A numerical study on near-limit extinction dynamics of dimethyl ether spherical diffusion flame

Fuel Processing Technology 185 (2019) 79–90

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Research article

A numerical study on near-limit extinction dynamics of dimethyl ether spherical diffusion flame

T

⁎

Jie Chena, Yinhu Kangb,c, , Yunpeng Zoua, Kaiqi Cuia, Pengyuan Zhanga, Congcong Liua, Jiangze Maa, Xiaofeng Lub, Quanhai Wangb, Lin Meid a

Key Laboratory of the Three Gorges Reservoir Region's Eco-Environment, Ministry of Education, Chongqing University, Chongqing 400045, China Key Laboratory of Low-grade Energy Utilization Technologies and Systems (Chongqing University), Ministry of Education of China, Chongqing 400044, China c Postdoctoral Station of Environmental Science and Engineering, Chongqing University, Chongqing 400045, China d Chongqing Special Equipment Inspection and Research Institute, Liangjiang District, Chongqing 401121, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Dimethyl ether Extinction Cool flame Spherical diffusion flame Computation

The near-extinction oscillatory dynamics and underlying mechanism of the dimethyl ether (DME) spherical diffusion flames in the hot- and cool-flame conditions were studied by the numerical approach with detailed chemistry and transport models. It was found that the self-sustaining spherical cool diffusion flame could be readily obtained in a wide range of parameters, and additionally, the flammability limit could be considerably extended by the cool-flame chemistry. Strong oscillations that can trigger flame extinguishment prior to the steady-state extinction turning point were observed in either the hot or cool flame regime. The DME hot flame near extinction was governed by a single oscillatory mode with a fixed frequency (1 Hz) that was irrelevant with the ambient oxygen level. By contrast, the cool-flame extinction was controlled by dual oscillatory modes which had rather distinct frequencies, and the oscillatory period of the high-frequency mode increased significantly when approaching the extinguishment. Additionally, the transient dynamics of cool flame near extinction was much more complicated, due to the existence of strong coupling between the high-frequency and low-frequency oscillatory modes which was affected by the phase difference. The governing reactions that controlled the oscillatory extinction in hot- and cool-flame regions were revealed using a logarithmic sensitivity index. It was found that the hot-flame oscillatory extinction was controlled by the competition of high-temperature exothermic/endothermic reactions with the chain branching/termination reactions involving small molecules. The cool-flame oscillation was controlled by the low-temperature branching and termination competition in the negative temperature coefficient regime.

1. Introduction Flame extinction is crucially important for the development of efficient, low-emission combustion devices or alternative fuels, so it is a hot topic in the field of combustion science and technology, and its relevant studies are of significance in the fundamental and application viewpoints [1–3]. Flame extinction is a rather complicated process, which depends on various factors including fluid dynamics, chemical kinetics, and flame geometry, etc. Flame extinction belongs to one of the most important near-limit phenomena in combustion science, which is strongly affected by the high- and low-temperature chemistries. It is important to various combustion systems such as transportation and power generation. Flame extinction is a typical limit combustion

phenomenon at which the chemical kinetics basically balances with the thermal/mass transport fluxes, i.e. the Damköhler (Da) number which is defined as the ratio of flow timescale to reaction timescale equals unity [4]. As a result, the interaction between chemical kinetics with transport plays a crucial role in the near-limit flame dynamics. For instance, the ignition and extinction limits are often utilized to verify the accuracy of chemistry and transport models. It is well accepted that flame extinction is caused by excessive leakage of heat and radicals from the flame front, which lowers the flame temperature and then extinguishes the combustion process. Flame extinction happens within a very small time interval or spatial domain, which makes its controlling mechanism rather complicated [5]. Flame extinction exists extensively in the industrial combustion

⁎ Corresponding author at: Key Laboratory of Low-grade Energy Utilization Technologies and Systems (Chongqing University), Ministry of Education of China, Chongqing 400044, China. E-mail address: [email protected] (Y. Kang).

https://doi.org/10.1016/j.fuproc.2018.12.006 Received 29 September 2018; Received in revised form 5 December 2018; Accepted 5 December 2018 0378-3820/ © 2018 Elsevier B.V. All rights reserved.

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Although the steady-state extinction dynamics was investigated extensively in various configurations, the near-limit transient dynamics due to the strong coupling between diffusive transport and detailed chemistry was not yet studied as thoroughly. The present study is focused on the oscillatory extinction dynamics for both hot and cool diffusion flames, which would be of fundamental and engineering importances since the oscillation-induced extinction may probably happen before the steady-state extinction point, as found in the PSR system [21]. Besides, the one-dimensional spherical diffusion flame with an unbounded boundary condition was selected as the target flame configuration. It is a well-defined model for many practical diffusion flames such as the droplet in spray flames, but it can avoid any thermal/mass feedbacks from the flame to the droplet surface that would change the vaporization rate, by supplying the fuel mixture from a spherical point source at a fixed mass flow rate. Furthermore, such strainless, reducedpressure flame is much prone to exhibit low-frequency oscillatory instability near extinction, which facilitates the observations considerably. In addition, dimethyl ether (CH3OCH3, DME) was used as the target fuel because of its promising potential as a surrogate to substitute the traditional hydrocarbon fuels such as diesel. DME has a much simpler fuel structure and thus smaller chemical mechanism size than that of other alkanes like n-heptane, but it has a strong NTC chemistry subset, which enables our investigation on its cool-flame dynamics.

