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Abstract

Structures of laminar non-premixed ethanol/air spray flames in the axisymmetric counterflow configuration are studied under fuel-rich conditions by means of numerical simulations. The monodisperse ethanol spray is carried by air and directed against an air stream. Both streams enter at 300 K, and the system is at atmospheric pressure. Up to three different structures of these flames for identical boundary and initial conditions are identified, and regime diagrams are presented that show their conditions of existence in terms of the gas strain rate on the spray side of the configuration, $a_{- \infty}$ , starting from 55/s at an initial spray velocity of 0.44 m/s. The equivalence ratio on the spray side, $E_{- \infty}$ , is varied between 1.1 and 1.6, and initial droplet radii, $R_{0}$ , from 10 to 50 $μ$ m are considered. The most stable spray flame structure is characterized by two chemical reaction zones. For some conditions, single chemical reaction zones on either side of the counterflow configuration are found. Conditions under which these different flame structures exist are analyzed. Previous studies identified only two different structures for non-identical boundary conditions, and in this study, three different structures are presented for the first time. Moreover, the transition mechanisms of one structure to another are analyzed. The competition between the energy-consuming spray evaporation and the exothermic chemical reaction rates as well as the location of the spray determines the existence of the different flame structures. This transition of the different flame structures may explain spray flame characteristics such as flame pulsation or flame instabilities.

Keywords

Triple structures fuel-rich spray flames counterflow configuration transition mechanisms

Introduction

Spray flames are relevant in many practical combustion systems such as industrial furnaces, liquid propulsion systems, household burners, and internal combustion engines.¹ In turbulent spray flames, the consideration of detailed chemical reactions is inevitable to study pollutant formation. Laminar flame structures are the basis of flamelet models,² developed for the simulation of turbulent gas flames in a computationally efficient way.^3–5 The advantage of using structures of laminar flames in the counterflow configuration is the possibility of transferring the two-dimensional gas equations into a one-dimensional formulation through a similarity transformation,² which allows the use of detailed chemical reaction mechanisms. These laminar flame structures are incorporated into turbulent flame simulations⁶ in which the turbulent nature of the flow field is accounted for through the use of probability density functions. In non-premixed combustion, the independent variables typically are the mixture fraction and its scalar dissipation rate.^2,6 The flamelet model is appropriate for high Damköhler numbers.

Flamelet models for turbulent combustion are beneficial if detailed chemical reactions are to be considered to predict pollutant formation. If turbulent spray flames are modeled, most often, laminar gas flamelets are incorporated.^7,6 The groundbreaking work of Continillo and Sirignano,⁸ who showed that the similarity formulation mentioned above for gas flames is also valid for dilute spray flames in the counterflow configuration, enabled their use in spray flamelet simulations.⁹ The advantage of this formulation is the consideration of the energy consumption due to the evaporation of the spray directly in the laminar spray flamelets. The formulation of Continillo and Sirignano⁸ was extended to account for variable gas properties and detailed chemical reactions by Gutheil and Sirignano.¹⁰ This new formulation allows for the consideration of pollutant formation in turbulent spray flames through the use of the spray flamelet model.^9,7,11

Flamelet models for spray combustion including spray flamelet libraries are more complex than gas flamelets since they do not only depend on the mixture fraction and its scalar dissipation rate (or the gas strain rate) but also on the initial gas and spray velocities, the equivalence ratio, and the initial droplet size of the spray.¹⁰ The effect of varying these parameters was studied for different fuels.^8,10,9,12,13 Moreover, partially premixed systems were investigated.¹⁴ All these investigations concerned stoichiometric conditions.

Continillo and Sirignano⁸ postulated that the governing equations of laminar counterflow spray flames might have multiple numerical solutions. Gutheil¹⁵ found two different solutions for the same boundary and initial conditions for stoichiometric methanol/air spray flames. One of these spray flame structures consists of two chemical reaction zones one of which resides on the spray side of the counterflow configuration and the second one on the gas side. The second spray flame structure has a single chemical reaction zone on the spray side, that is, that on the gas side does not exist. In that study, it was demonstrated that the gas-sided chemical reaction zone has a similar structure as a pure gas flamelet. Note that the chemical reaction zones are identified in terms of gas temperature, that is, the chemical reactions typically occur in the hot flame zone even though some reactions may also occur in regions that are off the peak gas temperature.

