Nonstationary Combustion of Methane with Gas Hydrate

Publication Date (Web): October 15, 2013 ... Dissociation is divided into several characteristic time stages with different rates of gas hydrate decom...
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Nonstationary Combustion of Methane with Gas Hydrate Dissociation S. Ya. Misyura* and V. E. Nakoryakov Institute of Thermophysics Siberian Branch, Russian Academy of Sciences, 1. Akad. Lavrentyev Avenue, Novosibirsk, 630090 Russia ABSTRACT: Kinetics of methane hydrate dissociation at combustion in the air atmosphere at the outer pressure of 1 bar was studied experimentally. Dissociation is divided into several characteristic time stages with different rates of gas hydrate decomposition. There is a very high temperature gradient along the granule radius. At combustion the movement velocities of dissociation front and pores are high. Thermal imaging of the powder surface demonstrates significant temperature maldistribution. Spatially nonuniform escape of methane from a sample leads to local temperature and concentration maldistributions in the flame, which cause thermal instability of the flame front and the loss of hydrodynamic stability. Instability of combustion generates two equivalent mechanisms of convective transfer: the shear laminar flow and large-scale vortices, which intensify fuel and oxidizer mixing. Experimental data have revealed not only the local spatial-time nonuniformities but also significantly nonstationary character of the averaged flow. The measured instantaneous and average velocities of methane on the powder surface allowed an estimate of the stoichiometric ratio. The maximal velocity of methane propagation corresponded to 1 mm/s in experiments without combustion and to 15 mm/s with combustion.

1. INTRODUCTION The problems of power and environmental safety forced us to be engaged into the search for alternative energy sources, which will soon influence the stable development of human society. Huge deposits of natural gas in the form of gas hydrates are found in the marine sediments, which in the coming years will be used widely.1 Along with a great number of technological tasks, there are two common problems relating to natural hydrates: transportation and storage of raw material in large tanks and the question of safety.1,2 Dissociation of methane hydrate under the pressure of 1 bar starts at very low temperatures. From the economic point of view, it is advisable to store methane hydrate at the temperature as close to the melting point of ice as possible. In this regard, in the last decades increased attention is paid to the mechanism of selfpreservation: the abnormally low rates of clathrate hydrates dissociation at the annealing temperature. There are almost no experimental and theoretical data on dissociation rates at combustion. Long-term storage of methane hydrate in huge reservoirs can cause unexpected decomposition of clathrate hydrate, accumulation of critical methane concentration, and its ignition. The risk of such a disaster initiates experimental study of not only clathrate hydrate dissociation but also its combustion at organization of the free convective flow. Methane combustion in the oncoming laminar flow at dissociation of gas hydrate was studied in refs 3−6. Methane combustion and velocities of flame propagation in the framework of 1D heat model was studied numerically in ref 4. Combustion of methane in free convection was studied in refs 7 and 8. The rate of gas hydrate dissociation is determined by the driving forces: deviation of pressure and temperature from the equilibrium values; structural parameters of crystalline cavities (sI, sII, and sH cubic structures).9,10 The hydrocarbon guest molecules introduced into the skeleton of water molecules © 2013 American Chemical Society

often form sI cubic structure typical for methane hydrate. Together with the above key parameters the important role at dissociation is also played by the size of particles,11,12 structural characteristics of ice,13,14 initial defectiveness of material and interaction features within the “host” and “guest” lattice.9,10 The dissociation rate depends also on the ratio of granule diameter to the size of ice grain and sample layer height12 as well as on the value of external heat flux.15 The abnormally low rates of gas hydrate decay related to formation of a solid ice crust can occur at negative temperatures.12−14 The strength characteristics of the crust depend on its thickness and grain size16 as well as on the ratio of crust thickness to the ice grain diameter.12 The mechanism of self-preservation was studied in refs 7,8,11−21. The mechanism of methane combustion relates not only to clathrate hydrate dissociation but also to hydrodynamics and heat transfer in the flow of the gas mixture. The free convective air motion is organized on the powder surface at combustion. The rate of chemical reaction in the flame will be determined by the gas flow conditions (laminar or turbulent), diffusion and convective mixing. At transitional and turbulent flows it is important to take into account the mechanism of turbulent diffusion and the ratio of typical time scales. Approximation of a thin thickness of the flame (very short times and high rates of chemical reactions) allows the solution to the problems of combustion in 1D statement and under the quasi-equilibrium conditions.22 The standard approximations should be also added: laminar and turbulent numbers of Prandtl (Pr), Schmidt (Sc), and Lewis (Le) equal one, and this allows us to consider the equations of motion, energy, and diffusion as the similar ones and obtain the similar profiles of temperature, velocity, Received: August 13, 2013 Revised: October 14, 2013 Published: October 15, 2013 7089

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Stability at combustion can have different meanings: stability of the velocity field, which relates to the velocity gradient, temperature, concentration, and stability of flame front propagation, which is characterized, for instance, by the Damkohler number. These two concepts are interrelated, but they differ. The loss of hydrodynamic stability can both suppress and intensify combustion. Incomplete burning of the reaction products at methane combustion in the mixing layer leads to their bright glow. Radiation is absorbed partially by the ambient medium (CO2, water vapors, water film of a sample, methane hydrate powder), and this reduces the combustion temperature and decreases flame stability. The expressions for consideration of heat losses of radiation are presented in ref 35. The current study deals with combustion of methane hydrate at free-convective motion of mixed gas. As it was shown above, stability of combustion kinetics is directly connected with kinetics of gas hydrate dissociation. Correct understanding of combustion process features requires consideration of related scientific disciplines. Therefore, the article emphasizes the tight interconnection between combustion stability, kinetics of methane hydrate dissociation, and the problems of strength and creep of the formed ice crust.