devices such as boilers, internal engines, and gas turbines; and it plays a vital role in the combustion safety, efficiency, and stability. In addition, the studies on flame extinction are also relevant with fire safety and suppression on ground or in space capsule, since it could provide the fundamental knowledge for the design of high-efficiency fire suppressants and extinguishment devices [6]. Hence, studies on the extinction mechanism of hydrocarbon fuels are of fundamental and practical significances in the combustion science. Flame extinction usually happens within the low-temperature region, in which the combustion regime is governed by the cool-flame chemistry. The cool-flame chemistry indeed is the fuel cracking kinetics which occurs at low- or intermediate temperatures (T < 1000 K) [7]. It is the transitional region between the low-temperature chemistry and high-temperature chemistry. Despite of its indiscernible contribution to the overall heat release, the cool-flame chemistry is crucially important due to its significance in the start of ignition, lift-off stabilization, and overall ignition delay time, etc. In the cool-flame regime, the fuel oxidation pathways are dominated by the reactions involving fuel fragments and molecular oxygen, which are extremely more complicated than that of the high-temperature fuel oxidation in which the thermal decomposition reactions dominate [8]. For example, the fuel oxidation process at low temperatures exhibits the negative temperature coefficient (NTC) behavior, in which the fuel oxidative reactivity decreases with the increment in reaction temperature [9]. Furthermore, many relevant studies also showed the strong correlation of cool-flame chemistry with engine knock [10], two-stage autoignition [11,12], lean flammability limit [8,13], turbulent burning velocity [14], lift-off length [15] or even stabilization mechanism [16] of the main flame. Consequently, the extinction mechanism due to cool-flame chemistry particularly merits a deeper study. The controlling dynamics of flammability and extinction have been extensively studied in various configurations including diffusive droplets, counterflow flame, and propagating flame front, etc. Recently, the micro-gravitational n-alkane droplet flame experiments in free space [17,18] reported dual modes of combustion and extinction, including high-temperature visible flame extinguishment followed by a long period of quasi-steady cool-flame droplet burning mode without any luminescence. Additionally, the self-sustaining cool flames were also observed in the counterflow geometry with or without ozone sensitization [8,13], or in freely-propagating flame configuration [19]. The experimentally measured ignition and extinction limits demonstrated that the flammability limits could be substantially extended by the transition from hot to cool flames [8,13,19]. The balance of chemistry with transport at the extinction turning point would lead to interesting oscillatory dynamics, especially in the cool-flame regime which is characterized by a strong coupling between the low-temperature reactions and transport or solid boundary [13,19]. The oscillatory dynamics behavior near extinction has been extensively observed in various combustion configurations, including the homogenous systems such as heated closed vessels [20] and perfectly-stirred reactors [21], as well as the inhomogeneous diffusive systems such as the counterflow flame [22], micro-gravitational spherical diffusion flame [23], and droplet flames [25–27]. The oscillatory cool-flame ignition as observed in homogenously-stirred reactors was attributed to the thermo-kinetic instability in which the low-temperature chainbranching reactions govern the heat release and loss. The experiments and simulations conducted by Farouk et al. [25–27] showed that the nalkane droplets at elevated pressures may possibly exhibit multi-stage oscillatory transient behavior after falling into the cool-flame region, which was ascribed to the coupling of low-temperature branching kinetics with diffusive heat loss [25–27]. The study on the n-heptane counterflow cool diffusion flame with ozone sensitization suggested that the instability behavior of the cool flame was a thermo-kinetic instability, which was triggered and controlled by the chemical kinetics associated with the OH radical population in the NTC regime that was coupled with heat production and loss [22].

2. Details of the numerical simulation 2.1. Governing equations with boundary conditions The spherical diffusion flame was utilized to examine the diffusion flame structure and extinction dynamics [23,24]. As shown in Fig. 1, the micro-gravitational spherical diffusion flame was established by issuing fuel mixture with initial temperature 300 K and exit velocity 13.0 cm/s from a porous spherical burner surface to the ambient quiescent oxidizer domain at temperature 450 K. The spherical burner is 1.0 cm in diameter, supported by a capillary fuel pipe. The fuel mixture stream was heavily diluted by argon (30%DME/70%Ar), and the ambient oxygen was diluted by varying fractions of helium. A heavy

