Sub Topic: Reaction Kinetics
Paper: 114RK-0017
9th U. S. National Combustion Meeting Organized by the Central States Section of the Combustion Insitute May 17-20, 2015 Cincinnati, Ohio
Two-dimensional Numerical Simulation of the Transition between Slow Reaction and Ignition J. Melguizo-Gavilanes∗1 , P.A. Boettcher2 , A. Gagliardi2 , V.L. Thomas3 , R. M´evel1 1
2
Graduate Aerospace Laboratories of the California Institute of Technology (GALCIT), Pasadena, CA 91125, USA
Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA 19104, USA 3
Mechanical Engineering, John Hopkins University, Baltimore, MA 21218, USA ∗
Corresponding Author Email:
[email protected]
Abstract: Thermal ignition of flammable gaseous mixtures from a hot surface is a major concern for a wide range of industrial activities. When homogeneous hydrocarbon fuel/oxidizer mixtures are heated in a closed vessel from ambient temperature to values near their autoignition temperature these mixtures undergo either a slow reaction or an ignition, when heat transfer to the surroundings is accounted for. Experiments and 0-D simulations using detailed chemical mechanisms have shown that this behavior is a function of the heating rate, initial composition, and pressure. The present study develops a onestep reaction model suited to capture the transition from slow to fast reaction that is appropriate for the wide range of conditions typically encountered in realistic industrial applications. A vertical crosssection of a cylindrical reactor is modeled to gain insight into the sequence of events leading to a slow reaction or an ignition. The heating of the reactor’s wall induces a very dynamic buoyancy driven flow in which the mixture rises along the walls and separates at the centerline creating two well defined vortical structures. Chemical heat release starts at the top of the vessel (where the mixture is hottest), and the flow reverses: the gas then rises along the centerline, separates and descends along the walls. Depending on the heating rate, the mixture undergoes slow oxidation or ignition whereby a flame that emanates from the separation point fully consumes the mixture. Keywords: ignition, slow reaction, simplified chemical kinetics model, multi-dimensional simulations
1
Introduction
Thermal ignition of flammable gaseous mixtures from a hot surface is a major concern for a wide range of industrial activities including commercial aviation, nuclear power plants and industrial chemical processes [1–3]. Typical ignition sources include concentrated hot surfaces (e.g. glow plugs) [2, 3], moving hot particles [4] and extended hot surfaces (e.g. heated vessels) [1]. Minimizing the risk of accidental combustion events through updated safety regulations and improved engineering design calls for a deep understanding of thermal ignition. In a previous study on the autoignition of n-hexane/air mixtures [1], it was experimentally demonstrated that a flammable mixture subjected to a range of heating rates is consumed in two very different ways. If the heating rate is low, a slow reaction occurs, whereas if the heating rate is high an ignition event takes
1
Topic: Reaction Kinetics
Paper: 114RK-0017
place. The slow reaction is characterized by a slow consumption of the reactants at essentially constant temperature and pressure, while the ignition event is associated with a thermal runaway and a significant pressure increase. The mixture is observed to transition to ignition with increasing heating rate, pressure, and equivalence ratio (0.7 ≤ φ ≤ 1.4). This behavior was also investigated computationally using detailed kinetics [1]. The computational model extends the classical theory of Semenov [5] by including an extra term accounting for the rate at which the mixture is heated. Detailed analyses of the chemical reaction pathways and energy balance have demonstrated that the competition between the chemical energy release rate and heat losses rate when close to the auto-ignition temperature can induce a composition change, so that an initially flammable mixture can be turned into a non-flammable mixture. In spite of the insight provided by the detailed simulations, the large number of chemical species and reactions involved during low-temperature oxidation of hydrocarbon fuels makes the computation very expensive to carry out comprehensive parametric studies, and/or study complex industrial configurations using multi-dimensional numerical simulations. The present study aims at assessing the applicability of the aforementioned model in multi-dimensional simulations, hence, the experiments performed in [1] were simulated in two-dimensions to investigate the flow structure and gain further insight into the reaction regimes mentioned above. The paper is structured as follows: in Section 2 and 3 the thermal ignition model and methodology employed to derive the one-step reaction parameters are described. Section 4 introduces the governing equations, numerical approach and simulation parameters. The results of the multidimensional simulations are explained in detail in Section 5. Section 6 includes concluding remarks. 2