and concentration in the dimensionless coordinates. The modern models widely use the synthesis of kinetics equations and turbulent statistical closure models. One of the common methods of turbulence closure is achieved by means of a largeeddy-simulation (LES) technique.23−25 The effect of subgridscale motions are treated using the model.26 Application of the turbulent integral scale and Kolmogorov length scale, and dimensionless criteria: turbulent Reynolds number (Ret), Damkohler number (Da), and turbulent Karlovitz number (Ka), allowed plotting the maps of flame propagation regimes depending on the values of similarity criteria. The regimes of nonpremixed combustion are considered in ref 27 depending on Ret and Da numbers: regime “Flamelet” with very thin flame and short time of chemical reaction (τc) in comparison with the characteristic time of turbulent physical process (for instance, the lifetime of large eddies (τt), at that, Da number is much higher than one; and regime “Unsteady effects”: characteristic time scales τt and τc are the values of one order, Da number is also about one; regime “Extinction”: the chemical time is much higher than the characteristic physical time scale (Da < 1). Slow chemical reactions are typical for this case; both local and total stop of combustion is possible. For the case of turbulent premixed combustion the Gibson scale and the turbulent Karlovitz number Ka are added to the above similarity criteria Da and Ret; the phase diagrams plotted by Ret, Da, and Ka are also added and suggested.28−30 The following regimes occur depending on the given criteria: “Flamelet”, which is divided into the wrinkled and corrugated ones; “Distributed reaction zones”; and “Well-stirred reactor”. Despite efficient application of the LES technique, the accuracy of the physical model is often determined by the closure models. Usually we consider small perturbations of combustion front, which cause distortion of the surface of flame front and formation of cellular structures stabilizing combustion and making it more stable. These small local instabilities were connected with hydrodynamic factors (the hydrodynamic model): Landau model,31 and with thermodynamic factors: Zeldovich model (the diffusion-thermal model of instability).32 It will be more correct to call these models quasi-unstable because small surface distortions are stabilized by the local growth of thermal and diffusion flows, which restore the balance. The existing paradigms in turbulent combustion research are considered in reviews,30,33 and a good example of noncompliance of the widely used assumptions in theoretical models is presented. For instance, even at a small distance from the nozzle (dimensionless coordinate H/D > 10, where H is the lift-off height, D is the nozzle diameter) the experimental values of mean scalar dissipation rate (χst) exceed significantly the theoretical predictions. The prediction is based on the k-ε turbulence model using statistical independence of the scalar dissipation rate and the flow mixture fraction.34 The macrostructural average characteristics (temperature, density, concentration of components) are connected statistically with the microstructural parameters (sizes of Kolmogorov’s and integral scale), especially for the problems with combustion. Small pulsations at Kolmogorov’s scale can lead to such pulsations of mixture component concentrations, when the local stoichiometric ratio (for small scales) is not satisfied, and as a result of combustion local nonuniformity of microregions will transform into nonuniformity of the macroflow and cause the growth of average kinetic energy of pulsation motion and, hence, the growth of the rate of scalar dissipation.

2. EXPERIMENTAL DATA AND ANALYSIS The scheme of experimental setup and experimental methods are presented in ref 12. Before the experiment the air in the shell was dewatered by SiO2 sorbent during a long period. The height of the gas hydrate layer in all experiments was constant (h = 6 mm). The samples of artificial methane hydrate meet the following parameters: initial mass concentration of methane was 12.14 mass%, initial average diameter of the powder granule was d = 2.1 mm. The formula for elementary cell is 2D 6T 46H2O (cubic sI) or 2(512)+6(51262).10 The characteristics of the powder of methane hydrate were determined by the weight and volumetric methods and by the X-ray analysis. The process of gas hydrate dissociation occurred under the nonisothermal conditions: the temperature of powder increased continuously from the initial temperature of liquid nitrogen to the temperature of ice melting under the atmospheric pressure. Sample heating occurred due to the temperature difference between sample surface Ts and outer medium T∞ = 23 °C. Since Ts increased with time, the outer heat flux decreased (until combustion beginning). Therefore, both heat flux and sample temperature changed. Ignition was activated at different points by means of an electric spark. Spark sources were located along the axis at the distances of 3 and 6 mm above the powder surface. After the ignition combustion occurred independently on the outer source. Combustion starts at certain initial concentration of mixture and average temperature of granule surface Ts of about −38 °C. The curve of gas hydrate dissociation at methane combustion on the powder surface is shown in Figure 1 (m0 is the initial mass of methane hydrate). Temperature Ts (the average temperature of powder surface) was measured by thermocouples up to Ts < −50 °C