Fig. 1. Schematic diagram of the spherical diffusion flame. 80

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dilution was designed for the fuel inlet stream to lower the global flame temperature, so that the hot-to-cool flame transition can happen at a considerable oxygen level condition. The flame was operated at a reduced pressure (0.3 atm), which could help to increase the flame instability and thus facilitate the investigation on extinction dynamics. Additionally, such flame configuration could provide us a feasibility to select the direction of convection from fuel to oxidizer (normal flame) or from oxidizer to fuel (inverse flame) for given fuel and oxidizer compositions. Since the velocity decreased dramatically out of the spherical burner surface (u~r−2), the spherical diffusion flame has extremely low scalar dissipation rates and approaches the pure diffusion flames that excludes the convection transport. To obtain the high-fidelity structure and transient dynamics of the DME laminar spherical diffusion flames, the computational simulations were conducted using detailed fuel chemistry and transport models with radiative heat loss. The numerical models derived from an appropriate modification of the PREMIX model proposed by Kee et al. [28]. The governing equations are shown in formulas (1)–(3), respectively, which solve the coupled equations of mass, energy, and all species. Mass equation:

∂ρA ∂Ṁ + =0 ∂t ∂r

ambient side, mass fractions of the ambient species (including O2 and He) were constrained; while for the other species, vanish of their gradients was applied, as expressed in formula (5). Extensive simulation studies in the literature [23,24,30] validated predictability and accuracy of the current models in reproducing the flame structure and dynamics of the steady-state or transient spherical diffusion flame.

2.2. Simulation methodology An adaptive mesh spacing scheme was utilized for discretization of the governing equations. The transient terms were discretized using the forward difference scheme, the convective terms using an upwind difference expression, and all diffusive terms using the central difference formula. The chemical reaction rates, thermodynamics data, and species diffusivity were evaluated using the Chemkin and Transport packages [31]. Since the solution of the current 1-D models belongs to a two-point boundary value problem, the Sandia's TWOPNT package [32] with the Newton's iteration algorithm was used to solve the steady-state and transient governing equations. A detailed chemical mechanism of DME proposed by Zhao et al. [33], consisting of 55 species and 290 reactions, was used for the current simulations. Its reliability and accuracy in predicting DME flame characteristics in terms of flow reactor and burner-stabilized flame profiles, ignition delay time, laminar flame speed, and jet diffusion flame structure and pollutant emissions were extensively validated in the literature [33–36]. In addition to the Zhao mechanism, the Fischer mechanism [37] composed of 79 species and 351 reactions, as well as the San Diego mechanism [38] consisting of 63 species and 284 reactions, was also commonly employed for detailed DME flame simulations in the literature. Fig. 2 and 3 show comparisons of these three distinct mechanisms in predicting the hot and cool DME

(1)

Species equation:

ρA

∂Yk ∂Y ∂ + Ṁ k + (ρAYk Vk ) − Aω̇k Wk = 0 ∂t ∂r ∂r

(k = 1, 2,…, K )

(2)

Energy equation:

ρA

∂T ∂T 1 ∂ ⎛ ∂T ⎞ 1 + Ṁ − + Aλ ∂t ∂r cp ∂r ⎝ ∂r ⎠ cp +

1 cp

K

∑ (Aω̇k hk ) + k=1

K

∑ ⎛ρAYk Vk cpk ∂T ⎞ k=1

∂r ⎠

⎝

A q̇ = 0 cp r

(3) 2

where r is the radius, t is the time, A = 4πr is the spherical surface area at radius r, Ṁ = ρuA is the conserved mass flow rate across a spherical surface. ρ, cp, and λ are density, specific heat capacity, and thermal conductivity of the mixture, respectively. K is the total number of species involved in the system, and Yk, cp,k, hk, and Wk are the mass fraction, specific heat capacity, enthalpy, and molecular weight for species k, respectively. ω̇k , Dk,m, and Vk are the volumetric reaction rate, mixture-averaged diffusivity, and diffusion velocity for species k, respectively. q̇r designates the volumetric radiative heat loss, which was estimated by summing up the contributing radiative loss of the participating species CO2, H2O, CO, and CH4, on the basis of optically-thin assumption [29]. For the sake of expression, the origin of coordinate x starts from the spherical burner surface, i.e. x = r-r0 where r0 is the spherical burner radius. An enough large computational domain (x = 0–250 mm) was selected to eliminate the interaction of ambient boundary with the inner flame. Our simulations verified that the scalar gradients at the ambient boundary were negligibly small at such finite domain width. The governing equations were solved subject to the following boundary conditions:

r = rb, T = Tb, Yk (u + Vk ) = u·Yk, b (k = 1, 2,…, K )

(4)

r = + ∞ , T = T∞, if (k = O2 , He ), Yk = Yk, ∞; otherwise, ∂Yk / ∂r = 0 (5) where the subscripts b and ∞ refer to the burner surface and ambient air, respectively. A fixed temperature and fixed mass flux fractions of each species were applied at the porous burner surface. As shown in formula (4), the inclusion of inlet diffusion (Vk) at the burner surface could guarantee that the mass flow rate of each species into the domain equals its mass flow rate supplied into the burner source. At the