Thermal Ignition Model
The thermal ignition process is described using a thermodynamic and chemical kinetic model. The thermodynamic model is given by the conservation of energy equation for a stationary constant volume and constant mass system. The system exchanges energy with its surroundings through heat transfer at the wall, Qw , which can be into or out of the system depending on the relative temperatures of the gas and the wall. ∆U = Qr + Qw
(1)
Assuming that the gas obeys the ideal gas equation and has constant specific heat, simplifies the equation to, dT = Q˙ r + Q˙ w , (2) mcv dt where the energy release rate due to chemical reactions, Q˙ r , is in competition with the energy transfer rate at the wall, Q˙ w . The heat transfer with the wall is expanded from the original Semenov theory to include the initial temperature of the wall, Tw0 , and the rate at which it is heated, α [1, 5]. ¯ T0 + αt − T , Q˙ w = S h (3) w
2
Topic: Reaction Kinetics
Paper: 114RK-0017
where S is the surface area and h is the heat transfer coefficient. The energy release rate is found by assuming that the reaction progresses in one irreversible step from reactant, R, to product, P : R → P.
(4)
The rate at which this reaction progresses depends on the temperature, T , and the molar concentration of reactant , [R] [6]. d[R] d[P ] =− = ω˙ = k(T )[R]n (5) dt dt In equation 5, ω˙ is the molar production rate per unit volume, k is the reaction rate, and n is the effective reaction order. Using the approach from [7], we express the molar concentration using the ideal gas law, pV = nRu T , [i] =
ni pi Xi p Xi = = = ρ, V Ru T Ru T Wi
(6)
where ni , pi , Xi , and Wi are the number of moles, the partial pressure, the mole fraction, and the molecular weight, respectively, of either the reactant, R, or the product, P . The variables p, and ρ are mixture pressure and density, respectively. The reaction is ultimately expressed in the form of the reaction progress variable, λ, which is equivalent to the mass fraction of the products, YP . WP W Equation 5 can be expressed in terms of mass fraction using Equations 6 and 7 YP = XP
dXP ρ dYP W d[P ] = = ρ. dt dt WP dt WP2
(7)
(8)
Solving for dYP /dt yields dYP W2 = P k(T )[R]n dt Wρ n 2 XR dYP WP 1 = k(T ) ρ . dt W ρ WR The rate of progress of the chemical reaction is governed by an Arrhenius rate law [6], Ea k(T ) = A exp − , Ru T
(9) (10)
(11)
where A is the pre-exponential, Ea is the activation energy and Ru is the universal gas constant. Substituting Equation 11 into Equation 10, the evolution of the mass fraction of P becomes dYP WP2 XRn Ea n−1 = A ρ exp − . (12) dt W WRn Ru T Combininig the terms in parenthesis in Equation 12 and assigning them to the pre-exponential factor Z [7]. In addition, restating the equation in terms of progress variable, λ, and introducing the consumption term, (1 − λ) yields. dλ Ea n−1 = Z(1 − λ)ρ exp − (13) dt Ru T 3
Topic: Reaction Kinetics
Paper: 114RK-0017
When λ = 0 the mass fraction of R is 1; when λ = 1 the mass fraction of P is 1. The energy release rate in equation 2 is the products of the energy stored in the chemical mixture, Qc , and the rate at which this energy is released, dλ/dt, dλ E dλ a n−1 = mqc = mqc Z(1 − λ)ρ exp − , (14) Q˙ r = Qc dt dt Ru T where qc is the chemical energy content per unit mass of R and m is the mass of the gas. The complete coupled equations describing this thermal ignition model are ¯ dT qc Ea Sh n−1 = Z(1 − λ)ρ exp − + Tw0 + α t − T , dt cv Ru T mcv Ea dλ n−1 = Z(1 − λ)ρ exp − . dt Ru T
(15) (16)
Table 1: Model Parameters
Parameter V S ¯ h cv Ru Ea qc
3
Value
Units −4
4.27 ×10 3.36 ×10−2 10 980 8.314 1.48 ×105 2.461 ×106
Description
3
m m2 W m−2 K−1 J kg−1 K−1 J mol−1 K−1 J mol−1 J mol−1
volume surface area heat transfer coefficient specific heat at constant volume universal gas constant activation energy chemical energy content
Calculation of Model Parameters