Figure 1. Changes in the mass of methane hydrate at combustion. 7090

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and thermal imager NEC-San Instruments for Ts > −50 °C. Combustion starts in 138 s and stops in 185 s. The curve with combustion (Figure 1) has four characteristic regimes: 1) initial dissociation; 2) with partial self-preservation. There is no methane ignition because of the low rates of methane escape and low Ts; 3) at some certain methane concentration in air the combustion process occurs. Under the given conditions the temperature of powder surface increases fast because of a high heat flux, which increases 10-fold. 4) The rate of gas hydrate dissociation is maximal and constant almost for the whole range. When combustion is completed almost all methane leaves the powder. The important feature of regime 4 is a constant tilt of the mass curve. For the case of dissociation without combustion7 the last stage of sample decay is characterized by a continuous and significant decrease in the dissociation rate, which reduces from the maximal value to zero. The constant tilt of the mass curve occurs, when the granule temperature approaches the temperature of ice melting, and this excludes self-preservation. Moreover, the high dissociation rate can be caused by two reasons: 1) At gas combustion at the pore outlet the high gradient of methane concentration in the mixing layer and, respectively, the high diffusion flow occur; 2) Perhaps, the high temperature and ice creep increase with time the average pore radius in the granule volume. An increase in pore crosssection reduced hydraulic resistance and increases the gas flow. In reality there two more factors, which can reduce the rate of methane escape. The total time of hydrate conversion into ice is significantly less than the total time of methane escape.15 Therefore, the rate of dissociation should decrease with time due to the pressure drop in pores. It is also important to take into account the granule curvature.15 Combination of all mentioned factors will determine the resultant rate of dissociation. A bulk of experimental data was analyzed in ref 19, where they compare about 14 types of gas hydrates with different “guest” molecules and sI and sII structures. Self-preservation is mainly typical to hydrates with the high dissociation pressure (P > 1 MPa at 273 K).19 The exclusion is CH3F (P = 0.2 MPa). Decay occurs through the metastable state during the period of about tens ns. The transitions (hydrate)→(water-gas in the metastable state)→(gas-ice) are more profitable from the point of energy activation than the direct transition hydrate→ice. The gas nanomicrocavities and ice grains are formed in the metastable state. Geometry of ice structures will be determined by the ratio of the rate of gas diffusion in water and the rate of ice crystallization. At annealing temperatures the rate of diffusion increases and crystallization decreases, and this facilitates formation of Ih ice with low concentration of defects. The high dissociation pressure also increases gas diffusion in the metastable state. Another important feature of self-preservation is formation of a solid ice crust. At high dissociation pressure not only gas but also water vapors flow intensively from the pores. As a result, a layer of oversaturated vapor and its sublimation is formed above the granule surface. For gas hydrates with low equilibrium pressure (P < 1 MPa) the vapor flow through the pores is significantly less, and vapor oversaturation will be probably lower. As it is known, the rate of ice formation depends both on supercooling degree and vapor oversaturation. Together with formation of the screening surface crust pore blocking inside the granules is possible. At annealing some portion of water is supplied to the pores and blocks them at crystallization because of the increased gas pressure in the metastable cavity (water-gas). The guest CH3F molecules have relatively low potential of interaction with water molecules in the metastable (transient) state. As a result, gas diffusion in water is high, and self-preservation occurs in this case. Previously, we have indicated the important connection between the crust strength and its thickness. Therefore, another important characteristic will be the ratio of the rate of surface ice crystallization (the rate of crust thickness growth) to the rate of ice creep because of the inner pressure of gas in pores. The processes of restructuring of the ice surface crust at annealing play the important role because the ice strength depends on this. Activation energy is connected with the inner structures, e.g., the size of grains, and this is proved by numerous experiments with metal alloys.36,37 Different scenarios of ice crust formation are possible (Figure 2).

Figure 2. (a) Formation of pores at dissociation; (b) formation of the surface screening ice crust (no combustion); (c) formation of the inner ice crust at methane combustion. As it was mentioned before, hydrate decay (without combustion) at the last stage is accompanied by a constant decrease in the rate of the sample mass change. At the beginning of this regime ice creep facilitates an increase in the surface density of pores (Figure 2 (a)). Total time of hydrate conversion into ice is significantly less than the total time of methane escape.15 Therefore, the rate of dissociation should decrease with time because of the pressure drop in pores. The temperature in the range of annealing contributes to partial pore blocking (partial self-preservation) and considerable reduction of dissociation rate (Figure 2 (b). In the case of combustion the decay scenario differs absolutely. The process of dissociation at combustion is accelerated by a factor of 10 and there is a very high temperature gradient over the granule radius. At slow dissociation the temperature gradient is significantly lower. At combustion near the surface the temperature approaches the melting point, and this excludes formation of the crust without pores on the surface. However, the crust is formed inside the granules (the range of annealing temperatures) (Figure 2 (c)). When the pressure in pores starts decreasing in the upper part of the granule, a rise of temperature in the granule center forms porosity in the inner crust. Destruction of the inner crust and dissociation in the center of the granules compensates the gas pressure drop, i.e., the total rate of methane dissociation stays constant. The curves of ΔVi/V0 (%) depending on time are shown in Figure 3. Values ΔVi are the increase in the volume of escaped methane; V0 is