Fig. 2. Simulated results for the DME spherical diffusion hot flame that was established at ambient oxidizer composition of 50.0%O2/50.0%He using the Zhao, Fischer, and San Diego detailed mechanisms, respectively. HRR is the abbreviation of heat release rate. Xi is the mole fraction of species I. 81

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Fig. 4. Response of the maximum flame temperature of the steady-state DME spherical diffusion flame with respect to the ambient oxygen mole fraction (XO2⁎). Circles designate the starting point of oscillation, and squares the critical point of extinction due to oscillation.

in Fig. 4 which illustrates the response of maximum steady-state flame temperature with respect to the variation in ambient oxygen mole fraction (XO2⁎). The S-curve of the DME spherical diffusion flame consists of stable hot and cool flame branches, and physically unstable branches. It is shown that the strongly burning flame with high-temperatures could be established at the pure oxygen condition. With continuous decrement in XO2⁎, the maximum temperature and thus combustion intensity decreased gradually until XO2⁎ = 23.2%, beyond which the maximum flame temperature dropped sharply and the flame fell into the cool flame region (T = 550–650 K). The bifurcation point of the upper stable hot-flame branch (XO2⁎ = 23.2%) where the maximum flame temperature equals nearly 1124 K was defined as the steady-state extinction turning point. After falling down to the cool-flame branch, the maximum temperature continued to decrease upon further decrement in XO2⁎ until XO2⁎ = 5.3%, which is defined as the steady-state cool-flame extinction turning point. Additionally, there existed a strong synergistic effect of XO2⁎ over cool-flame combustion, i.e. the cool flame could not jump up to the hot-flame branch until XO2⁎ > 77.9%, even though XO2⁎ was within the hot-flame flammability range (XO2⁎ = 23.2%–100%). Consequently, it was suggested that in the current flame conditions, the DME hot-flame flammability range was XO2⁎ = 23.2%–100%, while the cool-flame flammability range was XO2⁎ = 5.3%–77.9%. The coupling of detailed low-temperature chemistry with thermal/ mass transports was responsible for the existence of cool flames [13,19]. It is evident that the existence of cool flames could extend the DME flammability limit considerably, from XO2⁎ = 23.2% to a fairly low level XO2⁎ = 5.3%. In addition to the present study, the extended flammability limit in terms of stoichiometry for the n-heptane/air spherical propagating premixed cool flame versus the hot flame, as well as the extended extinction strain rate versus that of hot flame for the DME counterflow flame, was also reported in the literature [13,19]. It is well accepted the flame extinction mechanism can be interpreted using the Da number. Flame extinction usually occurs at the critical condition Da = 1 where the residence timescale cannot overwhelm that of chemistry, such that the incomplete fuel oxidation and successive leakages of heat and radicals through the flame front will most probably happen, which eventually leads to the extinguishment. It is analyzed that when the hot flame fell into the cool-flame regime at the hot-flame extinction turning point, although the chemistry timescale increased due to its low flame temperatures, the residence time increased more quickly because of the sudden expansion of cool-flame diameter, as Fig. 5 shows. As a result, the Da number increased instead to support a

Fig. 3. Simulated results for the DME spherical diffusion cool flame that was established at ambient oxidizer composition of 20.5%O2/79.5%He using the Zhao, Fischer, and San Diego detailed mechanisms, respectively. HRR is the abbreviation of heat release rate. Xi is the mole fraction of species I.

spherical diffusion flames, respectively. It is clearly shown that the Fischer mechanism results agree pretty well with the Zhao mechanism results, in either hot or cool flame condition. For the San Diego mechanism, its hot flame results are identical with the Zhao and Fischer mechanism results. With respect to the cool flame results, the San Diego mechanism agrees well with Zhao and Fischer mechanism results in the main reaction zone (x < 8.0 cm), and has a marginal deviation from the latter two mechanisms in the post-flame zone (x > 8.0 cm). In overall, the three mechanism results are basically identical with each other. Hence, the current simulation results obtained with the Zhao mechanism were reliable. During the simulation, the environment pressure (0.3 atm), fuel composition (30%DME/70%Ar) and exit velocity (13.0 cm/s) and temperature (300K), and ambient temperature (450 K) were kept as constants, while reducing the ambient oxygen mole fraction (from pure oxygen) gradually to evolve to flame extinction. First, the steady-state simulations were conducted to obtain the steady flame structure and Scurve [4] which consists of hot and cool stable flame branches and unstable branches. The structure dynamics, flammability limit, and extinction mechanism of the DME flames can be examined by performing a marching analysis along the S-curve. Then, the transient flame evolution dynamics were analyzed by performing transient simulations with a small temperature perturbation imposing to the steady solutions. 3. Results and discussion 3.1. S-curve behavior of the DME spherical diffusion flames The steady-state ignition and extinction dynamics of the DME spherical diffusion flame will be examined using the S-curve, as shown 82