In the current thermal ignition model a number of parameters are fixed by the physical system used in the experimental study [1], such as the surface area and volume (see Table 1). Other parameters, such as the activation energy and chemical energy content, are determined with the use of a detailed chemical model and the available experimental data. 3.1
Heat transfer
¯ was determined by peforming an inert two-dimensional The average heat transfer coefficient, h, simulation of the buoyancy driven flow induced by the heating of the reactor’s wall (See section ¯ = q 00 /(Tw (t) − T ), where q 00 is the wall heat flux, 4 for details). Using the standard definition, h and the difference (Tw (t) − T ) is the time dependent thermal gradient, with Tw (t) = Tw0 + αt. The average heat transfer coefficient was found to be 10 W/m2 K which is consistent with free convection of gases [8]. 4
Topic: Reaction Kinetics
3.2
Paper: 114RK-0017
Thermo-chemical parameters
In order to include realistic thermodynamic and chemical parameters in the one-step reaction model, a number of 0-D calculations were carried out using a detailed reaction mechanism referred to as the updated Ramirez et al. model [9, 10]. It includes 531 chemical species and 2,628 reactions. 3.2.1
Energy content
The first parameter calculated from the detailed mechanism is the energy content per unit mass, qc , assuming an adiabatic systems where all of the stored chemical energy is used to heat the product species until chemical equilibrium is reached. The energy content of the system can then be extracted by cooling the system back to its initial temperature and assessing the difference in internal energy. In other words, the final temperature of the products can be reached either by chemical reaction from reactants to products, or by heating the products from the initial to final temperature assuming negligible changes in specific heat. This approach is implemented numerically using Cantera with MATLAB [11] by equilibrating the mixture under constant specific volume and internal energy, and then resetting the temperature to the initial temperature and computing the difference in internal energy. Calculations were performed for φ = 0.4 − 3, To = 298 K and po = 20 − 200 kPa as shown in Figure 1 and only minor changes are observed relative to previous calculations [12].
Energy Content [kJ/kg]
2600
20 kPa 60 kPa 100 kPa 200 kPa
2400 2200
, spline(0,20,2) , spline(1,20,2) , spline(2,20,2) , spline(3,20,2)
2000 1800 1600 1400 1200 1000 0.5
1
1.5
2
2.5
3
Equivalence Ratio - ɸ
Figure 1: Evolution of energy content per unit of mass as a function of equivalence ratio. Conditions: To = 298 K and po = 101 kPa
3.2.2
Reaction Rate
The second parameter that is determined using the detailed mechanism is the activation energy, Ea . To obtain the activation energy as a function of pressure and equivalence ratio, a series of ignition delay-time calculations were performed assuming a constant volume reaction. The ignition delaytime is defined as the time to maximum pressure-gradient. Computations were carried out over 5
Topic: Reaction Kinetics
Paper: 114RK-0017
a period of 2000 s for φ = 0.4 − 3, po = 20 − 200 kPa; and To = 500 − 1100 K. Previous work [2, 12] shows how the activation energy is calculated by fitting the low temperature ignition delay time and calculating the slope of this curve. In this study the changes in activation energy with initial pressure and equivalence ratio were negligible and the activation energy was calculated to be approximately constant at 148, 000 J/mol. 3.2.3
Pre-exponential Factor
1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300
1
ɑ = 10.80 K/min ɑ = 10.75 K/min 530 520 510 500 490 480 1030
1050
Reaction Progress - λ
Temperature [K]
The pre-exponential factor, Z, is calibrated so that the one-step model reproduces the transition from slow reaction to ignition observed in the detailed simulations at 101 kPa for a stoichiometric mixture as the heating rate is increased from 10.75 K/min to 10.80 K/min. This transition behavior is shown in the one-step results for a stoichiometric mixture at atmospheric pressure in Figure 2. Figure 2 (Left) shows the very slight temperature increase above the heating rate for the slow reaction case as well as the large temperature increase for the ignition case. Similarly, the reaction progress variable shows a steep increase in the reaction rate at the point of ignition in Figure 2 (Right). The pre-exponential factors are individually fit for two different effective reaction orders n = 1, and n = 2 and determined to be Z = 8.75 × 1010 and Z = 7.25 × 1010 , respectively.
