Characteristic time correlation for lean blowoff of bluff-body-stabilized flames

COMBUSTION A N D F L A M E 35: 61-80 (1979 )

61

Characteristic Time Correlation for Lean Blowoff of Bluff-Body-Stabilized Flames S. L. PLEE* and A. M. MELLOR The Combustion Laboratory, School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 4 790 7

Characteristic times associated with turbulent mixing, homogeneous chemical kinetics, liquid-droplet evaporation, and fuel injection are quantified for lean blowoff. The model linearly correlates variations in combustor pressure, inlet temperature, air velocity, flame-holder geometry, fuel type, and injector size using data obtained from three different bluff-body stabilizers that simulate the same fundamental combustion processes of both conventional and advanced prevaporizing-premixing gas-turbine combustors. The characteristic time model does not attempt to solve for the entire combustor flow field but rather considers only key regions of the flow that are important to the flame-stabilization process. Lean blowoff is viewed as the competition between a fluid mechanic and chemical time evaluated in the shear-layer region between the hot recirculation zone and the free stream. Heterogeneous effects associated with fuel evaporation and injection represent perturbations on this process that are correlated using a dropletevaporation time and fuel-injection mixing time, respectively.

INTRODUCTION The combustion process in the primary zone of an actual gas-turbine combustor consists of hightemperature chemical reaction, turbulent mixing, heat and mass transfer, and thermal radiation. In addition, three-phase flow is usually present since soot particles, liquid fuel, and gaseous species exist simultaneously. To obtain a direct mathematical solution, coupled elliptic differential equations describing conservation of mass, species, momentum, and energy must be solved numerically. Caretto [1] has recently reviewed finite-difference techniques as applied to gaseous pollutant emission and concluded that their potential is currently limited by insufficient knowledge of the governing mechanisms of turbulence and chemical kinetics, especially the kinetics of practical hydrocarbon fuels such as Jet A or #2 diesel fuel. Because of the difficulties involved with continuum techniques at the present time, several * Present address: General Motors Research Laboratories, Warren, Michigan. Copyright © 1979 by The Combustion Institute Published by Elsevier North Holland, Inc.

investigators [2-5] have chosen to view the combustion process in terms of characteristic times, following DamkiShler [6]. MeUor [5] separates the combustion process into heterogeneous, fluid mechanic, and chemical effects, each of which is characterized by an appropriate time scale and easily computed from combustor inlet conditions. Rather than attempt to solve the entire flow field, the characteristic time model considers only regions of the burner that are important to combustor performance and emissions. The key to the model's success is the identification of the important physical processes occurring in a particular region of the flow field through either detailed probing or examination of relevant literature. This technique has already proven successful in correlating NO:, and CO emissions from both laboratory [7] and practical gas-turbine combustors [8-10]. In the present study, flame stabilization is also interpreted via the characteristic time approach. For laminar premixed gaseous (Bunsen burner type) flames, the model is essentially equivalent to the Peclet number correlation proposed by

0010-2180/79/040061+20501.75

62

S.L. PLEE and A. M. MELLOR

Putnam and Jensen [11 ] ; it also represents a quantitative application of the Zukoski and Marble [2] theory for predicting lean-blowoff limits in turbulent bluff-body-stabilized flames. In all cases, flame stabilization involves the competition between fluid-mechanic and chemical effects. After formulation of the model in the next section, lean-blowoff data obtained recently by Ballal and Lefebvre [12] and Plee and Mellor [13] under operating conditions typical of both conventional and advanced prevaporizing-premixing gas-turbine combustors are correlated using the characteristic time approach. FORMULATION OF FLAME-STABILIZATION MODEL Following Mellor [5], at least four characteristic times are expected to be important in the stabilization of gas-turbine-type flames. These times are associated with liquid-droplet evaporation, turbulent mixing, fuel injection, and homogeneous chemical reaction. Charactedstic Times The first characteristic time, the fuel droplet lifetime (%b), incorporated heterogeneous effects associated with the fuel spray and vaporization rate. The lifetime reu is defined from the "d 2 law" of Godsave [14] as do 2 feb =

(1)

where do is the initial droplet diameter and /3 is the evaporation coefficient. As an approximation to the overall evaporation rate, the initial droplet diameter is taken to be the initial Sauter mean diameter (SMD) using an equation described by Hunter et al. [15]. Following discussions in Tuttle et al. [7, 16, 17], the evaporation coefficient is corrected for forced convection; in addition, an ambient temperature of 1000°K surrounding the fuel and droplet relative velocity of 50 m/sec have been assumed. Fuel properties are evaluated at the 50% distillation point in the manner described by Colket et al. [18].