Xie et al.¹⁶ exhibited the coexistence of collocated, distributed, and cool flames when they numerically studied canonical counterflow spray flames at relatively low strain rates using different low-temperature chemical reaction mechanisms in a one-dimensional configuration. Carpio et al.¹⁷ studied two-dimensional dodecane spray flames with carrier nitrogen against an air stream using a one-step chemical reaction, and they identified two solutions where one is a diffusion flame and the second is flameless. Vié et al.¹⁸ identified the bifurcation of a monodisperse $n$ -dodecane spray and revealed multi-modal spray flame structures in a three-dimensional configuration, which directly affect flamelet-based tabulation methods. Sirignano¹⁹ used a three-dimensional formulation without the use of the boundary layer approximation for partially premixed gas flames, and identified even three different solutions under high strain rates.

Ying et al.²⁰ studied fuel-rich ethanol/air spray flames in the counterflow configuration. They also found two different spray flame structures for a monodisperse spray with a fixed initial droplet radius of 50 $μ$ m. One of these structures also shows two chemical reaction zones on the spray and the gas side of the counterflow configuration. However, the newly found spray flame structure with a single chemical reaction zone is qualitatively different from that found in the earlier study¹⁵: the flame structure shows physically separated vaporization and chemical reaction zones.

The existence of multiple (spray) flame structures may lead to flame instabilities as reported by Karaminejad et al.²¹ in an experimental study of flame spray pyrolysis for nanoparticle formation where flame pulsation and consequently flame stability strongly depend on spray characteristics. Kiran and Mishra²² performed an experimental investigation of flame stability of a simple liquefied petroleum gas (LPG) jet diffusion flame, and they showed that flame stability affects the pollutant emission characteristics in these flames. For real applications, the coexistence of multiple flame structures is related to the phenomenon of flame pulsation, which may occur in situations in which abrupt transitions between the different flame structures occur.

The phenomena of droplet reversal and oscillation were found in experiments by Puri and Libby.²³ They studied the trajectories of $n$ -heptane droplets in two different configurations: one referred to non-evaporating droplets in two identical isothermal nitrogen streams and the second one concerned a non-premixed methane/air flame in the counterflow configuration. In the flame, the droplets reversed and oscillated, crossing the stagnation point and the chemical reaction zone twice. The experimental results served to study models of drag, and it was concluded that lift forces may affect the droplet trajectories as well. Chen et al.²⁴ used the experiments of Puri and Libby²³ to validate their numerical simulations in two-dimensional configurations, and these results confirmed the reversal of the droplets, which was called “push-back” rather than droplet reversal. It seems that there is evidence about the phenomena of droplet reversal and oscillations, however, multiple structures of these flames have not yet been identified in experiments.

In the present study, the work of Ying et al.,²⁰ who identified two different spray flame structures for the same initial and boundary conditions, is extended to cover a wide range of fuel-rich ethanol/air spray flames for monodisperse sprays with different initial droplet radii, equivalence ratios, and strain rates. The structures for these conditions sometimes show unique solutions and for some conditions, two or three different spray flame structures for the same initial and boundary conditions are identified. Conditions under which these structures exist or coexist and the underlying physical processes for the transition of a structure into another one are analyzed and discussed. Also, the influence of these findings on the spray flamelet formulation of turbulent spray flames is outlined.

Mathematical formulation

The steady two-dimensional governing gas equations for a viscous laminar flow at a low Mach number with source terms to account for the dilute spray are non-dimensionalized and transformed into one-dimensional equations using a similarity transformation.^10,8 The transport properties of the ethanol/air spray are variable,²⁵ and a detailed chemical reaction mechanism is used.²⁶ The monodisperse ethanol spray is carried by air and directed against an air stream at atmospheric pressure, both streams are at 300 K. Depending on the initial spray conditions, the initially monodisperse spray may reverse and oscillate around the gas stagnation point, leading to a locally polydisperse spray.¹⁰ The resulting equations and the numerical solution procedure are reported earlier for stoichiometric spray flames in the same setting.^8,10

The general conservation equations of mass, momentum, mass fractions of chemical species, and energy under consideration of spray source terms accounting for the interaction of the gas and the spray are written as¹⁰

\frac{\partial ρ}{\partial t} + \frac{\partial (ρ v_{g, i})}{\partial x_{i}} = S_{v}

(1)

ρ \frac{\partial v_{g, j}}{\partial t} + ρ v_{g, i} \frac{\partial v_{g, j}}{\partial x_{i}} = - \frac{\partial p}{\partial x_{j}} - \frac{\partial τ_{i j}}{\partial x_{i}} - v_{g, j} S_{v} + S_{m, j}