1070
ɑ = 10.80 K/min ɑ = 10.75 K/min
0.8
0.7 0.6
0.6
0.5 0.4
0.4
0.2
0.3 1030 1040 1050 1060 1070
0 0
200
400
600
800
1000
1200
0
Time [s]
200
400
600
800
1000
1200
Time [s]
Figure 2: Simulation results for n = 1, po = 101 kPa, and φ = 1.0 showing the temperature evolution (Left) and progress variable evolution for a slow reaction and an ignition case with a small heating rate increase of 0.05 K/min (Right)
In summary, solutions to the one-step thermal ignition model are obtained for a pressure range of po = 20 − 200 kPa over the flammability range, φ = 0.5 − 3, at standard temperature and pressure [14]. The heating rate is varied from α = 1 K/min to 30 K/min to investigate the transition behavior. An ignition case is distinguished from a slow reaction case by the temperature overshoot, ∆T , above the prescribed by the heating rate. The transition is marked by a threshold temperature overshoot of 50 K as verified in previous work [12]. Also, in [12], it was shown how equivalence ratio, heating rate, initial pressure, and the effective reaction order influence the transition from slow reaction to ignition in a 0-D reactor. 6
Topic: Reaction Kinetics
4
Paper: 114RK-0017
Multi-dimensional Simulations: physical model, numerical approach and parameters
The ignition problem is governed by the compressible, reactive Navier-Stokes equations with temperature dependent transport properties ∂ρ + ∇ · (ρu) = 0 ∂t
(17)
∂(ρu) + ∇ · (ρuu) = −∇p + ∇ · τ + ρg (18) ∂t Dp ∂(ρh) + ∇ · (ρuh) = ∇ · (κ/cp ∇h) + qc ΩR + (19) ∂t Dt 2 ¯ with p = ρRT, τ = (p + µ∇ · u)I + µ[∇ · u + (∇ · u)T ] (20) 3 The Sutherland Law, the Eucken Relation and the JANAF polynomials are used to account for the functional temperature dependence of mixture viscosity (µ), thermal conductivity (κ) and specific heat (cp ) respectively. The chemistry is modeled using an irreversible one-step scheme (R → P ) in which the kinetic parameters are found using the procedure described in the previous sections. Using subscript R for reactants, species mass conservation can then be written as: ∂(ρYR ) + ∇ · (ρuYR ) = ∇ · (ρDR ∇YR ) + ΩR with ΩR = −ρYR Z exp (−Ea /Ru T ) (21) ∂t where ρ is density, u is the velocity vector, p is pressure, h is the mixture enthalpy, g is the gravitational acceleration, qc is the stored chemical energy, YR is the mass fraction of reactants, Z is ¯ are the universal and spethe pre-exponential factor, Ea is the activation energy, and Ru and R cific gas constants respectively. Lastly, the Lewis number is assumed to be unity which results in κ/cp = ρDR , hence, the dynamic thermal diffusivity of reactants is used to model its mass diffusivity. The governing equations are integrated in two dimensions using the Open source Field Operation And Manipulation (OpenFOAM) toolbox [13]. The spatial discretization of the solution domain is done using Finite Volumes, and the pressure-velocity coupling is achieved using the PIMPLE (PISO+SIMPLE) algorithm. The geometry simulated corresponds to a vertical cross section of the cylindrical reactor used in [2] (5 cm in diameter and 22 cm in length). Figure 3 shows the mesh, full O-grid with square center piece to avoid singularities and small angles at high resolutions. There are approximately 10k points in the computational domain, compressed near the wall of the reactor, with a minimum cell size of 60 µm to allow for enough resolution to resolve the thermal and hydrodynamic boundary layer. Initial conditions are po = 101 kPa, To = Tw0 = 300 K, Uo = 0 m/s, and YRo = 1, with non-slip condition and a prescribed temperature ramp on the vessel’s wall given by Tw (t) = Tw0 + αt with α = 15 K/min and 12 K/min. The chemical kinetics parameters used for modeling a slightly lean (Φ = 0.9) n-hexane-air mixture are qc = 2.461 × 106 J/kg, Ea = 148, 000 J/mol and Z = 8.75 × 1010 s−1 . 5 5.1
Results Time Evolution - slow reaction and ignition event
