al. [5] studied various reflection patternsof planar shock waves from straight wedges in dust–gas suspensions. The resultsprovided a clear picture of whether and how the presence of dust particles affectsthe shock-induced flow field.

However, a study of two-phase reacting flow has been very limited until now.Among others, Sichel et al. [6] measured the ignition delays of various dusts such asPittsburgh seam coal, graphite, diamond, oats, and RDX (cyclotrimethylene trini-tramine). This study showed the detonability as well as flammability of variousdust=oxidizer mixtures, based on the comparative ignition delays of the dust parti-cles when suddenly exposed to a high-temperature environment. Elperin et al. [7]and Igra et al. [8] analyzed the interaction of a normal shock wave with carbon-particle laden oxygen gas. These articles investigated a variety of physical phenom-ena in the relaxation zone, provided the properties of carbon particles were constantduring reaction. Gokhale and Bose [9] extended a nonreacting two-phase flow to areacting one in a one-dimensional nozzle by using the explicit MacCormack scheme.Recently, Park and Baek [10] described the interaction phenomenon when a movingshock wave hits a two-phase medium of gas and particles with=without chemicalreaction. In their study, a transient development of thermofluid mechanical charac-teristics was predicted mainly for the case without chemical reaction and discussedfor various particle mass densities and particle specific heats. An example result withchemical reaction was also presented, but a more detailed description could not bediscussed therein due to the paper length.

Therefore, the purpose of the present study is to further describe an unsteadytwo-dimensional ignition phenomenon when a moving shock wave hits a two-phase medium of gas and particles, especially considering the effects of particle

NOMENCLATURE

Cp specific heat at constant pressure and

volume respectively, J=kgK

D particle diameter, m

L reference length, m

Pr Prandtl number

t time, s

T temperature, K

x, y Cartesian coordinates, m

q gas density, kg=m3

j thermal conductivity of gas, J=Kms

m dynamic viscosity, kg=ms

r mass density of particle per unit

volume, kg=m3

/ mass fraction ratio of particles,

[¼qp=(qþ qp)]

Subscripts

g gas phase

p particle phase

1 ambient state

270 J. S. PARK AND S. W. BAEK

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combustion. A mixture of carbon dust and pure oxygen is assumed to be uniformlydistributed in the whole domain. The detailed transient variations of chemical as wellas thermofluid mechanical behavior are illustrated for various initial gas tempera-tures, particle mass fractions, and particle diameters.

GOVERNING EQUATIONS

The current study deals with ignition phenomena of a carbon particle-ladenoxygen gas initiated by a moving shock wave, when a two-dimensional 27� rampexists in the flow passage as indicated in Figure 1. When the shock wave hitsa two-phase medium that is initially at rest, the gas-phase velocity is instantly accel-erated while its temperature rapidly increases. However, the particle velocity andtemperature do not immediately respond to a physical change in gas because ofthe inertia and heat capacity of the particles. These effects incur the velocity slipand temperature difference between the gas and particle phases, whereby themomentum and energy exchanges occur. Through these interactions, the gas andparticle phases finally reach the equilibrium state in the long run. Since the ratesof momentum and energy exchanges are quite limited within the initial period of50 ms after shock-wave interaction with the mixture, the gas and particle phasesare still out of equilibrium. The carbon particle temperature increases due to the heat

Figure 1. Schematic view of particle-laden gas medium with a moving shock wave.

PARTICLE-LADEN GAS BEHIND A SHOCK WAVE 271

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feedback from gas while going downstream. Once its temperature reaches the ignitiontemperature, carbon particles will burn. Therefore, the flow field variation for thecase with chemical reaction will show an entirely different pattern than the nonreactingcase, due to the high-temperature environment incurred by the carbon reaction [10].

For the present analysis, the following assumptions are introduced. The gasand particle flows are two-dimensional and continuous, which represents a Eulerianapproach. The particles are assumed to be spherical and uniform in size at each localposition. It is reasonable that the volume occupied by the particles is negligible sothat they do not interact with each other, since the particle sizes considered in thepresent study are smaller than 10 mm. And the inner temperature distribution insidethe particle is assumed to be uniform. Moreover, the effect of the radiative heattransfer is not taken into consideration, and the thermal and Brownian motions ofthe particles are neglected. Before the shock wave interacts with the two-phase mix-ture, it is in a state of thermodynamics and kinematic equilibrium.