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(x = 0–3.0 cm), with the peaking temperature and heat release rate (HRR) located at x = 2.37 cm. On the contrary, once the hot flame dropped into the cool-flame region, the flame temperature dropped sharply to the low-temperature range (550–650 K), and at the meantime, the main reaction zone situated itself to occupy a much wider domain, with much gentler spatial gradients and smaller scalar dissipation rates. At XO2⁎ = 18.9%, the domain of cool-flame zone was expanded to x = 0–10.0 cm, which implies that the DME cool flame had the nature of low activation energy; its thermochemical dynamics would be rather insensitive to the variations in flame parameters such as XO2⁎, pressure, or stoichiometry, etc., in comparison with the hot flame. A parameter study indicated that the change of DME cool-flame structure was fairly indiscernible with the variation in XO2⁎, while even a small perturbation in XO2⁎ could change the hot flame temperature and structure dramatically. This can be also demonstrated by the Scurve as shown in Fig. 4. Response of the maximum flame temperature with respect to XO2⁎ was much more insensitive for the cool-flame branch than the hot-flame branch. Fig. 5 also shows that the products of DME hot flames were basically CO2 and H2O. While with respect to the cool flame, large fractions of unburned hydrocarbons including CO, H2, and fuel radicals were also available in the product composition. This is due to the low-temperature fuel decomposition reactions through which the large fuel molecules are decomposed to small hydrocarbon species or radicals. The fuel decomposition reactions belong to a subset of the cool-flame chemistry. They produced unburned hydrocarbon species (including CO and H2) and radicals, which cannot be completely oxidized further to the final products (CO2 and H2O) in the low-temperature condition. Consequently, these species existed as major constituents in the product composition. The transitional dynamics from hot to cool flames coincides with the experimental observation on the droplet combustion in microgravity as reported in [17,18]. In the following part, the transient dynamics and governing mechanism near extinction of the DME spherical diffusion flame will be clarified by diagnostics of the numerical solutions.

Fig. 5. Hot and cool flame structures at ambient oxidizer compositions of 42.1%O2/57.9%He (a) and 18.9%O2/81.1%He (b), respectively.

physically-stable cool flame. As reviewed in [13], the extended cool-flame flammability limit compared with the hot flame has important implication in the viewpoint of fire safety. More specifically, after the extinguishment of hot droplet flame, reactions may not cease completely, instead, the dark, indiscernible cool flame may be still occurring at very weak reaction rates. If some enhancement conditions are applied to the cool droplet flame, say recovery of high oxygen concentration or elevated pressure, the cool flame may jump to hot flame immediately. However, the current study indicated that it was much difficult to shift the DME spherical diffusion cool flame to hot flame. As shown in the S-curve of Fig. 4, the minimum XO2⁎ that could activate the shift from cool flame to hot flame was at least 77.9%. Furthermore, Fig. 4 demonstrates that the DME spherical diffusion cool flames could be readily established over a wide range of XO2⁎ (XO2⁎ = 5.3%–77.9%), implying the promising potential of spherical diffusion flame burner apparatus in the cool-flame experiments and combustion science, even without the assistance of ozone sensitization.

3.3. Transient dynamics of the DME hot and cool flames near extinction Fig. 6a displays the transient evolution dynamics of maximum flame temperature after a small perturbation in temperature was imposed to the steady-state solution, for three typical near-extinction steady-state hot flames which were established at XO2⁎ = 24.02%, 24.04%, and 24.10%, respectively. It is shown that the transient flame at XO2⁎ = 24.10% developed to the original steady state before perturbation after several cycles of oscillations, so it was physically-stable. While with respect to the flame at XO2⁎ = 24.02%, its oscillatory amplitude increased gradually cycle by cycle, until extinguishment where the instantaneous peak temperature dropped below the steady-state extinction temperature (1124 K or so) and could not recover back. Hence, the flame in this condition indeed was physically-unstable (although it was located on the stable hot-flame branch as shown in Fig. 4), since any perturbation would trigger extinguishment induced by oscillations. For instance, as shown in Fig. 6b, the transient flame at XO2⁎ = 24.03% with a negligibly small temperature perturbation (+0.03%) finally evolved to extinction through about 162 cycles of oscillations. The oscillatory flame at XO2⁎ = 24.04%, which did not evolve to extinguishment for a long period of time with a quasi-constant oscillation amplitude, can be regarded as the practical extinction turning point. Moreover, the present study indicated that the oscillation frequency near the hot-flame extinction was basically unchanged (1 Hz), regardless of the level of ambient oxygen concentration. Fig. 7 shows the transient response of maximum flame temperature upon small temperature perturbations for the steady-state cool flames established at XO2⁎ = 6.0%, 6.1%, and 6.2%, respectively. We detected the coexistence of dual oscillatory modes with rather distinct frequencies near the cool-flame extinguishment, which was particularly

3.2. Structure dynamics of the DME hot and cool flames The steady-state hot flame established at XO2⁎ = 42.1% and cool flame at XO2⁎ = 18.9% were selected to present the hot and cool flame structure behaviors, and the results are shown in Fig. 5a and b respectively. It is shown that the inner combustion process was supported by the right ambient side oxygen molecules by diffusive transport. Due to the restricted oxygen diffusion flux into the domain, even the hotflame temperatures and thus reaction intensity were rather depressed compared with the traditional jet flames, where the oxygen supplement is governed mainly by strong convection. As shown in Fig. 5a, for the hot flame at XO2⁎ = 42.1%, the main combustion reactions were distributed within a fairly narrow region 83