Simulation of the heating of a reactive gas inside a cylindrical reactor is performed. The heating rate was increased at 3 K intervals starting from 9 K/min. A drastic change in the reaction behavior 7
Topic: Reaction Kinetics
Paper: 114RK-0017
Figure 3: Computational domain for cylindrical reactor - O-grid with square centerpiece and cell compression near walls
occured when the rate was increased from 12 to 15 K/min, namely, from initially having slow comsumption of the mixture at 12 K/min to an ignition event at 15 K/min. Cylindrical reactor - heating rate: 12 K/min & 15 K/min k = 8.75e+10 exp(148,000/RuT)
2400 2100
650 625 600 575 550 525 500
Temperature (K)
1800 1500 1200
2400 2100 1800 1500 1200 900 600 1200
1400
1114 1115 1116
1600
900 600 300
0
250
500
750
1000 1250 Time (s)
1500
1750
2000
Figure 4: Cylindrical reactor temperature histories for heating rates of 12 K/min and 15 K/min
The simulation is initialized with the combustion vessel completely filled with reactive mixture (YRo = 1), and a time dependent temperature ramp is imposed on the wall. The temperature maximum in the computational domain and reactor wall temperature are monitored during the course of the numerical integration to accurately determine the time to ignition. Figure 4 shows their evolution. The dramatic change noted above is clearly seen. The insets show in detail the two behaviors observed. In both cases a weak departure from the temperature ramp imposed at the wall occurs around 500 K. For the 12 K/min case, chemical activity starts after 1125 s of heating, reaction proceeds very slowly to completion over 200 s and returns to Tw (t) over ∼ 400 s. The maximum temperature difference between the reactor’s wall temperature and maximum inside of vessel at the peak of reaction is 33 K and occurs at 1404 s. For the 15 K/min case, a very different evolution is observed. Chemical activity starts around 937 s, reaction takes place violently with full consumption of the combustible mixture over 5 ms, and equalization with Tw (t) over 3 s. The temperature peaks to ∼ 2300 K during ignition (t ∼ 1114.73 s). Next, the two-dimensional fields 8
Topic: Reaction Kinetics
Paper: 114RK-0017
obtained from the simulation are analyzed to get additional insight into the beahavior of the gas inside the reactor. 5.2
Flow analysis - α = 15 K/min t = 504s
t = 804s
t = 1080s
t = 1114s
Figure 5: Temperature (top), velocity (middle) and product mass fraction (bottom) fields - early times (heating) for α = 15 K/min
Figure 5 shows temperature, velocity and product mass fraction fields together with velocity vectors during the initial heating of the reactor for 15 K/min. As late as 804 s, the temperature maximum occurs at the wall and the gas is heated more or less uniformly throughout the vessel (the difference between wall temperature and gas temperature is of 1 K only). The heating of the wall induces a buoyancy driven flow in which the mixture rises along the walls of the reactor and separates at the top of the vessel creating two well-defined vortical structures. Note that from 504 s to 804 s there is a decrease in velocity from 0.006 m/s to 0.003 m/s. This is counterintuitive as one would think that as long as the heating continues the gas inside the reactor should continue to accelerate. However, since the mixture at hand is reactive, once the temperature is high enough to surpass the initial energy threshold for initiation of chemical reaction, heat addition to the gas starts. Competition between the wall’s heating and chemical reaction ensues. The chemical heat release rate is higher than the wall’s heating rate, diffusive and convective losses combined, which results in the flow being slowed down and reversed: the gas then rises along the center of the reactor, separates and descends along the walls (see velocity fields at t = 1080 s). 9
Topic: Reaction Kinetics
Paper: 114RK-0017