Figure 3. Dissociation of methane hydrate: curve 1 corresponds to the low rates of dissociation;13 curve 2 for the high rate of the sample decomposition during combustion of methane hydrate. the maximal volume of escaped methane. Points 1 correspond to the regime of quasi-self-preservation.13 The curves for the quasi-isothermal case (1) and for the nonisothermal hydrate dissociation (2) are significantly different. The tilt of the curve (2) is substantially higher than for (1). The curves of alteration of instantaneous methane velocity in time are shown in Figure 4. The velocity was determined for every narrow time period as the average value for the surface area of the reservoir with powder (the values of gas mass flux Δm are divided by area S, mixture density ρ, and time period Δτ, S corresponds to reservoir diameter of 44 mm). The velocity was determined by two methods: 7091

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rate36,38 in the presence of high strains. Curve 2 corresponds to the “Climb of edge dislocations” mechanism.39 Mechanism 2 should arise at very high dislocation density. However, in practice, the density of ice dislocations is several orders lower than it is required by the “Climb” law. The stresses in pores are determined according to the temperature and pressure of the curve of methane gas hydrate equilibrium. Point 3 corresponds to the strain rate at T = 260 K, point 4 is typical to T = 273 K, and point 5 is the case for the gas pressure drop in pores with time. The power law of deformation is written in the general form as

⎛ Q ⎞ ⎟ εṡ = aρv = bρσ n exp⎜− ⎝ RT ⎠

(1)

where a and b are the constants, dε/dτ is the strain rate, ρ is initial density of dislocations, σ is strain, Q is activation energy, and v is velocity of dislocation. The pores inside the granule volume are formed because of phase transformation of hydrate into ice. The motion of pores at hydrate decay is directed from the granule surface to their center. Therefore, velocity of pore motion vp equals the velocity of dissociation front motion. According to experimental data of ref 15, the rate of the growth of ice crust thickness and, respectively, velocity of pore motion vp has the order of 1 × 10−4 m/s. The surface pore density by the SEM measurements (confocal scanning microscope) after completion of combustion is 1011÷1012 m−2. Actually, it is reasonable to consider two characteristic velocities: vp and vd, where vd is the velocity of ice dislocations (vd is not connected with the phase transition). Kinetics of these two mechanisms will be determined by activation energy. However, the presence or absence of phase transition makes these mechanisms absolutely different. Perhaps, Figure 5 characterizes only the behavior of vd. At self-preservation the pressure of gas in pores under the ice crust is very high, and this corresponds to point 4. The strength of the crust is violated because of ice creep and growth in temperature Ts, the pores become open, and gas pressure in pores decreases with time. At a decrease in the inner stresses in pores the rate of deformation will correspond to curve 5 (pore density is 1012 m−2), when the stresses decrease from 2.6 to 0 MPa. Points 4 and 5 correspond to the same temperature and, hence, to the same limit strain. However, for point 5 (combustion), the rate of methane escape and the rate of gas hydrate dissociation are an order higher than those for point 4 (without combustion). The question is as follows: what is the parameter in relationship (1) corresponding to this drastic increase in the density of methane flow at combustion. If dislocation velocity v increases due to combustion or dislocation density ρ becomes higher, then, deformation rate dε/dτ increases also. Nevertheless, point 5 can hardly be higher than curve 1, and it should coincide with point 4, i.e., the rate of deformation for points 4 and 5 will be constant. There are the maximal pore density (pore saturation) and the limit maximal velocity of pore motion. Then, an increase in the velocity of methane flow is caused not by the growth of dε/dτ but an increase in the average diameter of pores (a change in the function of size distribution of pores), and this is caused by fast material heating (high heat flux). At an increase in the average diameter of pores the hydraulic resistance drops (the resistance is reversely proportional to the pore diameter) and the velocity of gas motion in pores increases. To increase the total methane flow by a factor of 10, the average pore diameter should be increased 10 times. The process of methane hydrate dissociation occurs under the nonisothermal conditions. The powder temperature increases from the temperature of liquid nitrogen to the temperature of ice melting. The pictures obtained by a thermal imager at different time moments are shown in Figure 6. It can be seen (Figure 6 (a),(b)) that the motion of heat front is nonuniform: there is temperature maldistribution on the surface and there are high local and average temperature gradients over the reservoir diameter. When combustion is over, the temperature field becomes uniform. A high heat flux at combustion eliminates significant temperature nonuniformities. The curves of a change in sample surface temperature at different time moments are shown in Figure 7. At the moment of combustion completion the surface temperature was 2 °C, then it decrease to 0 °C.

Figure 4. The velocity of methane hydrate: curve 1 − without combustion; curve 2 − with combustion; points 3 − measurements by the hot-wire anemometer (without combustion). curves 1 and 2 were obtained by the data on a change in the powder mass; points 3 are the measurements by a hot-wire anemometer. According to the diagram, the maximal velocity of methane at combustion equals 13−15 mm/s, and without combustion it is 1.5 mm/s. It can be seen in Figure 1 by the temperature curve that during combustion the sample temperature approaches the point of ice melting fast. Combustion increased the density of methane flow by a factor of 10. Data on methane velocity will help us to perform an estimate by a change in concentration in the air-methane mixture and evaluate disturbance of a stoichiometric ratio. When the solid screening ice crust is formed, the gas pressures of dozens of atmospheres are generated under the crust, and they cause the inner mechanical stresses. Significant mechanical stresses are formed even at the open pores in the inner part of granules because of the inertial character of gas flow from the pores,15 and they can lead to dislocation creep. Certainly, macrodescription of ice deformation, consisting of large grains (the size of more than 1 mm), will differ significantly from microobjects: deformation of the surface of pores with the diameter of 0.5−5 μm. The important difference from coarse ice is the size of grains of 5−50 μm, formed at dissociation. Since there are almost no data on dislocation creep at gas hydrate dissociation, it is reasonable to consider the diagram of deformation rates for coarse ice. The inner mechanical stresses at hydrate dissociation will be determined by the dissociation pressure, which depends on temperature. The maximal pressure (maximal inner stresses) will be near the melting point (above 2.6 MPa). A change in the rate of polycrystalline ice deformation vs strains is shown in Figure 5. During the process of clathrate hydrate dissociation the whole sample converts from the hydrate state to hexagonal ice. The velocity of dislocations relates to the laws of deformation, shown in Figure 5: curve 1 corresponds to the power law of deformation