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Fig. 6. (a) Transient response of the maximum flame temperature upon a small temperature perturbation for the steady-state hot-flame solutions established at XO2⁎ = 24.02%, 24.04%, and 24.10%, respectively. (b) Transient response of maximum temperature upon δT = +0.03% temperature perturbation for the steady hot flame of XO2⁎ = 24.03%. δT is the perturbation ratio of temperature that is imposed to the steady-state solution.

extinction point of the cool flame. Besides, it is also interesting to note that the oscillation period near the hot-flame extinction was always constant (about 1 s), irrelevant with XO2⁎. While the high-frequency oscillation period of cool-flame extinction increased significantly with the decrement in XO2⁎, from 1.3 s at XO2⁎ = 10.5% to 8.3 s at XO2⁎ = 6.1%. In overall, the transient cool flames near extinction was featured with low frequencies of oscillation compared with the hot flames. As Fig. 4 illustrates, the near-limit oscillation could exist in a wider range for the cool flame than the hot flame. In addition, the critical oscillatory extinction limit for hot or cool flame was a bit higher than the corresponding turning point of the S-curve, however, the practical oscillatory extinction point would depart considerably from the S-curve turning point with the increase of pressure [21].

different from the hot flame extinction that was featured with a single oscillatory mode. The high-frequency oscillatory mode was shown by solid curve, and the low-frequency one by dashed curve. As shown by the curves corresponding to XO2⁎ = 6.1% and 6.2%, the oscillation period of the high-frequency mode was about 8.3 s, while that for the low-frequency mode was significantly large (about 53 s). In addition, oscillations of both the low- and high-frequency modes attenuated with the increase of XO2⁎. Consequently, the flames established at high oxygen condition (XO2⁎ = 6.1% and 6.2%) were oscillatorily stable, however, the flame at low oxygen condition (XO2⁎ = 6.0%) was unstable since its peak temperature dropped below the cool-flame steadystate extinction temperature (548 K or so) after a few cycles. It is also noted that the high-frequency oscillation amplitude was enhanced in half period of the low-frequency oscillation where temperature increased with time; while in the next half period, the high-frequency oscillation amplitude was suppressed. This implies a strong interaction between the low-frequency and high-frequency modes that depended on the phase difference. The low-frequency oscillation of the flame at XO2⁎ = 6.1% vanished after a sufficiently long time, while its highfrequency oscillation increased firstly and then maintained at a fixed amplitude after 200 s, so it can be regarded as the practical critical

3.4. Oscillatory extinction mechanism of the hot and cool flames The present simulations demonstrate that the oscillation amplitude of the near-limit transient flames increased or decayed with a basically equal growth ratio cycle by cycle. The so-called growth ratio (γ) is defined as the ratio of oscillation amplitude in the previous cycle to that 84

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Fig. 7. Transient response of the maximum flame temperature upon a small temperature perturbation (δT = +1.5%) for the steady-state cool-flame solutions established at XO2⁎ = 6.0%, 6.1%, and 6.2%, respectively.

oscillatory stability. Additionally, enhancement of (R1: H + O2 = O + OH) can also increase the equilibrium temperature, while the impact of (R53: CH3 + H(+M) = CH4(+M)) on the equilibrium temperature was rather ignorable. With respect to the cool-flame extinction with oscillation, this study found that it was mainly controlled by the low-temperature chain branching and termination competition in the NTC region. It was found that the OH radical played a crucial role in the acceleration of coolflame branching, which initiates from reaction (R240) that generates a fuel radical R (CH3OCH2). Then, the fuel radical was transformed to the final products through reactions (R264), (R271), (R273), (R274), (R275), and further oxidations in sequence. The above reaction pathway is chain branching since it consumes one OH molecule in beginning and produces two at the end.

in the next cycle. It is clear that the stability/instability of the oscillatory flames depends on its γ, i.e. γ < 1 represents stable and γ > 1 unstable, γ = 1 is the critical point for oscillatory extinguishment. It is suggested that the reaction with more significance for γ is more important for the oscillatory extinction. Thus, the logarithmic sensitivity index defined based on γ, as expressed in formula (6), was utilized for the sensitivity analysis to reveal the controlling reactions in the oscillatory extinction. Ar is the pre-factor for the rth reaction, and Ar′ is the perturbed pre-factor. γ and γ′ designate the oscillatory growth ratio before and after the pre-factor perturbation.