As the heating continues, heat addition by chemical reaction increases and the flow is accelerated further to 0.05 m/s at t = 1114 s. Note that the heat release due to chemistry together with buoyancy, keep the hottest gas confined to the top of the reactor (see fields at 1080 and 1114 s). Consistent with the region where the temperature is highest, the product mass fraction fields show that chemical activity is stronger at the top of the combustion vessel. These fields also reveal the extent of mixing that takes place inside the reactor. Both vortices transport hot partially reacted mixture from the top and convect it to the bottom. The concentration of product mass fraction is highest at the top, close to where separation occurs, because it is precisely in this region where the hottest gas sits, and where the flow is significantly slowed down before separating to start its descent. t = 1114.7s
t = 1114.72s
t = 1114.73s
t = 1114.735s
Figure 6: Temperature (top), velocity (middle) and product mass fraction (bottom) fields - shortly before ignition for α = 15 K/min
Closer to ignition, at t = 1114.7 s in Fig. 6, mixing and velocity continue to increase, but as time progresses (specifically over 35 ms), a region of localized chemical activity and associated temperature increase appears close to the region where the flow separates. The temperature of the gas in this zone is 243 K higher than that of the wall (see Fig. 6 at t = 1114.735 s). The mixture has reached a point where the exponential dependence of the reaction rate has taken over. Shortly after, at t = 1114.7362 s, the main ignition event takes place bringing the temperature of the gas to ∼ 1222 K over 1.2 ms. A flame kernel forms at t = 1114.7375 s and starts to propagate downwards to the bottom of the reactor fully consuming the mixture in 2.5 ms as can be seen in Fig. 7.
10
Topic: Reaction Kinetics t = 1114.7362s
Paper: 114RK-0017 t = 1114.7375s
t = 1114.73875s
t = 1114.74s
Figure 7: Temperature (top), velocity (middle) and product mass fraction (bottom) fields - Ignition, flame kernel formation and flame propagation for α = 15 K/min
5.3
Flow analysis - α = 12 K/min
The case in which a heating rate of 12 K/min is imposed on the reactor’s wall is analyzed next. At early times, the evolution is similar to that described above for 15 K/min. As long as the temperature maximum remains at the wall, the buoyancy flow induced consists of two vortices in which the gas rises along the walls and descends along the centerline of the reactor. This can be observed in Fig. 8 for times 804 s and 1008 s. At 1128 s flow reversal occurs, the temperature maximum now lies inside the reactor. The velocities induced in this case are significantly lower than those of the previous case (5 times lower) when comparing t = 1200 s for 12 K/min and t = 1114 s for 15 K/min. Also the mixing and extent of chemical activity is incipient with concentration differences within the reactor of 1 × 10−4 (see mass fraction field at t = 1200 s). Figure 9 shows that at 12 K/min, the mixture undergoes very slow consumption rather than an ignition event as was pointed out above when analyzing the time histories for both cases. At t = 1344 s there is a 20 K difference between the hottest and coldest region inside the reactor. Velocity continues to increase from 0.01 m/s to 0.03 m/s between t = 1200 and 1344 s, and the product mass fraction also increases from 0.0334 to 0.2245 over the same time frame. The peak in velocity, temperature difference between reactor’s wall and gas, and product mass fraction difference inside the reactor is reached at 1404 s, with values of 0.039 m/s, 33 K and 8.9 × 10−3 respectively. The flow structure continues to consist of two vortices with the hottest gas confined at the top of the vessel. Subsequently, as the energy stored in the chemical bonds of the mixture is depleted, the system relaxes, returning to its original configuration with the temperature maximum located at the wall, hence the gas rises along the walls, separates and descends along the centerline. The mixture is now fully consumed (product mass fraction P = 1) as can be observed in Fig. 9 at t = 1740 s.