(2)

ρ \frac{\partial Y_{k}}{\partial t} + ρ v_{g, i} \frac{\partial Y_{k}}{\partial x_{i}} + \frac{\partial}{\partial x_{i}} (ρ V_{k, i} Y_{k}) = M_{k} {\dot{ω}}_{k} + (δ_{F k} - Y_{k}) S_{v}, k = 1, \dots, N

(3)

\begin{aligned} ρ C_{p} \frac{\partial T}{\partial t} + ρ v_{g, i} C_{p} \frac{\partial T}{\partial x_{i}} & = - \sum_{k = 1}^{N} h_{k} M_{k} {\dot{ω}}_{k} + \frac{\partial p}{\partial t} + v_{g, i} \frac{\partial p}{\partial x_{i}} \\ + \frac{\partial}{\partial x_{i}} (λ \frac{\partial T}{\partial x_{i}}) - ρ \frac{\partial T}{\partial x_{i}} \sum_{k = 1}^{N} C_{p, k} Y_{k} V_{k, i} \\ - τ_{i, j} \frac{\partial v_{g, i}}{\partial x_{j}} - S_{v} \int_{T_{0}}^{T} C_{p, F} d T + S_{e} \end{aligned}

(4)where

v_{g, i}

is the gas velocity in

i

direction,

ρ

is the gas density,

Y_{k}

denotes the mass fraction of species

k

C_{p}

C_{p, k}

, and

C_{p, F}

are the specific heat capacities at constant pressure for the mixture, the species

k

, and the liquid fuel F, respectively.

T

is the gas temperature,

p

is the static pressure,

h_{k}

is the specific enthalpy of species

k

λ

stands for the heat conductivity, and

τ_{i j}

denotes the viscous stress.

S_{v}

S_{m}

, and

S_{e}

are the mass, momentum, and energy source terms accounting for the interaction between the gas and liquid phases,

V_{k, i}

is the diffusion velocity

V_{k}

of species

k

i

direction in the mixture.

δ

denotes the Kronecker symbol, the subscript F represents the fuel, and

{\dot{ω}}_{k}

and

M_{k}

describe the molecular chemical reaction rate and the molecular weight of species

k

k = 1, \dots, N

, respectively.

The diffusion velocity $V_{k}$ of species $k$ is given by Kee et al.²⁵

V_{k} = - \frac{ρ D_{k}}{\bar{M}} \frac{\partial (Y_{k} \bar{M})}{\partial x_{i}} - \frac{D_{k}^{T}}{T} \frac{\partial T}{\partial x_{i}}

(5)

D_{k}

and

D_{k}^{T}

denote the diffusion coefficient of species

k

into the mixture and the thermal diffusion coefficient, respectively.

\bar{M}

is the mean molecular mass of the mixture. Thermal diffusion is considered for the light species H and H

_{2}

The source terms for mass $S_{v}$ , momentum $S_{m}$ , and energy $S_{e}$ are written as¹⁰

S_{v} = \sum_{k = 1}^{K} n_{k} {\dot{m}}_{k}

(6)

S_{m} = \sum_{k = 1}^{K} [- n_{k} m_{k} \frac{d v_{d, k}}{d t} + n_{k} {\dot{m}}_{k} v_{d, k}]

(7)

S_{e} = \sum_{k = 1}^{K} [- n_{k} [{\dot{q}}_{k} + {\dot{m}}_{k} L_{v} (T_{l, k})] + n_{k} {\dot{m}}_{k} \int_{T_{0}}^{T_{s, k}} C_{p, F} d T]

(8)In the above equations

{\dot{m}}_{k} = d m_{k} / d t

is the mass evaporation rate with

m_{k} = (4 / 3) ρ_{l} π R_{0, k}^{3}

, and

v_{d, k}

is the droplet velocity.

L_{V}

is the temperature-dependent latent heat of vaporization. In equation (8),

T_{l, k}

refers to the liquid temperature of component

k

. The integration ranges from

T_{0}

T_{s, k}

, which denote the initial liquid temperature and the surface temperature of droplet size group

k

, respectively. The energy transferred to the liquid of component

k

is denoted by

{\dot{q}}_{k}

The droplet number density, $n_{k}$ of droplet size group $k$ yields

\frac{\partial n_{k}}{\partial t} + \frac{\partial (n_{k} v_{d, i, k})}{\partial x_{i}} = S_{n, k}

(9)

S_{n, k}

denotes the source term accounting for changes in the droplet number density due to droplet reversal or oscillation.¹⁰ The convective droplet evaporation model of Abramzon and Sirignano²⁷ is used where the diffusion-limit model accounts for droplet heating. Note that possible negative mass transfer from the liquid to the gas phase and variable liquid properties allow for condensation and droplet expansion, respectively.