In addition to these assumptions, the following simplifications are furtherintroduced to derive the governing equations. The gas mixture consists simply oftwo species such as oxidant gas, O2, and product gas, CO2. Any external forces suchas gravity or electromagnetic force are neglected. Based on these assumptions, thegoverning equations, i.e., the two-dimensional time-dependent compressible Eulerequations including the species conservation equation due to the chemical reactionbetween gas and particles, are given in the conservative form in the authors’ previousarticle [10]. The detailed forms of the governing equations are omitted here for brev-ity. In order to prevent the numerical calculation from being unstable, all variables inthe above equations are nondimensionalized using the reference flow and physicalproperties such as L1, q1, T1, a1, and m1.

For a calculation of reacting flows, the accurate evaluation of thermophysicalgas properties is of vial importance. In this article, the specific heat, Cp, and thermalconductivity, j, for each species are determined by the third order polynomials oftemperature [11], and the visocsity, m, is evaluated using the Sutherland formulafor each species [12]. The thermophysical properties of the mixture are estimatedusing Wilke’s mixing rule [12].

Additionally, the carbon burning rate needs to be determined for calculatingchemical reaction. In this work, the carbon reaction takes place in a pure oxygenenvironment. Furthermore, the initial amount of carbon is much smaller than thatof oxygen, so that it is reasonable to assume that the carbon is fully oxidized tocarbon dioxide. Therefore, the following single-step chemical reaction model is usedhere to describe the carbon reaction rate:

CðsÞ þO2ðgÞ ! CO2ðgÞ þ 3:937� 105 kJ=kmol ð1Þ

as referred to by Baek and Seung [13].

NUMERICAL METHOD

As mentioned in the previous section, the two sets of equations which governthe behavior of the gas and particles are of hyperbolic type. Each variable is closelycoupled with the others, since there occur strong momentum and energy exchanges


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between gas and particles. Hence, in order to get the correct as well as stable sol-ution, it is important to choose adequate numerical methods for solving these typesof equations. In the present analysis, a coordinate transformation from a physicaldomain to a computational one is employed to enhance numerical efficiency andto conveniently apply the physical boundary conditions. Also, the finite-volumemethod (FVM), which easily satisfies the conservation rule and is computationallystable at the surface of discontinuities, is applied.

In the course of interacting developments, there is some zone where particlesare not observed, that is, the particle density becomes zero. However, the gas densitycannot always be zero. Therefore, two different approaches are introduced for eachphase. For the gas phase, the second-order total variation diminishing (TVD)scheme [14] with van Leer limiter and Roe’s average is applied, while for the particlephase the monotonic upstream scheme for conservation laws (MUSCL) withminmod limiter [15] as well as the solution of the Riemann problem for the particlemotion equations [16] is used.

For the accurate time integration of the above governing equations, the LUdecomposition as proposed by Jameson and Turkel [17] is adopted to reduce thecomputational effort in calculating the inverse matrices. Simultaneously, a fully con-servative unsteady, implicit second-order, time-accurate subiteration method [18] isused.

INITIAL AND BOUNDARY CONDITIONS

In this analysis of the unsteady two-phase flow over a ramp, one of the mostimportant problems in view of convergence as well as numerical accuracy is treat-ment of the initial and boundary conditions. At the initial time, the gas and the par-ticles are presumed to be in thermal equilibrium and at rest. And the particles areuniformly distributed along a ramp as illustrated in Figure 1, whereas the shock-wave front is initially positioned at the inlet.

Figure 1 shows a schematic of the particle-laden gas medium with a movingshock wave and a ramp of which the inclined angle is 27�. While the x coordinateis in parallel to the shock wave moving direction, the y coordinate is in the verticaldirection to the x axis. The plane orthogonal to the x–y plane is assumed to beinfinite so that, in the present study, the computational domain is limited to onlythe x–y plane. The domain is divided into a 151� 81 grid system and the smallestcells are located near the wall. The time step used is 0.01 ms after many preliminarycalculations are performed with different grid sizes and time steps.