11
Topic: Reaction Kinetics t = 804s
Paper: 114RK-0017 t = 1008s
t = 1128s
t = 1200s
Figure 8: Temperature (top), velocity (middle) and product mass fraction (bottom) fields - early times (heating) for α = 12 K/min
5.4
Energy equation analysis
In order to get additional understanding of the complex physics taking place inside the reactor, the contributions of each of the terms in the energy equation and temperature are plotted along the vertical centerline (diameter) of the reactor. Distance is measured taking the top of the vessel as a reference. The solid lines are the convective and diffusive heat losses, and the chemical source term given respectively by hConvection = −∇ · (ρuh), hDif f usion = ∇ · (κ/cp ∇h), and hSource = qc ΩR . The dashed line is the sum of the above terms, and the dashed-dotted line is the temperature. The plots are taken at four different instances corresponding to times shown in the two-dimensional fields to allow for a direct comparison. For 15 K/min, Fig. 10 at t = 804 s, the temperature distribution clearly shows the maxima located at the wall. On this ordinate scale (108 W/m3 ) all the terms remain in balance. Note also that the temperature minimum does not occur at the center of the reactor but is located slightly below closer to 3 cm. Shortly before ignition (t = 1114.735 s) the competition between convective and diffusive losses, and chemistry is distinctly illustrated. The increase in temperature is brought about by having a higher heat release rate than the rates at which heat is diffused back to the wall and convected inside the reactor. The plot at t = 1114.7362 s shows that ignition occurs close but away from the wall at 0.5 cm. The structure of a nascent flame can also be observed. Lastly, at t = 1114.7375 s, the reaction wave that emanates from the ignition center propagates back towards the top wall and downwards to the bottom consuming the fresh reactive mixture left in the reactor. 12
Topic: Reaction Kinetics t = 1344s
Paper: 114RK-0017 t = 1404s
t = 1464s
t = 1740s
Figure 9: Temperature (top), velocity (middle) and product mass fraction (bottom) fields - slow consumption for α = 12 K/min
For the case with a lower heating rate (12 K/min - see Fig. 11), the energy contributions along the diameter of the reactor for the duration of the consumption of the reactive mixture have the same order of magnitude. Therefore, the competition between the terms is more clearly seen at each location along the vessel. For instance, at the wall (0 and 5 cm) the balance is kept by diffusion and the heat release rate due to the chemistry. Convective losses at these locations are zero because of the non-slip condition imposed. Further down, diffusion is counteracted by convection and chemistry, then around the center of the reactor convection is balanced by diffusion and chemical heat release. As the bottom of the vessel is reached, the convective term decreases to zero and diffusion counteracts the chemical source term. The distribution of the terms remains essentially unchanged except that at t = 1404 s the peak in heat release due to chemistry is more pronounced at 1 cm. Overall, during the entire consumption process, chemical reaction occurs uniformly within the vessel, which is in stark contrast with the sequence of events seen for 15 K/min where localized chemical activity created an ignition center whereby a flame that emanated from the separation region of the reactor consumes the mixture rapidly. 6
Conclusion
The present study develops a one-step reaction model suited to capture the transition from slow to fast reaction that is appropriate for the wide range of conditions typically encountered in real13
Paper: 114RK-0017
Temperature
0.75
501
0
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0.25
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0
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-8
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1 2 3 4 5 Vertical centerline from top of vessel (cm)
7
498
hTerms x 10 (W/m ) @ t = 1114.7375s
1250
3.5
3
1000 0 750
-8
3
hTerms x 10 (W/m ) @ t = 1114.7362s
-8
0.5
-3.5 -7
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1 2 3 4 5 Vertical centerline from top of vessel (cm)
500
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Temperature (K)
502
hTerms x 10 (W/m ) @ t = 1114.735s
1.5
3
hTerms x 10 (W/m ) @ t = 804s
Cylindrical reactor - α = 15 K/min
hConvection hDiffusion hSource Sum
Temperature (K)
Topic: Reaction Kinetics
500
Figure 10: Evolution of ignition event for α = 15 K/min: contributions of each term in energy equation and temperature along vertical certerline from top of vessel. Top Left: at t = 804 s - early times. Top Right: at t = 1114.735 s - shorly before ignition. Bottom Left: at t = 1114.7362 s - ignition/flame kernel formation. Bottom right: at t = 1114.7375 s - early stages of flame propagation.