The physical gas properties are calculated following the work of Kee et al.²⁵ In particular, the multicomponent diffusion model is used. Variable liquid properties are provided by Gutheil.¹²

The governing equations are non-dimensionalized with specific reference values,¹⁰ and a similarity transformation for the two-dimensional gas phase equations is used so that the gas equations become one-dimensional.^8,10

The resulting equations are solved numerically, and in the remainder of the paper, the gas strain rate, $a_{- \infty}$ and the equivalence ratio $E_{- \infty}$ both of which are evaluated on the fuel side of the configuration, will be used to characterize the spray flame structures. Moreover, the spray flame structures depend on the initial droplet size $R_{0}$ of the monodisperse spray. The initial spray velocity $v_{d, - \infty}$ is identical to the initial gas velocity $v_{g, - \infty}$ and equals 0.44 m/s at $a_{- \infty} = 55$ /s; it increases with gas strain rate.^8,10

Results and discussion

Numerical simulations are performed for a monodisperse ethanol spray with carrier gas air directed against an air stream. The system is at atmospheric pressure with initial gas and spray temperatures of 300 K and an equivalence ratio $E_{- \infty}$ on the spray side of the configuration of 1.1 through 1.6. Structures of fuel-rich spray flames with initial droplet radii of 10 to 50 $μ$ m and an initial spray velocity $v_{g, - \infty} = v_{d, - \infty} = 0.44$ m/s at a spray-sided gas strain rate of 55/s are investigated. Gas strain rates up to the values beyond which no solutions are achieved are studied. First, regime diagrams are presented to show the different conditions under which the single or multiple spray flame structures exist, followed by a detailed analysis and discussion of the flame structures affected by initial droplet radii, equivalence ratios, and gas strain rates to assess the transition and breakdown mechanisms of the individual spray flame structures.

Regime diagrams

Diagrams for the existence of the multiple spray flame structures for the same initial and boundary conditions for initial droplet radii, $R_{0}$ of 10, 30, and 50 $μ$ m are shown in Figure 1(a) through 1(c). The symbols in Figure 1 indicate the existence of spray flames in dependence of equivalence ratios $E_{- \infty}$ from 1.1 through 1.6, that is, fuel-rich conditions on the spray side of the configuration, versus the gas strain rate $a_{- \infty}$ on the spray side of the configuration for the different initial droplet sizes of the monodisperse sprays. Both the equivalence ratio $E_{- \infty}$ and the gas strain rate $a_{- \infty}$ on the spray side of the configuration are varied. Triangles left (blue symbols) and triangles right (red symbols) display the existence of a spray flame structure with a single chemical reaction zone on the spray or the gas side of the counterflow configuration, respectively, whereas circles (black symbols) denote structures with two chemical reaction zones that locate on either side of the configuration. These individual structures will be discussed further below. No symbols show that no spray flame exists under the respective condition.

A comparison of the structures displayed in Figure 1 shows that the flame structures with two reaction zones are more stable than those with single chemical reaction zones since they persist to high strain rates. The larger the initial droplet radius, the smaller the range of strain rates in which numerical solutions are obtained. If multiple structures exist, the single chemical reaction zone on the gas side of the configuration is preferred for the larger initial droplet sizes, whereas the spray structures with a spray-sided single chemical reaction exist mainly for the spray flames with the smallest initial droplet radius of 10 $μ$ m. Triple structures exist only for small initial droplet radii at moderate strain rates. A detailed analysis of these different structures and their existence is discussed next in terms of the initial droplet size, the equivalence ratio, and the gas strain rate.

Figure 1.

Regime diagrams of spray flame structures with single spray- or gas-sided reaction zones or two reaction zones for different initial droplet radii $R_{0}$ , depending on the gas strain rate and the equivalence ratio on the spray side of the configuration. (a) $R_{0} = 10 μ m$ ; (b) $R_{0} = 30 μ m$ ; and (c) $R_{0} = 50 μ m$ .