Four types of boundary conditions are required for the computation of theflow field, i.e., upper and lower walls, inlet and outlet conditions. At the upperand lower walls, a slip as well as adiabatic condition is imposed, since the flow isassumed to be inviscid and impermeable. At the outlet, the outflow condition ofthe first-order extrapolation is used, for an incident shock wave moves at supersonicspeed. At the inlet, the boundary conditions for gas are determined using the postconditions behind the given shock wave by equating the Fanno line equation andthe Rayleigh line equation. For the particle phase, different conditions are applied.At the walls, the slip condition (neither adherence nor reflection condition) as well asthe impermeable condition is provided, while the other properties are extrapolated


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from the inner ones. The inlet and outlet boundary conditions are initially specifiedsuch that there are no particles at the inlet. A numerical computation is finishedbefore the shock-wave front reaches the outlet of the domain.

RESULTS AND DISCUSSION

The effects of the reacting dust suspension (carbon particle-laden oxygen gas)on the flow properties are investigated numerically in the following by changing theinitial gas environment temperature and the particle parameters such as particle di-ameter and particle mass fraction. In order to discuss these parameters compara-tively, the case with reacting particles is compared with the pure gas case withoutany particle in this study under the operating conditions listed in Table 1.

Before the numerical results for the shock-induced combustion of the carbonparticle-laden oxygen gas are discussed, it is necessary to understand the fundamen-tal structure of the shock wave. Its main structure in the computational domain witha ramp comprises the slip line, i.e., a weak discontinuity that separates the thermo-dynamic regions with different densities, Mach stem, and triple point at which theincident shock, the reflected shock, and the Mach stem merge with one another asreferred to in [10].

In the following, the results obtained with the current code developed here willbe presented and discussed.

Effects of Initial Gas Temperature

The variations of the gas and particle velocities are examined in Figure 2a whenthe initial gas temperature is varied from 600 to 1,000K. In general, the propagationspeed of a shock wave is seen to increase at 50 ms, as shown in Figure 2a, since theinitial gas temperature is different even if the incident shock-wave Mach number isthe same for all, as listed in Table 1. Whereas the gas hit by the shock wave isinstantly accelerated due to shock interaction, the particles are still stagnant. Theyare instead accelerated by a momentum transfer between gas and particles. There-fore, the rates of momentum and energy exchanges between the two phases is muchhigher for an elevated initial gas temperature. Through a comprehensive examin-ation, a unique region is observed for Tg,initial ¼ 800 and 1,000K, in which theinitially slower particles become faster than the gas velocity. This is, so called, theshock-wave expansion region, which is located at the front of the slip line. Thisphenomenon results from a strong adverse pressure gradient caused by chemical

Table 1. Operating conditions for reacting particle-laden gas flow

Gas temperature (K) 600.0–1000.0

Gas pressure (Pa) 33,325.0

Mach number 2.03

Angle (�) 27.0

Particle diameter (mm) 1.0–10.0

Particle mass fraction 0.09–0.33

Particle mass density (kg=m3) 2,248.2

Particle specific heat (J=kgK) Third-order polynomial for each species [11]


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Figure 2. Effect of initial gas temperatures on the velocity and pressure distributions along the wall at

t¼50 ms.


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reaction as shown in Figure 2b. In comparison with the case of ‘‘no particle’’ atTg,initial ¼ 800K, it is also observed that the shock-wave propagation speed fortwo phases becomes slower due to the interaction between gas and particles.

Figure 3 shows the transient particle concentration distributions along the wallfor the initial gas temperatures of 600 and 1,000K. As the initial gas temperatureincreases, the particles are observed to spread out over a wider region and are sweptaway far downstream due to stronger momentum interaction. The detailed spatialdistribution of the particle concentration near the shock wave is given in Figure 4,which illustrates that the gradient of the particle concentration is lower for the higherinitial gas temperature.

Based on the above discussion, the high initial gas temperature environmentinduces a higher relative velocity between gas and particles due to stronger carbonreaction, so that the gas temperature behind the shock wave also becomes higher,as confirmed in Figure 5a, which shows a rapid increase in gas temperature forTg,initial ¼ 800 and 1,000K. For the case of Tg,initial ¼ 600K, no ‘‘chemical reaction’’occurs because the shock-wave speed is not high enough to ignite particles. Conse-quently, the gas temperature decreases through heat transfer to the particles andits temperature finally becomes the same as the particle temperature. The mass frac-tion distributions of CO2 and O2 at t ¼ 50 ms are depicted in Figure 5b. Obviously,more CO2 gas is produced for Tg,initial ¼ 1,000K, whereas no CO2 is available forTg,initial ¼ 600K.