istic industrial applications. The kinetic model developed was used to perform a two-dimensional simulation of the heating of a reactive gas inside a cylindrical reactor. Both reaction regimes were captured. A drastic change in the reaction behavior occured when the rate was increased from 12 to 15 K/min, namely, from initially having slow comsumption of the mixture at 12 K/min to an ignition event at 15 K/min. The evolution for the ignition case, α = 15 K/min, proceeds as follows: (1) buoyancy driven flow induced by heating of the reactor’s walls with the gas ascending along the walls and descending along the centerline; (2) once the heat release rate is high enough and significant energy is deposited in the gas, the gas slows down, comes to rest and the flow reverses: the gas then rises along the centerline of the reactor and descends along the walls; (3) chemical activity continues to increase and is confined to the top of the reactor, because the temperature of the gas is high in this region, and convective losses are small since the flow is slowing down as it approaches the top of the vessel before it separates; (4) ignition occurs close to the separated region and a flame propagating downwards consumes the rest of the mixture over a few milliseconds. The slow consumption scenario, α = 12 K/min , evolves in the following fashion: (1) initial heating of reactor’s wall induces a buoyancy driven flow with the gas rising along the wall and descending along the centerline of the vessel; (2) upon increase in chemical activity, the heat release due to the 14
Paper: 114RK-0017
Temperature
501.5 3.5
501 500 499 498.5 0
1 2 3 4 5 Vertical centerline from top of vessel (cm)
7
498
hTerms x 10 (W/m ) @ t = 1464s
-7
3.5
590
0
580
-3.5
570
-4
-4
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600
3
500.5 0
7
620
3.5
0
1 2 3 4 5 Vertical centerline from top of vessel (cm)
7
560
620
3.5 600
3
600
-7
580 -3.5 -7
0
1 2 3 4 5 Vertical centerline from top of vessel (cm)
0
-4
-4
0
560
580 -3.5 -7
0
1 2 3 4 5 Vertical centerline from top of vessel (cm)
Temperature (K)
502
3
hTerms x 10 (W/m ) @ t = 1404s
hTerms x 10 (W/m ) @ t = 1344s
7
3
hTerms x 10 (W/m ) @ t = 1008s
Cylindrical reactor - α = 12 K/min
hConvection hDiffusion hSource Sum
Temperature (K)
Topic: Reaction Kinetics
560
Figure 11: Evolution of slow reaction for α = 12 K/min: contributions of each term in energy equation and temperature along vertical certerline from top of vessel. Top Left: at t = 1008 s - early times. Top Right: at t = 1344 s - initial slow chemical activity . Bottom Left: at t = 1404 s - peak of slow chemical activity. Bottom right: at t = 1464 s - decay in slow chemical activity.
chemistry moves the temperature maximum away from the wall resulting in flow reversal; (3) the chemical heat release rate is rather weak but still marginally stronger than diffusive and convective losses, and the mixture is fully consumed over a significantly longer time scale (hundreds of seconds instead of a few milliseconds). These results emphasize the importance and applicability of developing simplified kinetic models that reproduce the physical behavior of reactive mixtures for use in multi-dimensional simulations of realistic geometries. Acknowledgments J. Melguizo-Gavilanes is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Postdoctoral Fellowship Program
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Topic: Reaction Kinetics
Paper: 114RK-0017
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