Characteristics of the different spray flame structures

First, a condition is selected for which all three different spray flame structures exist so that their characteristics may be analyzed and discussed.

Considering Figure 1, all three different spray flame structures exist for an initial droplet radius $R_{0} = 10 μ m$ , an equivalence ratio $E_{- \infty}$ of 1.2, and a gas strain rate on the spray side of the counterflow configuration, $a_{- \infty}$ of 400/s, see Figure 1(a). These structures are displayed in Figure 2.

Figure 2.

Different flame structures for $R_{0} = 10 μ m$ , $E_{- \infty}$ = 1.2, $T_{l, 0} = T_{g, 0}$ = 300 K, $a_{- \infty}$ = 400/s: normalized droplet radius $R / R_{0}$ , gas temperature $T_{g}$ , mass fraction of fuel vapor $Y_{C_{2} H_{5} OH}$ , mass evaporation rate $S_{v}$ , and gas and droplet velocities, $v_{g}$ and $v_{d}$ . (a) Two chemical reaction zones; (b) gas-sided reaction zone; and (c) spray-sided reaction zone.

The monodisperse spray flame structures with a relatively small initial droplet radius of 10 $μ$ m are characterized by a vaporization zone that resides completely on the spray side of the configuration since the droplets do not reach the stagnation plane which locates at the axial position of $y = 0$ mm. This is also reflected in the profile of the source term in the continuity equation (1) which extends over the range of about $- 2$ mm to $- 0.2$ mm in which the droplets evaporate. The profile of the gas temperature shows a dip at the location just prior to the stagnation plane where a peak in the profile of $S_{v}$ occurs and all droplets eventually evaporate causing strong energy consumption related to spray evaporation. The profile of the fuel vapor correlates with that of $S_{v}$ , and it reflects the fuel consumption in the hot regions of the flame which can also be seen in the molar chemical reaction rate ${\dot{ω}}_{C_{2} H_{5} OH}$ , which exhibits (negative) peaks at about $- 1.8$ mm and at $- 0.2$ mm where most of the fuel vapor has been produced through vaporization.

The gas and droplet velocities $v_{g,_{- \infty}}$ and $v_{d,_{- \infty}}$ , which are equal at the spray boundary reflect the strong influence of the drag force on the droplets. As the droplets evaporate and become smaller, the droplet and gas velocities merge.

Two more different spray flame structures are found with single chemical reaction zones. In Figure 2(b), the reaction zone resides on the gas side of the configuration whereas in Figure 2(c), it locates on the spray side. The principal spray flame structure seen in Figure 2(b) was identified in an earlier study by Ying et al.²⁰ for $a_{- \infty} = 55$ /s for which the third structure displayed in Figure 2(c) does not exist at this low gas strain rate and an initial droplet radius of 50 $μ$ m at $E_{- \infty} = 1.5$ , see Figure 1(a).

A comparison of the spray flame structures displayed in Figure 2(a) and (c) shows that the spray-sided structure in Figure 2(a) is almost the same as that in Figure 2(c) whereas the structure in Figure 2(b) is qualitatively different. Gutheil¹⁵ identified the two different spray flame structures seen in Figure 2(a) and (c) for methanol/air sprays that show two chemical reaction zones and one on the spray side of the configuration, and it was argued that the spray-sided chemical reaction zones are similar or even identical to the gas-sided reaction zone being extinguished in the situation with a single chemical reaction zone on the spray side, which is also valid here. However, the spray flame structure with separated chemical reaction zones and vaporization zones displayed in Figure 2(b), is new and qualitatively different since it shows separated vaporization and chemical reaction zones as discussed by Ying et al.²⁰ However, this is the first study to show that three different spray flame structures may exist for the same initial and boundary conditions for fuel-rich spray flames in the counterflow configuration.

The existence of the different spray flame structures may be useful to explain spray flame pulsation that may occur in spray flames.²⁸ The spray flame structure displayed in Figure 2(b) is very interesting in the context of micro-explosion of droplets which may occur in multicomponent droplets that reside in a relatively hot environment outside the main combustion zone. The scenario shown here would provide the perfect conditions for puffing and micro-explosions²⁹ to happen.

Small droplets evaporate entirely on the spray side of the counterflow configuration as can be seen in Figure 2. As the initial droplet radius is increased, the flame structures change considerably.