SIr =

(γ − γ ′)/ γ ∂ ln (γ ) ≈ (Ar − Ar′)/ Ar ∂ ln (Ar )

(6)

The transient simulations were repeated reaction by reaction with a perturbation in its pre-factor, to obtain the SIr value of each reaction. As indicated by its definition, the reactions with largest amplitudes of SIr are dominated in the oscillatory extinction, and in addition, positive SIr means the reaction is favorable to oscillatory extinction and instability, while negative SIr favorable to stability. Fig. 8 shows the governing reactions with their SIr values at the hot- and cool-flame oscillatory extinction limits, respectively. It is shown that the endothermic reactions (R1: H + O2 = O + OH) was most influential for the hot-flame oscillatory extinction. Enhancement of its reaction rate can significantly increase the flame stability, due to its dominance in the chain-branching pathways. Enhancing the high-temperature exothermic reaction (R29: CO + OH = CO2 + H) could also extend the hot-flame flammability limit, since heat release was most important for the flame stability near extinction. (R53: CH3 + H(+M) = CH4(+M)) and (R56: CH4 + OH = CH3 + H2O) with strong exothermic effect, however, was unfavorable to the hot-flame oscillatory stability, because of their chain-termination nature. In overall, the hot-flame oscillatory extinction was controlled by the competition of high-temperature exothermic/endothermic reactions with the chain branching/termination reactions involving small molecules. Fig. 9a displays the results of a parameter study where the prefactors of governing reactions (R1: H + O2 = O + OH) and (R53: CH3 + H(+M) = CH4(+M)) were multiplied by 1.2 individually for the transient simulations of near-extinction hot flame established at XO2⁎ = 24.02% with a perturbation δT = +0.3%. It is verified that the enhancement in reaction (R1: H + O2 = O + OH) could significantly improve the oscillatory stability of hot flames near extinguishment, while reaction (R53: CH3 + H(+M) = CH4(+M)) was adverse to the

CH3OCH3 + OH = CH3OCH2 + H2O

(R240)

CH3OCH2 + O2 = CH3OCH2O2

(R264)

CH3OCH2O2 = CH2OCH2O2H

(R271)

CH2OCH2O2H + O2 = O2CH2OCH2O2H

(R273)

O2CH2OCH2O2H = HO2CH2OCHO + OH

(R274)

HO2CH2OCHO = OCH2OCHO + OH

(R275)

Furthermore, there also existed a chain termination pathway including reactions (R272) and (R44) that produce one OH but consume two OH, which compete with the branching pathways for the OH radical. In consistence with the expectation, Fig. 8b verifies that the chain-branching reactions (including (R240), (R264), (R271), (R273), (R274), and (R275), most of which were exothermic) had negative SIr values with a favorable effect on the oscillatory stability. On the contrary, the chain-termination reactions ((R272) and (R44)) with positive SIr values were unfavorable to oscillatory stability. CH2OCH2O2H = OH + CH2O + CH2O CH2O + OH = HCO + H2O

(R272) (R44)

Fig. 9b displays the parameter study results where the pre-factors of governing reactions (R240) and (R272) were multiplied by 1.08 individually for the transient simulations of a near-extinction cool flame (XO2⁎ = 6.1%, δT = +1.5%). It verifies the effectiveness of reaction (240) to improve the cool-flame oscillatory stability near extinguishment, and the adverse effect of (R272) to stability. However, the 85

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Fig. 8. The controlling reactions with decreasing importance to oscillatory instability revealed by the logarithmic sensitivity index for the (a) hot- and (b) cool-flame oscillatory extinctions, respectively.

the red circle, around the steady-state solution (namely the intersection point of the two orthogonal dashed lines). The maximum reaction rates and temperature deviated from the steady-state solution cycle by cycle, until extinguishment in the end where the rotation of phase function curve cannot continue. Additionally, each subfigure can be divided into four quadrants by the two orthogonal dashed lines, with the upper right and left denoted as I and II, and the lower left and right as III and IV, respectively. It is shown that for the high-temperature reactions that were favorable to oscillatory stability (including R1, R29, R30, and R48), its reaction rates decreased with the decrement in temperature from the peak points in quadrant I to the corresponding trough points in quadrant III, along the clockwise direction without a phase difference. Due to the small chain termination rate and heat loss at the trough point, the reactions rates and temperature turned to increase from this point. In the next half cycle, the maximum rates of branching reactions R1, R29, R30, and R48 increased with the increment in temperature, however, they peaked prior to the maximum temperature. This was because the chain-termination rate of (R53) that was most unfavorable to oscillatory stability increased further with temperature at this point; the maximum rates of (R53) and maximum temperature oscillated rather synchronously, without a phase difference. As a result, after the

simulation reveals that the oscillation frequency did not change with the variation in (R240) reaction rate. Besides, it is worth to mention that the above elementary reactions are closely correlated with the oscillatory stability/instability. Many previous studies [21,39–42] mathematically showed that for the spatially inhomogeneous reacting flow system, the near-limit oscillatory dynamics was due to the existence of complex eigenvalues of the Jacobian matrix of its governing equations. The complex eigenvalues came from the chemical Jacobian matrix with stiff chemistry, rather than from the mixing Jacobian matrix. Consequently, the above elementary reactions are responsible for the oscillatory extinction. The flame extinction was determined by the complex coupling between chemical branching and termination, and heat release and loss, with distinct phase differences. Fig. 10 shows the phase function between the maximum temperature and maximum reaction rates that controlled the hot-flame oscillatory extinction, which was established at XO2⁎ = 24.02% with a temperature perturbation δT = +0.3%. Herein, the phase function curve was obtained by plotting time series of the maximum reaction rate against time series of the maximum flame temperature. It can be seen that during the oscillatory extinction process, the phase function curve revolved in the clockwise direction from 86