Figure 5. Dependence of the rate of polycrystalline ice deformation on strains: (1) − power law; (2) − “Climb of edge dislocations” law; (3) − T = 260 K; (4) − T = 273 K; (5) −T = 273 K (for the gas pressure drop in pores with time from 2.6 to 0 MPa (A)→(B)→(C)→(0 MPa)). 7092

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thickness of the near-wall boundary layer. Therefore, a high heat flux occurs because of the high temperature gradient near the wall. As a result, at combustion the temperature of powder surface approaches 0 °C fast, and the high velocity of pore motion and high pore density intensify clathrate hydrate dissociation. It can be seen in Figure 4 that combustion starts and stops at methane velocity of about 1 mm/s, and this corresponds to the average methane concentration in the layer of gas mixture of about 10%. Methane combustion is determined by average methane concentration in the combustion zone, which is moved away from the wall by several millimeters, but not by methane concentration directly on the wall. The scheme of motion of convective gas flows governed by combustion and temperature and density gradients is shown in Figure 8. The origin point of axes is in the center of reservoir with powder.

Figure 8. Free-convective motion of gas at combustion. Figure 6. Thermal images of the surface of methane hydrate powder at different time moments: (a) 80 s; (b) 120 s (before combustion); (c) 186 s (completion of combustion); (d) 211 s (25 s after combustion).

The boundary layer consisting of the near-wall and jet zones is located along the horizontal axis. Thickness of the near-wall zone δ corresponds to a change in the sign of velocity derivative dux/dy. The thickness of jet zone is significantly higher than that of the nearwall zone. Value ux increases from zero to the maximum. The jet boundary layer develops along the vertical axis. At first, velocity uy increases (flow acceleration) because of the high gradient of gas density, but it decreases then continuously because of friction in the shear layer at some distance from the wall (flow deceleration). Maximal Grashof number Grx corresponds to the laminar flow, and Gry corresponds to the turbulent flow only at the height above 100−150 mm. Up to Y = 100 mm there is also the laminar flow. Distribution of gas temperature across the boundary layer is shown schematically in Figure 9, where δm is the thickness of the near-wall layer with maximal temperature Tm. In the near-wall zone 0 < Y < δm the temperature profile is close to the linear one.

Figure 7. A change in the temperature of the surface of methane hydrate powder along the reservoir diameter: (1) − τ = 1 s; (2) − τ = 5 s; (3) − τ = 20 s (τ = 0 s corresponds to the time of combustion completion). According to visualization of measurements, during combustion and during the first seconds after combustion completion there was a thin liquid film on the granule surface. The heat fluxes were determined by the linear temperature profile of the gas mixture above the sample surface. At combustion the heat flux was 13700 Wm2−. The maximal thickness of the water film at combustion was determined by the balance of heat fluxes in the liquid film and in the gas layer; it was 0.05−0.1 mm. The thickness of the gas thermal boundary layer was 4 mm (without combustion) and 7−8 mm (at combustion). It was assumed in ref 4 that the high velocity of flame propagation was caused by a thin diffusion layer and high concentration gradient. For the airmethane mixture the numbers of Prandtl Pr and Lewis Le are close to one; therefore, the thickness of dynamic, thermal, and diffusion boundary layers are close to each other. According to experimental results, at combustion the thicknesses of boundary layers were almost doubled. The thickness of diffusion boundary layer δd is proportional to (Dτ)1/2 (D is diffusion coefficient, τ is time). With a rise of gas temperature from 295 K to 1200−1500 K diffusion increases by a factor of 3−4, and this corresponds to the double increase in the

Figure 9. The scheme of a change in the gas mixture temperature along the vertical coordinate. Actually, the experimental data show that the process of combustion is very nonuniform over the powder surface. According to thermal imaging (Figure 6 (b), the temperature field is nonuniform. This maldistribution relates to different granules diameter and their different inner structures formed at sample synthesis. Different temperatures of particles generate spatial and temporal nonuniformity for the dissociation rate and, hence, for the density of methane flow. A 7093

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change in the density of methane flow along coordinate x is shown schematically in Figure 10 (a) (j ̅ is the average density of methane flow