Fluid-mechanic effects on the combustion process are characterized by the second and third times. The turbulent mixing process occurring in the shear layer region between the fresh incoming air and the recirculation zone is characterized by %1. Tuttle et al. [7, 17] recently quantified this mixing time in terms of a characteristic large-scale eddy lifetime. However, since turbulence parameters are not easily measured in practical systems, %1 is assumed directly proportional to a geometric dimension and inversly proportional to a convective velocity. The other fluid-mechanic time is associated with the turbulent mixing of fuel and oxidizer in the region of the fuel injector (rfi), which is expected to be particularly important in combustors utilizing air-blast and air-assist nozzles [5]. For pressure atomizers, rfi is only of limited importance since very little oxygen is available near the point of injection. However, should fuel penetrate the rich central recirculation zone as a result of high-injection momentum or relatively slow droplet evaporation, rfi is expected to characterize the mixing process in this free-stream region. The final characteristic time (the) represents the homogeneous chemical reaction of fuel and oxidizer. This process occurs after the fuel and oxidizer have been mixed on a molecular level and is a function of local equivalence ratio, inlet temperature, and pressure. Thus The describes an ignition delay time for flame stabilization that is an exponential function of temperature where the apparent activation energy is determined empirically. Note that because of the various approximations used in the definition of these characteristic times, they are only expected to be order-ofmagnitude estimates that hopefully model the important physical processes of spray combustion. The model is thus semiempirical in nature since the relationship between the characteristic times is determined from first principles; however, certain empirical correlating parameters are also required. The advantage of the model over traditional loading parameters is that detailed changes in geometry and fuel type can be accommodated through the inclusion of appropriate characteristic times. For flame stabilization, the model is first

CHARACTERISTIC TIMES IN LEAN BLOWOFF

63

demonstrated using simple laminar Bunsen burnertype flames.

square root of the laminar flame speed, an ignition delay time can be defined as

Laminar Premixed Flames

rue ~ (reaction rate) - 1 ~ SL - 2 .

For laminar flow in a tube, flame stabilization involves the competition between fluid mechanics and chemical kinetics in the boundary layer. For flashback, the limit is represented by the velocitygradient theory of Lewis and von Elbe (see discussion in their book [19] ), SL

gfb

de'

(2)

where g is velocity gradient at wall, SL is laminar flame speed, and d e is quenching distance. Blowoff is represented in a similar manner, except that the velocity gradient at the wall is replaced by the gradient at the boundary between the tube flow and outside ambient conditions (i.e., in the shear layer). Assuming the quenching distance to be proportional to the thermal diffusivity (a) divided by the flame speed [ 11], equation (2) becomes SL 2

g~

ot

(3)

Substituting for the velocity gradient, which can be computed from the friction factor ( 6 4 / R e ) i n laminar flow, and multiplying by d2/a (d = tube diameter), yields Pe~ ~ Pe s L 2.

(4)

(5)

Furthermore, assuming that the velocity gradient at the wall (flashback) or in the shear layer (blowoff) is proportional to a fluid-mechanic time for laminar flow, r~m~g-1.

(6)

Thus (3) involves the competition between a fluidmechanic (T~m) and chemical (rhc) time at the limit. The fuel-ignition delay time is computed from the general overall rate expression for hydrocarbon oxidation assuming a first-order fuel dependence. Since most practical'flows of interest are fuel lean, the oxygen concentration should remain approximately constant so that [HC] e- E / R T I/ _moles _ ~ ) .

--d[HC] at

\ c m 3 sec

(7)

Dividing by the overall density of the system, (7) in terms of mole fraction becomes -~XH C

XHce--E/RT(sec--1),

(8)

dt

which has units of sec- 1 . Defining the ignition delay time as the inverse of the hydrocarbon reaction rate yields e E/R T

Equation (4) is the Peclet number correlation of Putnam and Jensen [11] where the Peclet number is based on the flame speed (PesL) and approach flow velocity (Pea). An example of the Peclet number correlation is shown in Fig. 1 for the flashback data of Khitrin et al. [20] at elevated inlet temperatures; common symbols represent datum points corresponding to inlet temperature variations at a given equivalence ratio. An alternate approach is to view the flamestabilization process in terms of characteristic times. Since the reaction rate is related to the

rh c

(sec)

(9)

~s

since the product of the approach flow equivalence ratio (~) and stoichiometric fuel:air ratio (s) is directly proportional to the hydrocarbon mole fraction. Combining (9) and (6), the characteristic time model for flame stabilization in laminar flow becomes 1 .

g

eE/R T .