Figure 3. Effect of initial gas temperatures on the transient particle concentration distributions along

the wall.


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Figure 6 illustrates the temporal variation of the gas density with the initial gastemperature of 600 and 1,000K. Simply, as the initial gas temperature increases, thegas density decreases due to higher temperature incurred by the carbon reaction. Thedetailed spatial distributions of the gas density are given in Figure 7, which showsthat as the initial gas temperature increases, the gas density gradient becomes higherover a wider region where the particles exist, and its distribution is more complex.

Effects of Particle Diameter

Spatial and temporal variations of the particle concentration are represented inFigures 8 and 9 for two particle diameters of 1.0 and 10.0 mm. It is noted that theparticle mass fraction, / ¼ 0:23, is fixed here, so that the particle number densitydecreases when the particle diameter increases. Furthermore, when the particle diam-eter is varied, the velocity as well as the temperature relaxation time are also changedaccording to the following equations (2) and (3) as suggested by Rudinger [19]:

sv ¼rpD

2p

18mð2Þ

sT ¼ 3

2Prdsv ð3Þ

where d is the ratio of the particle specific heat to the gas specific heat. This revealsthat as the particle diameter increases, so does each relaxation time in proportion tothe square of the particle diameter. In other words, more time is required for the gasto accelerate and heat up the particles with larger diameter.

Figure 8 shows that for the case of 1.0 mm, particles move farther downstreamalong the ramp and are distributed more uniformly. To the contrary, particles withdiameter of 10.0 mm are almost stationary, getting denser only near the ramp corner.

Figure 4. Particle concentration contours near shock wave at t¼50 ms: (a) Tg,initial¼600K; (b)

Tg,initial¼1,000K.


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Figure 5. Effect of initial gas temperatures on the temperature and mass fraction distributions along the

wall at t¼50ms.


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This is more obviously manifested in Figure 9, which shows the temporal variationof particle concentration. Figure 10 shows that the gas density gradient in the relax-ation zone becomes higher for smaller particle diameters, since more gas momentumand thermal energy are transferred to the particles with more active chemical reac-tion. This tendency can be more clearly seen in Figure 11, which reveals that the wall

Figure 6. Effect of initial gas temperatures on the transient gas density distributions along the wall.

Figure 7. Nondimensional gas density contours at t ¼ 50 ms: (a) Tg,initial ¼ 600K; (b) Tg,initial ¼ 1,000K.


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density distribution changes abruptly in the relaxation zone for smaller particle dia-meters. This change is not caused simply by the physical interaction of oxygen gasand carbon particles. Additionally, the chemical reaction of oxygen gas and carbonparticles plays an important role.

Figure 8. Particle concentration contours at t¼50ms: (a) Dp¼1.0mm; (b) Dp¼10.0mm.

Figure 9. Effect of particle diameters on the transient particle concentration distributions along the wall.


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Gas and particle velocity distributions along the wall are illustrated in Figure 12a.When the particle diameter is larger (10 mm), the momentum interaction between gasand particles is weaker, so that the gas velocity is a little lower than that for the ‘‘noparticle’’ case and the particle velocity developed is much lower than that for 1 mm.

Figure 10. Nondimensional gas density contours at t¼50 ms: (a) Dp¼1.0mm; (b) Dp¼10.0mm.

Figure 11. Effect of particle diameters on the transient gas density distributions along the wall.


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On the other hand, for the case of particle diameter 1 mm, the particle velocity easilycatches up with the gas before ignition occurs. However, once ignition occurs, theparticle velocity becomes even higher than that of the gas due to the strong adverse

Figure 12. Effect of particle diameters on the velocity and transient pressure distributions along the wall.


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Figure 13. Effect of particle diameters on the temperature and mass fraction distributions along the wall

at t¼50 ms.