Figure 11. Thermal images of flame at methane combustion at different time moments. Figure 10. Instabilities of diffusion flame layer: (a) a change in the methane flow along coordinate X; (b) distortion of streamlines and vortex formation; (c) inclination of the axis of maximal transverse velocity uy. along the coordinate). Zero value of axis X corresponds to the center of a tank with powder. Coordinate axis Y is directed from the tank center normally to the powder plane. Spatial nonuniformity leads undoubtfully to nonuniformity of dissociation of methane hydrate in time. The granules, which at first had the lower dissociation rates, will demonstrate the higher rates at the final stage of combustion; therefore, the area with maximal combustion temperature will move over the surface continuously. The nonuniform character of methane flow leads to formation of the local temporal and spatial nonuniformities of mixture concentration (methane-air), and at combustion it leads to nonuniform rate of chemical reaction and nonuniform temperature. The high temperature and density gradients destabilize the averaged laminar flow and promote the loss of combustion front stability. The curvature of the flame front surface leads to formation of the vortex structures in the diffusion mixing layer. Since vortex sizes are comparable with the width of the mixing layer, the combustion conditions do not meet “Flamelets”, they are the “Unsteady effects” conditions, which turn to the “Extinction” conditions: at a decrease in the rate of dissociation, methane flow density and methane concentration in the mixture, combustion intensity becomes spontaneously lower, and then, combustion stops. Distortion of the streamlines and vortex formation are shown in Figure 10 (b). The dashed line indicates the stable streamline (without local curvatures). The loss of stability leads to formation of large-scale vortices despite the laminar Gr numbers. The large vortices affect not only instantaneous but also the average profiles of velocity, concentration, and temperature. Figure 10 (c) shows that the loss of stability leads not only to distortion of the streamlines and isotherms but also to the inclination of the axis, corresponding to the maximal value of transverse velocity uy. The tilt of the axis corresponding to maximal velocity uy is shown in Figure 10 (c) (dashed line). The thermal images of flame at different time moments are shown in Figure 11. Under the effect of vortex instability the flame tilt changes with time, the front bends, and diffusion thickness of the mixing layer changes. Figure 11 shows instantaneous profiles of the temperature field. Since the processes of heat, momentum, and concentration transfer are mutually connected, local curvatures of isotherms also speak about distortions of the concentration field and curvature of streamlines of the velocity field in Figure 10 (b, c). Inclination of the axial line in Figure 11 (170 s) corresponds to Figure 10 (c). Spatial nonuniformity of the flame is shown in Figure 12. Point 0 of coordinate x corresponds to the center of reservoir with powder. Experimental points correspond to the maximal combustion temperature. As it can be seen, the local zones of maximal temperature change their position in time. At combustion beginning and after its completion there is maximal deviation of a coordinate from the axial line (x = −7 mm and x = 10 mm). At mean times the combustion

Figure 12. A change in coordinate x, corresponding to the zone of maximal combustion temperature, with time. front is most stable: coordinate x is close to zero at τ = (15÷25) s. The total time of combustion is about 40 s. The nonstationary character of combustion is obvious in Figure 13. The instantaneous temperature profiles were measured along the

Figure 13. A change in temperature profiles at combustion with time: (1) − τ = 5 s; (2) − τ = 10 s; (3) − τ = 15 s; (4) − combustion stop (time τ = 0 s corresponds to the beginning of methane combustion). vertical coordinate for different time moments. These profiles were measured for different values of horizontal coordinate and correspond to x for the maximal gas temperature (coordinate x was changed because of the motion of combustion zone over the surface). Combustion of methane at methane hydrate dissociation is studied in ref 5. In contrast to the current study the air motion occurred due to the forced near-wall air flow but not to free convection (buoyancy). In Figure 14 (a) the velocity of oncoming undisturbed air flow is u0 = 0.1 m/s and in Figure 14 (b) u0 = 1.1 m/s. In case (a) velocity u0 has the same order as the average velocity of methane; as a result the surface gas inflow makes the diffusion layer unstable. It is shown above that injection is nonuniform by its coordinate and time. The large vortices in ref 5 are visualized clearly. In case (b) the air velocity increased by a 7094

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jet width. The free convective flow with methane combustion studied in the current work differs absolutely from the case in Figure 15 (a). Low Gr numbers determine whether this regime is laminar or transitional. On the other hand, there are large vortex motions caused by the loss of stability of the mean flow (see Figure 15 (b)). Transverse velocity gradients are not high. A lack of developed turbulence (an absence of small-scale turbulent diffusion) increases significantly typical scale l2: the lifetime of a large vortex will be determined only by molecular diffusion. Thus, in the considered flow the effect of vortex transfer and large scale (about the jet width) will lead to significant intensification of heat transfer and admixture diffusion and significant nonsimilarity of transfer equations. For nonpremixed turbulent flame in ref 27 it is suggested to take into account interaction of turbulent transfer with flame combustion, using two dimensionless criteria: turbulent Damkohler number (Dat) and turbulent Reynolds number Ret (Dat = τt/τc, Ret = (u′lt)/ν, where τt is the characteristic time for turbulent integral scale, τc is chemical reaction time, u′ is the characteristic velocity of vortices with size lt, and ν is the kinematic viscosity). It is suggested in ref 27 to plot the map of combustion regimes within the flame in the form of Dat = f (Ret). Three regimes of turbulent combustion are presented in Figure 16: 1) “Flamelet” (very thin flame and high Dat numbers), 2)