.

.

~s

(lO)

64

S.L. PLEE and A. M. MELLOR

5000 slope

2

~

2000

UNSTABLE 1000

Pe~

SO0

0¢

- 0.30 * 0.s0 f-l~ . 0.7s 0 ¢ - 1.33 < > ~ -- 1.60 /~

200

i00

I

100

A

200

I

SO0

1000

PesL Fig. 1. Peclet number correlation for CH4/O 2 flashback data [20] at various ¢ and Tin.

If the velocity gradient is evaluated at the same temperature as the ignition delay time, then 1

Ti,~ -

gT

,

(11)

Tg

where the ratio Tin/T accounts for the acceleration of the air flow as a result of the increased temperature. Finally, the fluid mechanics and

chemistry are again uncoupled by moving the temperature ratio to the right-hand side of the equation. 1

T

eE / R T

g

Tin

~bS

(12) The flashback data of Fig. 1 are now replotted in Fig. 2 using this equation.

CHARACTERISTIC TIMES IN LEAN BLOWOFF

65

S00

200

UNSTABLE 100

o11O/°

s0

o o

rfln x lO6 (sec)

o

O

20

0

10

O

for ¢ < 1.0 ~'~ r - .92

~

STABLE

o ° cr.~o',,~I

0.30

0

~-

A

¢ - 0.S0

[] ¢ - 0.7s 0

¢,,1.33

<> ¢ •1

.2

.5

1

2

5

10

1.60

20

50

100

!

rhc x 106 (sec)

Fig. 2. Characteristic time correlation for CH4/O2 flashback data [20] at various ¢ and Tin.

The characteristic time model of Fig. 2 is relatively accurate for equivalence ratios less than unity (rich flashback data are not expected to correlate based on the derivation of rhc), with a correlation coefficient of 0.92. Therefore, the Peclet number and characteristic time correlation for flame stabilization in laminar flow are essentially identical as the derivation predicts for equivalence ratios of practical interest. Using the experimental data of Khitrin et al. [20], the appropriate ignition delay time was found to have an apparent activation energy of 36,000 cal/mole with the temperature evaluated at the average between the inlet and the adiabatic flame temperature at flashback. This activation energy was selected for the ignition delay time because it gave a reasonably good correlation coefficient and was also in the range of 32-56 kcal/mole reported in the literature for methane combustion [21]. A similar plot is also available [22] for methane-air flashback and blowoff data of Grumer et al. [23].

Because the laminar flame speed in the Peclet number is extremely difficult to calculate for fuels, pressures, and inlet temperatures typical of continuous gas-turbine combustors [24], the authors believe that the characteristic time model provides a greater degree of flexibility for incorporating these operating conditions. Furthermore, the characteristic time model can also include heterogeneous effects associated with fuel droplet evaporation and penetration that may be encountered in modern air-breathing engine applications. Bluff Body Stabilized Flames

Problems of flame flashback in advanced prevaporizing-premixing combustors have been reviewed in a recent paper by Plee and Mellor [25]. The authors noted that the upstream flame propagation reported in the literature for noncatalytic combustors has been a direct result of flow disturbances in the fuel-preparation tube rather than classical