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Fig. 9. Transient response of the maximum temperature with respect to a perturbation in the pre-factor of a governing reaction, for the transient near-extinction hot flame at XO2⁎ = 24.02% and δT = +0.3% (a), and transient cool flame at XO2⁎ = 6.1% and δT = +1.5% (b), respectively. δAi is the relative perturbation in pre-factor of the i-th reaction.

increase after entering quadrant III. Hence, it was concluded that the cool-flame oscillation belonged to the kinetically-controlled type in the NTC region where the reactivity of cool-flame chemistry decreased with increasing temperature.

peaking values, the branching rates turned to decrease with further increasing temperature in quadrant me. Fig. 11 shows the evolution of phase function between the maximum reaction rates of the controlling reactions and maximum temperature, for the oscillatory flame established at XO2⁎ = 6.1% with temperature perturbation δT = +1.5%, from 200 s to 250 s where oscillations developed to the stage of quasi-constant oscillatory amplitude. It indicates that the branching and termination reactions had different phase function topologies. It is observed that the chainbranching reactions (including R240, R264, R271, R273, R274, and R275) had about +π/2 phase difference with respect to the maximum temperature. It was analyzed that the reactivity of these low-temperature chain-branching reactions increased with increasing temperature, but characterized by weak temperature dependence due to their small activation energies. Consequently, the low-temperature branching rate increased with temperature in quadrant II. However, with further increment of temperature in quadrant I, since the chain-termination rates surpassed that of branching due to the much larger activation energies of R272, the branching rate started to decrease because of OH sufficiency. In the transitional stage from quadrant IV to III, because of the continuous decrease in termination rate, the branching rate began to

4. Conclusions The near-limit oscillatory dynamics and extinction mechanism of the DME spherical diffusion flame in hot- and cool-flame conditions were numerically studied in this paper. The main findings include: (1) Self-sustaining cool flames can be readily established on the spherical diffusion burner in microgravity over a wide range of parameters. Additionally, the flammability limit of DME spherical diffusion flame could be considerably extended by the cool-flame chemistry. (2) The DME spherical diffusion flame exhibited oscillation behavior near the extinction limit, in either hot- or cool-flame regime, or oscillation in the latter regime was much stronger and more complex. The near-limit oscillations triggered the occurrence of flame extinguishment prior to the steady-state S-curve burning point. 87

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Fig. 10. Phase diagram between the maximum flame temperature and maximum reaction rates of the controlling reactions (R1), (R29), (R53), (R56), (R30), and (R48) respectively, for hot-flame oscillatory extinction at XO2⁎ = 24.02% and δT = +0.3%. The red circle designates the beginning of oscillation upon perturbation, and the intersection point of the orthogonal dashed lines designates the steady-state point before the perturbation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Additionally, the DME hot flame extinction was governed by a single oscillatory mode with a fixed frequency (1 Hz) that was irrelevant with XO2⁎, while the cool flame extinction was governed by dual oscillatory modes that had rather distinct frequencies. The oscillatory period of the high-frequency mode increased significantly when approaching the extinction point. In addition, a strong interaction between the high-frequency and low-frequency modes was observed which made the transient dynamics near extinction more complicated. (3) The hot-flame oscillatory extinction was controlled by the competition of high-temperature exothermic/endothermic reactions with the chain branching/termination reactions involving small molecules. Enhancement of (R1: H + O2 = O + OH) was most effective to improve the hot-flame oscillatory stability. The cool-flame

oscillation belonged to the kinetically-controlled type in the NTC region, and the competition of low-temperature branching (R240: CH3OCH3 + OH = CH3OCH2 + H2O) and termination (R272: CH2OCH2O2H = OH + CH2O + CH2O) was responsible for the underlying mechanism of its oscillations. Acknowledgements The present research was supported by the National Natural Science Foundation of China (Grant No. 51706027), China Postdoctoral Science Foundation (Grant No. 2016M590865), National Key Research and Development Plan (Grant No. 2016YFB0600201), Scholarship Fund of the China Scholarship Council (Grant No. 201608505020), Fundamental Research Funds for the Central Universities (Grant Nos. 88

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Fig. 11. Phase diagram between the maximum flame temperature and maximum reaction rates of the controlling reactions (R240), (R264), (R273), (R274), (R272), and (R44) respectively, for the cool-flame oscillatory extinction within [200 s, 250 s] which was established at XO2⁎ = 6.1% and δT = +1.5%. The red circle designates the time 200 s and red square 250 s, the intersection point of the orthogonal dashed lines designates the steady-state point before the perturbation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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