Figure 14. Instability of the near-wall boundary mixing lay: (a) u0 = 0.1 m/s; (b) u0 = 1.1 m/s. factor of 10. The rate of dissociation will also increase due to a rise of the heat flux density q = αΔT. Dimensionless Nusselt number Nu = (αl)/λ (α is heat transfer coefficient, l is typical size, λ is mixture heat conductivity, ΔT is temperature difference between the external flow and flow on the wall). For the laminar boundary layer the heat flux is associated with the velocity via the Re criterion by relationship Nu = c(Re)1/2(Pr)1/3, where Re = u0x/ν. The 10-fold increase in the velocity leads to a rise of α by a factor of 3.2. An increase in methane velocity will be significantly less than an increase in the rate of dissociation; therefore, for case (b) the ratio of the air flow velocity to methane velocity is u0/uch4 ≫ 1. As a result, the boundary layer for (b) is much more stable, and a change in the thickness of diffusion layer with time will be also lower. In ref 5 significant fluctuations of velocity of the front edge of the flame occur because of significant instability of the boundary layer for low velocities u0; the magnitude of these alterations in different experiments is up to 3. The Prandtl model, where the length of the mixing path is proportional to transverse coordinate l1 = ky, is widely used for the near-wall boundary layer (Figure 15 (a)). The high velocity gradient

Figure 16. Different regimes of combustion for nonpremixed turbulent flame. “Unsteady” (characteristic times of the chemical and physical processes of the same order of magnitudes), and 3) “Extinction” (at slow chemical reaction and low Dat numbers combustion diminishes). The regime with thin flame thickness allows us to solve the problem in a single-dimensional statement and use the flame-attached coordinate system, then, the equation for temperature becomes more simple (derivatives by other coordinates will be reduced).30 In the real technical devices the regime of the thick flame layer often occurs at low times of combustion. For instance, for free-convective combustion of methane at clathrate hydrate dissociation unstable diffusion combustion by type (b) occurs from the very beginning (Figure 15 (b)). The main type of mixing for the given problem is determined not by near-wall mixing but by the jet flow with low deformation rates at some distance from the wall. As a result the determining typical scale of the vortex will be not l1 (Figure 15 (a)) but l2. Low deformation rate in the mixing layer and absence of developed turbulence (absence of turbulent diffusion) increase the lifetime of a large vortex significantly and increase the length of transfer l2. The important feature of this type of combustion is the fact that the laminar regime occurs for most Rex. For the developed turbulent flow molecular diffusion has the infinitesimal order in comparison with turbulent transfer. Under unstable laminar conditions vortex energy ω (Figure 14 (b)) is scattered via molecular diffusion at low deformation rate of the mean flow. Therefore, the typical scale of vortex l2 will be much higher than l1 and comparable with the jet width. In this case superposition of large-scale stable vortices on the laminar flow will lead to more intensive heat flux and diffusion in comparison with momentum transfer l1. Similarity of equations of heat, momentum, and diffusion transfer as well as similarity of dimensionless profiles of temperature, velocity, and concentration can be violated for the case of l2. Another important feature of methane combustion in this study is a significant expansion of the flame thickness because of low air (oxidizer) velocities. As a result, single-

Figure 15. The length of mixing path for two types of transfer (a, b). near the wall does not allow existence of large vortices: due to high deformation rates the large vortices are divided into the small ones. In the external area of the boundary layer the deformation rates are low and the size of vortices is about the layer thickness; however, their effect on the near-wall area is insignificant. Thus, the transfer mechanism of pulsation component momentum relates to translational but not rotational transfer from point x1 to x2 because of extremely low l1. For the near-wall area the turbulent dimensionless profiles of velocity and scalar values are similar at Pr = 1, Le = 1. The pattern of the jet flow is different. According to Prandtl, the transfer mechanism connects l1 with longitudinal coordinate l1 = kx. An absence of the wall makes large vortices more stable, and the role of vortex component increases. Taylor suggested connecting the turbulent shear stresses not with momentum but with vortex transfer. In the 2D flow averaged vortex intensity is ω̅ =

1 ⎛ ∂u ̅ ∂ v ⎞ 1 ∂u ̅ − ̅⎟ ≈ ⎜ 2 ⎝ ∂y ∂x ⎠ 2 ∂y

(2)

According to Taylor, the length of mixing path (lT = √2kx) is √2 times higher than the Prandtl length. By Taylor transverse turbulent heat transfer is twice as high as this transfer by Prandtl, and dimensionless velocity and temperature profiles differ significantly.40 Therefore, the effect of vortex transfer disturbs similarity of the velocity and scalar fields (temperature and admixture concentration). In the model of Taylor typical scale lT is considerably lower than the 7095