66 flashback as defined in the previous section. Therefore, the primary flame-stabilization problem of interest in conventional and noncatalytic prevaporizing-premixing combustors is blowoff, specifically, the weak extinction limit since these combustors operate overall fuel lean. For bluff body stabilized flames, two different theories [2, 26] exist for blowoff that can be summarized in the framework of characteristic times. Altenkirch and Mellor [24, 27] have compared these two theories in detail and concluded that the DamkShler similarity group (residence time/ignition delay time) in the shear layer controis blowoff in agreement with Zukoski and Marble [2]. They found that the temperature in the recirculation zone is uniform (approximately 90% the adiabatic flame temperature) and that the ratio of the wake temperature to the adiabatic flame temperature remains approximately constant as the air velocity increases toward the blowoff limit [2]. Figure 3 summarizes the physical model for flame stabilization proposed by Altenkirch and Mellor [24, 27]. Using the technique of Gosman et al. [28], they concluded that the time a fluid particle spends in the recirculation zone is too long to be characteristic of blowoff; rather, it is the purpose of the recirculation zone to provide heat and free radicals to the shear layer. Furthermore, because the residence time in the wake region exceeds the shear-layer residence time, the recirculation zone should be extinguished after the shear layer. Since the recirculation zone represents only minor heat release relative to the main stream flame, this final blowout is only of secondary interest [29]. Finally, Altenkirch and Mellor [27], following Spalding [30], suggest that since most flows of practical interest are turbulent, the shear-layer residence time should be replaced by a turbulent mixing time in the shear-layer region between the recirculating burned gases and the free stream. This suggests that the hot turbulent eddies present in the shear-layer region must ignite before they are quenched by the relatively cold free stream; if they cannot ignite, the shear-layer flame blows off. Following Tuttle et al. [7, 17], the turbulent mixing time is assumed to be the characteristic breakdown time of the large-scale eddies. Unfortu-

S.L. PLEE and A. M. MELLOR nately, appropriate geometric length scales cannot be specified a priori but must instead be determined through analysis of the experimental data [17]. For blowoff, the turbulent mixing time is taken to be a ratio of the flameholder width (L), which is also a measure of the size of the recirculation zone, divided by the air velocity at the edge of the stabilizer (Va,T) , L %1' ~

Va,T

(msec)

(13)

(prime denotes temperature dependence in velocity), since this is the region where stabilization occurs. Note that the velocity is evaluated at the temperature in the shear layer (7'), which must be higher than the inlet temperature because it is adjacent to a hot recirculation zone; this accounts for the acceleration of the air flow as a result of the increased temperature. The ignition delay time for a lean reacting eddy is e E/R T

The

- -

(msec)

(14)

from the previous laminar flow analysis. Note that here the parameter s has been omitted since the stoichiometric fuel:air ratio is approximately constant for practical hydrocarbon fuels. Combining (13) and (14), the characteristic time model for lean blowoff becomes L

eE / R T

Va,T

¢

(15)

Now since Va, T is a function of the recirculationzone temperature (approach flow equivalence ratio), we write the ratio T/Tin on the right-hand side of (15) so that the fluid mechanics and chemistry are uncoupled, L

T

eE / R T

v~

Tin

¢

(16)

where L

T

rsl ~ Wa, /'he

Ti n

eT M

~b

CHARACTERISTIC TIMES IN LEAN BLOWOFF

67

Well Stirred Reactor, l~ng Residence Time, r---- Iligh Temperature, Source of Heat and Active Species

/ /

r

Fuel + Air

~ / f

/

/ Main Stream, Combustion Taking Place i f z___ Shear layer Residence Time Range Faiough for Ignition and Heat lip J Shear Layer, Residence Time Characteristic for Z_____ Stabilization, Source of Ignition is Recirculation Zone Fig. 3. Physical model of flame stabilization from Altenkirch and Mellor [24].

For the prevaporizing-premixing combustor shown in Fig. 3, the appropriate length in rsi should scale with the housing diameter (De) minus the baffle diameter (D) since this is a measure of the size of the recirculation zone. The temperature in 7hc' is assumed proportional to the adiabatic flame temperature at the approach flow equivalence ratio just prior to blowoff (an estimate of the thermal driving force between the recirculation zone and the shear layer) following Zukoski and Marble [2]. The model can also be extended to three other simplified flame-holder configurations shown schematically in Fig. 4. Each is characterized by a shear-layer flame that is ignited through transfer of heat and mass from a central recirculation zone (boundary shown with dashed lines). Configurations A and B shown in Fig. 4 are both prevaporizing-premixing combustors since fuel is introduced upstream of a baffle and mixed with air prior to combustion, and configuration C simulates conventional combustors. Variations in refer-

ence velocity, combustor pressure, inlet temperature, flame-holder blockage, and geometry have been examined (see Table 1). In addition, changes in housing diameter [12], fuel type [13], and injector size [13] have also been considered. Appropriate characteristic time-scaling parameters for each of these combustors are listed in Table 2. In each case the turbulent mixing time is the ratio of the flame-holder width divided by the annulus velocity ( F a). Thus the appropriate length scale for the tube-and-disk combustor is assumed to be the baffle diameter (/9) minus the tube diameter (d), which goes to the proper limit (D) as the tube diameter approaches zero. The ignition delay time is usually evaluated at the highest temperature in the system since regions characterized by this temperature are the last to be extinguished and thus control the flame stabilization process. Since the combustor of Fig. 4(a) is essentially identical to the flame holder shown in Fig. 3 except for the location of the bluff body,