As the initial droplet radius is increased from 10 to 30 $μ$ m, the droplets cross the stagnation plane and reverse as seen in Figure 3. The conditions in Figures 2 and 3 are the same except for the initial droplet size and the gas strain rate at the spray boundary which equals 400/s and 200/s, respectively.

Figure 3.

Different flame structures for $R_{0} = 30 μ m$ , $E_{- \infty}$ = 1.2, $T_{l, 0} = T_{g, 0}$ = 300 K, $a_{- \infty}$ = 200/s: Normalized droplet radius $R / R_{0}$ , gas temperature $T_{g}$ , mass fraction of fuel vapor $Y_{C_{2} H_{5} OH}$ , mass evaporation rate $S_{v}$ , and gas and droplet velocities, $v_{g}$ and $v_{d}$ . (a) Two chemical reaction zones; (b) gas-sided reaction zone; and (c) spray-sided reaction zone.

The regime diagrams in Figure 1(a) and (b) for initial droplet radii of 10 and 30 $μ$ m, respectively, reveal that the three different spray flame structures exist at lower strain rates and persist to the lowest strain rates under investigation. However, these spray flames do not exist at very high strain rates. As the gas strain rate is increased, the spray flames with the larger initial droplet size favor the single chemical reaction zone on the gas side of the configuration whereas, for those with the smaller initial droplet radius, the spray-sided single chemical reaction zone is preferred. This is a consequence of the different behavior of the larger droplets and their evaporation characteristics.

Figure 3 displays the three corresponding structures to those in Figure 2 for the larger initial droplet radius and the reduced gas strain rate at the spray boundary. The most important difference is the deep penetration of the larger droplets into the configuration: the droplets cross the stagnation plane and reverse due to the opposed airflow, and the initially monodisperse spray becomes locally polydisperse. The drag force of the larger droplets reflects in the profiles of the droplet velocities. At the positions of droplet reversal, there is enhanced evaporation due to the prolonged residence time of the droplets at that position as can be seen in the profile of $S_{v}$ .

The principal characteristics of the three different spray flame structures are maintained for the larger initial droplet radii. The spray flames become broader and the evaporation zone is enlarged. The location of droplet reversal resides inside the chemical reaction zone if there are two chemical reaction zones, see Figure 3(a), whereas, in the situation of single chemical reaction zones, they do not locate inside but closer to the spray or the gas side of the configuration for the gas- and spray-sided chemical reaction zones, respectively, as displayed in Figures 3(b) and (c). It is also found that the droplets in the structure with one chemical reaction zone on the spray side of the configuration cross the chemical reaction zone due to their larger momentum. At higher initial droplet radii, not only droplet reversal but also oscillation occurs and under that condition, triple flame structures are not found.

Scenarios for the breakdown of one or the other spray flame structures will be discussed in the next subsection.

Transition mechanisms of spray flame structures

The triple flame structures are lost for initial droplet radii of 50 $μ$ m and for smaller and larger equivalence ratios for the smaller initial droplet sizes as displayed in Figure 1. When the gas strain rate of 200/s is increased or decreased as compared to the conditions in Figure 3, one of the triple flame structures is lost and only two different spray flame structures exist. At an initial droplet radius of 30 $μ$ m, the flame structure with the gas-sided reaction zone persists in both cases. For the initial droplet radius of 10 $μ$ m, a reduction of $a_{- \infty}$ causes the flame structure with the gas-sided reaction zone to be maintained, and at higher strain rates, the structure with the spray-sided reaction zone is left over. In all situations, structures with two chemical reaction zones exist. The reasons for this will be discussed next.

If the gas strain rate is increased from 200/s to 400/s at $R_{0} = 30 μ$ m, the flame structure with one reaction zone on the spray side of the configuration breaks down as the peak gas temperature drops to below about 1700 K; this temperature was identified below which the flames extinguish.²⁰ Figure 4(a) and (b) shows the remaining spray flame structures at $R_{0} = 30 μ$ m and $E_{- \infty}$ 1.2 and $a_{- \infty} = 400$ /s. The spray flame structure displayed in Figure 3(c) is lost since it transfers into the spray flame structure with two chemical reaction zones, the strong vaporization zone on the spray side locates in a rather hot environment and the second chemical reaction zone develops and the structure shown in Figure 4(a) is recaptured at the higher strain rate.

Figure 4.