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pressure effects as shown in Figure 12b. Furthermore, since the momentum transferbetween gas and particles is stronger for smaller particle diameters, the shock-wavespeed becomes slower. Similar reasoning applies to the thermal interaction. After theshock wave hits the two-phase medium, the gas temperature rapidly increases,thereby warming up particles as drawn in Figure 13a. When the carbon particlesare ignited, as for the particle diameter of 1 mm, both gas and particle temperaturesrapidly increase again. However, for the case of particle diameter 10 mm, the particletemperature does not reach the carbon ignition temperature due to weak thermal in-teraction, so the gas temperature is not different from that for the ‘‘no particle’’ case.Figure 13b represents mass fraction distributions of oxidizer (O2) and product (CO2)gases along the wall. As explained in the preceding paragraph, for the smaller par-ticle diameter the chemical reaction becomes active so that the product gas is pro-duced whereas oxidizer is reduced. However, for the large particles, no reactionoccurs so that no variation in oxidizer and product gases is observed.

Effects of Particle Mass Fraction

A change of the particle mass fraction, which is related directly to the change ofthe number of particles in the suspension, will not change the velocity and tempera-ture relaxation times according to Eqs. (2) and (3). However, the chemical reactionrate and variation in the number of particles will affect the characteristics of thewhole flow field. When the particle mass fraction increases, the number of particleswill increase for fixed particle size. The effects of the particle mass fraction on thephysical variations when the particle diameter is kept at 1 mm will be discussed inthe following.

Figure 14 shows the particle concentration distributions for two cases of par-ticle mass fraction, / ¼ 0.09 and / ¼ 0.33. The gradient of particle concentration isseen to be much higher for / ¼ 0.33, which is quantitatively more apparent in Figure 15.Figure 16 represents the gas density distribution with variations of the particle mass

Figure 14. Particle concentration contours at t¼50 ms: (a) / ¼ 0.09; (b) / ¼ 0.33.


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fraction. As the particle mass fraction increases, the amount of carbon particlesinvolved in the chemical reaction increases. Hence, it is expected that the gastemperature distribution will be more complex for higher particle mass fraction

Figure 16. Nondimensional gas density contours at t ¼ 50 ms: (a) / ¼ 0.09; (b) / ¼ 0.33.

Figure 15. Effect of particle mass fractions on the transient particle concentration distributions along

the wall.


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and so is the gas density as in the figure. Transient variation of gas density is alsoplotted quantitatively and compared in Figure 17, which indicates that the gas den-sity is varied in a broader region for higher particle mass fraction.

Figure 18a shows the gas and particle velocity distributions along the wall.Since the velocity relaxation time does not depend on the particle mass fraction,as shown in Eq. (2), the velocity distribution is analogous. However, as the particlemass fraction increases, the momentum transfer between gas and particles isenhanced, so the gas as well as the particle velocity become slower. Similarly, thethermal exchange also becomes more active for higher particle mass fraction. Conse-quently, behind the shock wave the gas temperature for / ¼ 0:33 is lower than thatfor / ¼ 0:09 before ignition occurs. However, once ignition occurs, it is reversed, inFigure 18b.

Figure 19 indicates the mass fraction distributions of the chemical species CO2

and O2. Obviously, more CO2 and less O2 are shown in the figure for higher particlemass fraction.

CONCLUDING REMARKS

The conservation equations are solved numerically for an analysis of interac-tion phenomena of reacting carbon particle-laden oxygen gas behind a shock waveusing methods explained previously. Different from the previous literatures, this

Figure 17. Effect of particle mass fractions on the transient gas density distributions along the wall.


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Figure 18. Effect of particle mass fractions on the velocity and temperature distributions along the wall

at t¼50 ms.


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article focused on elucidating the general thermofluid dynamic behavior of a two-phase reactive mixture together with discussing the effects of the initial gas tempera-ture as well as the particle diameter and the particle mass fraction in two dimensions.Main conclusions are as follows.

1. As the initial gas temperature increases, the velocity difference between gasand particles becomes higher. Also, the particles are scattered over a widerregion and are dragged far downstream due to stronger momentum andenergy exchanges.

2. For a smaller particle diameter, behind the shock wave the particle velocityeasily follows the gas before ignition occurs. However, after ignition, theparticle velocity becomes higher than that of the gas due to the strongadverse pressure effects. Furthermore, due to the stronger momentumtransfer between gas and particles for smaller particle diameter, theshock-wave speed is slower.