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dimensional approximation is unacceptable, and the flow should be considered at least along two coordinates. Due to 3D character of vortices and comparable density gradients by three coordinates, it is probably necessary to take into account the third coordinate. A significant thickness of the diffusion combustion layer in this flow excludes even application of the one-step reaction model (CH4, O2, CO2, H2O) for the chemical reaction, and the reduced four-step mechanism for methane flame is more appropriate. Expansion of the combustion zone disturbs the stoichiometric ratio, and the temperature of combustion decreases drastically; this requires consideration of deviation from the equilibrium of chemical reactions. Since in the current study we know a change in the density of methane flow above the powder surface, mixture concentrations were estimated. Concentration of mixture depends on the rate of methane escape, molecular diffusion, and convection transfer, determined by free convection. Combustion started and stopped at methane concentration of about 10% in the mixture. In the presence of combustion the methane flow increased by a factor of 10. The temperature increase at combustion intensified molecular diffusion D is proportional to √T/ρ (T is temperature, ρ is density), and the velocity of convective motion u0 is proportional to (ΔT)1/2. Thus, combustion intensified both the density of methane flow and velocity of the air flow. As a result, the coefficient of air deficiency was 1.8, i.e., for implementation of the stoichiometric ratio the amount of air was understated at combustion by a factor of 1.8, in average. A significant fuel excess in comparison with the oxidizer led to a decrease in combustion temperature and reduced the reaction rate. Characteristic hydrodynamic time τt increases insignificantly at a decrease in temperature because the mixing layer expands and the temperature and concentration gradients reduce. Time τc, in contrast to time τt, increases by several orders because the reaction rate relates to the temperature through the exponential dependence. In the current work the Damkohler number corresponded to the transition boundary between the “Unsteady” and “Extinction” regimes (Figure 16), and, hence, it should be close to one (one order). Since hydrodynamic time (characteristic scale l2 = 5 ÷ 10 mm) corresponded to 0.2 s, then characteristic time of the chemical reaction of combustion is τc = 0.02 ÷ 0.2 s. There is strict differentiation of the laminar and turbulent flows by a vertical line at Ret = 1 in Figure 16. It is assumed that the forces of inertia, determined by the pulsation component, reduced to the viscosity forces, characterize the change of regimes. Actually, the transitional regimes are determined not by pulsation but average parameters. Thus, methane combustion at clathrate hydrate dissociation corresponds to the laminar flow (laminar Gr and Ri numbers). At that, transition from kinetic to diffusion combustion is absolutely associated with the loss of stability by the flame front and formation of large-scale structures ω (l2). The combustion process will be determined both by the molecular Damkohler Dam number and turbulent Dat number (Dam is proportional to L2/D, the width of the jet mixing layer L can be taken as the characteristic scale of the averaged flow, D is molecular diffusion). Then, with contribution of two types of diffusion the regime map can be determined by a generalized Das = (Dat + kDam) number, where k is the coefficient taking into account the contribution of molecular diffusion. For developed turbulence and at significant prevalence of turbulent diffusion over molecular transfer combustion will be determined only by turbulent transfer (k = 0). For the laminar conditions k = 1, Dat = 0. For combustion of gas hydrate considered in the current work, Ret number will be determined by scale l2 and pulsation of circumferential velocity u′ (Figure 15 (b)). As it was mentioned above, the experimental data demonstrate divergence between the measured rate of scalar dissipation and simplified theoretical model of combustion in a jet even at a small distance from the nozzle. Therefore, the connection of combustion with Re number, determined by the average flow velocity, is obvious. Moreover, the wall effect is often considerable in the problems: the presence of the near-wall and external jet flows, which should be also generalized by Re. Experimental data on the strong effect of Le number are presented

in ref 30. If Le = 0.5, then the characteristic rate of heat transfer becomes significantly lower than the characteristic rate of diffusion, and this causes the growth of local temperature gradients in the flame. The higher deformation rates caused by the density gradient make the flame front unstable and the additional circulation zones are formed: the cellular structures. The following dependence is suggested in ref 27 Dat = τt /τc is proportional to (Ret )n Dafl , Dafl = τf /τc

(3)

where n = 0.5, τf is the flow residence time, and τc is the chemical time. Actually, the Damkohler number will depend on Le, Re, Pr, and Ri (Richardson number, Ri = Gr/Re2) together with Ret. In relationship (3) power n will be also the function of the above dimensionless similarity criteria. Thus, the tile of the curves (separating combustion regimes) in logarithmic coordinates in Figure 16 will depend on the similarity criteria of the averaged flow.

3. CONCLUSIONS It is shown that under the nonisothermal conditions dissociation of methane hydrate is divided into several characteristic time stages with different rates of gas hydrates dissociation. There are two characteristic velocities: the velocity of pore motion and the velocity of ice dislocations. The presence or absence of phase transition makes these mechanisms absolutely different. Nonuniform spatial escape of methane from the sample led to the nonuniform character of the scalar fields and a loss of stability of the flame front. Unstable combustion caused two equivalent mechanisms of convective transfer: the shear laminar flow and large-scale vortices, which intensify fuel and oxidizer mixing. The experimental data have revealed not only the local time-spatial maldistributions but also significant nonstationary character of the averaged flow. Methane combustion at dissociation changes significantly the boundary conditions, and it is necessary to solve the joint problem on gas hydrate dissociation and kinetic equations for the chemical reactions with consideration of the nonstationary character of the averaged flow. The measured instantaneous and average methane velocities on the powder surface allowed an estimate of the stoichiometric ratio. The coefficient of air deficiency was 1.8, i.e., the amount of air required for satisfaction of the stoichiometric ratio was understated at combustion by a factor of 1.8. A significant fuel excess over the oxidizer led to a significant decrease in the combustion temperature and reduction of the reaction rate. The maximal methane velocity on the powder surface without combustion corresponded in experiments to 1 mm/s and at combustion it was 13−15 mm/s.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +7 383 3356577. Fax: +7 383 3356577. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work was financially supported by the grants of RF Government for the state support of research conducted under the guidance of leading scientists No.14.B25.31.0030 (the leading scientist − Yoshiyuki Kawazoe), under the guidance of leading scientists in Russian universities No.11.G34.31.0046 (the leading scientist − K. Hanjalich, NSU). 7096

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