Flame structures at $E_{- \infty}$ = 1.2, $T_{l, 0} = T_{g, 0}$ = 300 K, $a_{- \infty}$ = 400/s. (a) Two chemical reaction zones, $R_{0} = 30 μ m$ ; (b) gas-sided reaction zone, $R_{0} = 30 μ m$ ; and (c) two chemical reaction zones, $R_{0} = 50 μ m$ .

At the larger initial droplet radius of 50 $μ$ m, only two different spray flame structures are found, see Figure 1(c), and an increase in gas strain rate from 55/s to 100/s at $E_{- \infty} = 1.2$ leads to a unique spray flame structure with two chemical reaction zones, see Figure 4(c). The disappearance of the single spray flame structure is associated with the build-up of a shoulder in the gas temperature profile on the spray side of the configuration that develops into a separate chemical reaction zone,²⁰ leading to the structures with two chemical reaction zones that dominate the regional diagram for $R_{0} = 50 μ$ m as displayed in Figure 1(c).

Thus, droplet reversal and oscillation caused by the increased momentum of larger droplets lead to the transition to the reduced number of multiple flame structures and eventually to a unique structure. The droplet penetration into the chemical reaction zone and toward the gas side of the configuration caused by the droplet reversal and possible oscillation around the stagnation plane enhance the combustion of the spray flame which again feeds the flame with fuel vapor. Thus, the strong bond between the vaporization of the spray, which on the one hand requires energy from the combustion zone and on the other hand feeds the reaction zone with fuel vapor, and the chemical reactions, which consume the fuel vapor and deliver the energy for the vaporization, characterizes the spray flames and the transition between the different spray flame structures. Moreover, the droplet motion and location of the droplets within the counterflow configuration determine the location of the vaporization and combustion zones, and the droplet motion differs for monodisperse sprays with different initial droplet radii.

The transition mechanism of three different spray flame structures to two at $R_{0}$ = 10 $μ$ m is presented in the context of Figure 5. Figure 5(a) and (b) displays the two structures at a strain rate of 900/s where the structure with the gas-sided chemical reaction zone no longer exists. As the strain rate is increased, the chemical reaction time decreases and chemical extinction occurs. A comparison of the maximum gas temperatures for the different spray flame structures at 400/s for otherwise fixed conditions displayed in Figure 2 shows the lowest temperature for the gas-sided chemical reaction zone which is the least stable out of the three. A further increase of the gas strain rate to 1300/s leads to the breakdown of the second spray flame structure and the one with two chemical reaction zones is left, see Figure 5(c) for the same reason as discussed in the context of Figure 4 above. A further increase in strain rate leads to a stronger narrowing of the spray flame, a considerable reduction in the peak gas temperature and chemical reactions cannot be sustained so that no spray flames exist beyond 1300/s.

Figure 5.

Flame structures at $E_{- \infty}$ = 1.2, $T_{l, 0} = T_{g, 0}$ = 300 K, $R_{0} = 10 μ m$ . (a) Two chemical reaction zones, $a_{- \infty}$ = 900/s; (b) spray-sided chemical reaction zone, $a_{- \infty}$ = 900/s; and (c) two chemical reaction zones, $a_{- \infty}$ = 1300/s.

Figure 6 shows the loss of the spray-sided flame structure at $E_{- \infty}$ = 1.5, $R_{0} = 10 μ$ m for a reduction of the gas strain rates from 900/s to 600/s and to 250/s. The equivalence ratio is increased from 1.2 to 1.5 in Figure 6 for a better visibility of the physical processes. With a decrease in strain rate, the chemical reaction zone widens up whereas the width of the vaporization zone reduces and completely falls into the chemical reaction zone for the lowest strain rate as is evident from the sequence of subfigures. Note that in Figure 6, the range of the axial position is varied for a better resolution of the results. At the lowest strain rate, the gas temperature exhibits a shoulder on the gas side of the counterflow configuration, building up the chemical reaction zone on that side of the configuration, and the structure transfers into the spray flame structure with two chemical reaction zones. For lower strain rates than 250/s, the spray flame structure displayed in Figure 6 no longer exists since it transfers into a structure exhibiting two chemical reaction zones. This is possible since the energy-consuming vaporization zone moves toward the spray side of the configuration, leaving over enough energy to build up the second chemical reaction zone at the gas side of the configuration.

Figure 6.

Spray-sided flame structures at $T_{l, 0} = T_{g, 0} = 300$ K, $E_{- \infty}$ = 1.5, $R_{0} = 10 μ m$ for different gas strain rates. (a) $a_{- \infty} = 900$ /s; (b) $a_{- \infty}$ = 600/s; and (c) $a_{- \infty}$ = 250/s.