3. For a higher particle mass fraction, the momentum transfer between gasand particles is so high that the gas as well as the particle velocity becomesmaller. Accordingly, the thermal exchange is also more active. This resultsin the gas temperature behind the shock wave becoming lower before ig-nition occurs. However, after ignition, it is reversed.

Figure 19. Effect of particle mass fraction on O2 and CO2 mass fraction distributions along the wall at

t¼50ms.


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REFERENCES

1. G. F. Carrier, Shock Waves in a Dusty Gas, J. Fluid Mech., vol. 4, pp. 376–382, 1958.2. G. Rudinger, Some Properties of Shock Relaxation in Gas Flows Carrying Small

Particles, Phys. Fluids, vol. 7, pp. 658–663, 1964.3. R. F. Hoglund, Recent Advances in Gas-Particle Nozzle Flows, ARS J., pp. 662–671,

1962.4. I.-S. Chang, One- and Two-Phase Nozzle Flows, AIAA J., vol. 18, pp. 1455–1461, 1980.5. G. Bendor, O. Igra, and L. Wang, Shock Wave Reflections in Dust-Gas Suspensions,

J. Fluids Eng., vol. 123, pp. 145–153, 2001.6. M. Sichel, S. W. Baek, C. W. Kauffman, B. Maker, J. A. Nicholls, and P. Wolanski, The

Shock Wave Ignition of Dusts, AIAA J., vol. 23, pp. 1374–1380, 1985.7. I. Elperin, O. Igra, and G. Ben-Dor, Analysis of Normal Shock Waves in a Carbon

Particle-Laden Oxygen Gas, J. Fluids Eng., vol. 108, pp. 354–359, 1986.8. O. Igra, G. Ben-Dor, and I. Elperin, Parameters Affecting the Postshock Wave Relaxation

Zone in an Oxygen Carbon Particle Suspension, J. Fluids Eng., vol. 108, pp. 360–365,1986.

9. S. S. Gokhale and T. K. Bose, Reacting Solid Particles in One-Dimensional Nozzle Flow,Int. J. Multiphase Flow, vol. 15, pp. 269–278, 1989.

10. J. S. Park and S. W. Baek, Interaction of a Moving Shock Wave with a Two-Phase Reac-ting Medium, Int. J. Heat Mass Transfer, vol. 46, pp. 4717–4732, 2003.

11. M. W. Chase, Jr., C. A. Davies, J. R. Downey, Jr., D. J. Frurip, R. A. McDonald, andA. N. Syverud, JANAF Thermochemical Tables, 3d ed., J. Phys. Chem. Ref. Data, vol.14, 1985.

12. R. C. Reid, J. M. Prausnitz, and B. E. Poling, The Properties of Gases & Liquids, 4th ed.,McGraw-Hill, New York, 1988.

13. S. W. Baek and S. P. Seung, Parametric Analysis on the Postshock Wave Relaxation Zoneof Carbon Particle-Laden Oxygen Gas, Combustion and Flame, vol. 80, pp. 126–134, 1990.

14. H. C. Yee, Construction of Explicit and Implicit Symmetric TVD Schemes and TheirApplications, J. Comput. Phys., vol. 68, pp. 151–179, 1987.

15. H. C. Yee, A Class of High Resolution Explicit and Implicit Shock-Capturing Methods,NASA TM-101099, 1989.

16. R. Saurel, E. Daniel, and J. C. Loraud, Two-Phase Flows: Second-Order Schemes andBoundary Conditions, AIAA J., vol. 32, pp. 1214–1221, 1994.

17. A. Jameson and E. Turkel, Implicit Schemes and LU Decompositions, Math. Comput.,vol. 37, pp. 385–397, 1981.

18. T. H. Pulliam, Time Accuracy and the Use of Implicit Methods, AIAA Paper 93-3360,AIAA 11th Computational Fluid Dynamics Conf., Orlando, FL, 1993.

19. G. Rudinger, Fundamentals of Gas-Particle Flow, Elsevier, New York, 1980.


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reacting carbon particle-laden oxygen gas behind a shock wave

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