In summary, the spray flame structures with two reaction zones exist for all cases under investigation and they are most stable compared to the spray flame structures with a single chemical reaction zone. Triple flame structures are only found at low initial droplet radii and moderate gas strain rates; the spray flames with the largest initial droplet radius of 50 $μ$ m do not show the structure with the spray-sided reaction zone since the high momentum of the droplets causes the droplets to penetrate the chemical reaction zone with possible reversal and oscillations and mainly evaporate on the gas side of the configuration.

Transition mechanisms for the loss or development of spray flame structures are dominated by the sensible interaction of energy-consuming vaporization, the location of the droplets within the counterflow configuration, and the exothermic chemical reactions.

The impact of the multiple spray flame structures on spray flamelet modeling is not as severe as anticipated since the spray-sided chemical reaction zones of the structure with two chemical reaction zones are more or less identical to those of the single spray-sided flame structure and the gas-sided part of the chemical reaction zone is essentially a gas flame as found in earlier studies.¹⁵ However, the finding of the single flame with the gas-sided chemical reaction zone²⁰ is new and requires special treatment. The novelty of the present investigation is the identification of triple spray flame structures for certain conditions found in fuel-rich spray flames.

Conclusions

Multiple structures of laminar fuel-rich ethanol/air spray flames in the counterflow configuration are studied numerically. A monodisperse ethanol spray with carrier gas air is directed against an air stream at atmospheric pressure for initial droplet radii of 10, 30, and 50 $μ$ m, equivalence ratios of 1.1 to 1.6, and gas strain rates on the spray side of the configuration ranging from 55/s at an initial spray velocity of 0.44 m/s.

Up to three different flame structures are found for the same initial and boundary conditions. Two different spray flame structures with a single chemical reaction zone either on the spray or the gas side of the counterflow configuration are identified, and the third spray flame structure consists of two chemical reaction zones on either side of the stagnation plane. Generally, spray flame structures with two chemical reaction zones are the most stable. For small initial droplet sizes, the flame with the spray-sided reaction zone is more stable than that on the gas side whereas for larger initial droplet radii, this is reversed due to the higher momentum of the larger droplets that penetrate deeper into the counterflow configuration.

The conditions under which the different flame structures coexist are analyzed and discussed. The existence of two different spray flame structures for fixed boundary and initial conditions was discussed in earlier studies,^15,20 but the identification of three different structures is new. For stoichiometric flames, the spray flame structure with a single chemical reaction zone on the spray side was identified before,¹⁵ and it was argued that that spray flame structure shows the same characteristics as the spray-sided structure of the flame with two chemical reaction zones, which is confirmed for fuel-rich combustion: the gas-sided chemical reaction zone is extinguished in the situation with the single reaction zone compared to the structure with two chemical reaction zones. In this study, the existence of the third structure is demonstrated, and for the first time, the coexistence of up to three different spray flame structures is found for the same boundary and initial conditions, depending on the gas strain rate, the equivalence ratio, and the initial droplet radius.

Regime diagrams are presented and discussed to display the range of existence of the multiple structures. Mechanisms for the transition of the structures and the breakdown or appearance of the different types of spray flame structures are analyzed.

The spray flame structures with two chemical reaction zones and that with a spray-sided chemical reaction zone can easily be incorporated into spray flamelet models of turbulent spray combustion.

The structure with the gas-sided chemical reaction zone shows quite distinct vaporization and chemical reaction zones, which provides suitable conditions for the explanation of flame pulsation or conditions under which micro-explosion may occur. Abrupt transition between the different spray flame structures found in the present study may be responsible for flame instabilities identified in experimental studies.^22,21

Footnotes

Acknowledgements

The financial support of the German Research Foundation (DFG) through SPP 1980, grant GU 255/13-2, and HGS MathComp is gratefully acknowledged.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the German Research Foundation (DFG) through SPP 1980, grant GU 255/13-2, and HGS MathComp.

ORCID iD

Eva Gutheil

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Multiple structures and transition mechanisms of laminar fuel-rich ethanol/air counterflowing spray flames

Abstract

Keywords

Introduction

Mathematical formulation

Results and discussion

Regime diagrams

Characteristics of the different spray flame structures

Transition mechanisms of spray flame structures

Conclusions

Footnotes

Acknowledgements

Declaration of Conflicting Interests

Funding

ORCID iD

References