68

S.L. PLEE and A. M. MELLOR __ recirculation zone / ~ - ~ t m d a r y fuel + air

"42-J

shear layer flame

Configuration A

Ballal and Lefebvre [12]

recirculation zone

-

~r

L

Configuration B

Tube-and-Disc

Plee and Mellor [13]

•• )

m ---~ air

recirculation zone

._--__Sly~l~

I___JL/ -~-Configuration C

shear layer flame

Disc-in-lk,ct Plee and Me'llor [13] (negligible heterogeneous effects}

Fig. 4. Schematic of three simplified bluff-body-stabilized flames.

the appropriate equivalence ratio and flame temperature are evaluated in the same manner. However, for configuration B, the inner shear layer [shear layer beginning at the edge of the tube, Fig. 4(b)] should be responsible for flame stabilization since high fuel concentrations are present at this point. The equivalence ratio representative of the tube-and-disk combustor is assumed to be Ct, which is based on the air-flow split through the tube rather than the total air flow rate (¢ov). However, the is not based entirely on the equiva-

lence ratio in the fuel preparation tube since this does not take into account the influence of the relativly cold inlet air flowing around the outside of the disk. Obviously there exists a fundamental difference between the configurations studied by Ballal and Lefebvre [12] and the tube-and-disk flame holders. In the latter case, the recirculation zone is not completely surrounded by a hot reacting shear layer; instead, the outer shear layer at the edge of the disk is almost entirely free of fuel and should be relatively unreactive. Therefore, a better

CHARACTERISTIC TIMES IN LEAN BLOWOFF

69

TABLE 1 Summary of Burner Operating Conditions for Three Simplified Bluff Body Stabilized Flames Conditions

Configuration A [Fig. 4(a)]

Configuration B [Fig. 4(b)]

Configuration C [Fig. 4(e)]

10-100 m/see 0.2-0.9 atm 300-575 ° K

12-75 m/see 2.3-8 atm 600-800 ° K (vitiated) 14.6 cm 44-70%

21-86 m/see 2.3-6 atm 500-800 °K (vitiated) 14.6 cm 23-6 1%

Reference velocity Pressure Inlet temperature Pipe diameter Flame-holder blockage ratio Fuel

10 cm, 15 em, 25 cm 4-34% Call 8 (gas)

Call a (liquid) and Jet A

C3H 8 (liquid), Jet A, Jet A blended, JP 4, and DF 2

Injector

Gas

Geometry

Three different simplex pres- Simplex pressure-atomizing sure-atomizing nozzles nozzle

45 ° Conical baffle

TABLE 2 Parameters for Characteristic Time Model [Equation (16)] a Configuration

V

L

Fig. 3 Fig. 4(a) Fig. 4(b) Fig. 4(c)

Va Va Va Va

Dc - D D D- d D

~

T

~bov T~ov ¢~ov T¢~ ~t (Tq~t + 9"~'m)12 ~ov T~=1

a Variables: D e = combustor housing diameter, D = baffle diameter, d = tube diameter, V a = annulus velocity, ~ = equivalence ratio, TO = adiabatic flame temperature at 0, Tin = inlet temperature; subscripts: ov = overall, t = tube. representation o f the temperature o f the inner shear layer is an average between TOt and the inlet temperature, which allows for the quenching effect o f the cold outer recirculation zone [see Fig. 4(b)]. If the disk diameter equals the housing diameter, the appropriate temperature would then to TOt (equal to T % v ) as in the case o f Fig. 3. Configuration C differs substantially from the previous flameholders studied because the fuel is injected at the center of the recirculation zone. Tuttle et al. [7, 17] successfully treated this configuration as a turbulent diffusion flame for modeling NOx formation. Their results show that very high fuel:air ratios exist in the recirculation zone due to fuel vaporization and then decrease

Tube-and-disk

Disk-in-duct

into the main stream. As a result, rapid reaction rates and very high temperatures are present in the shear-layer region located between the fuel-rich recirculation zone and the free stream. Because stoichiometric eddies in this region are the last to be extinguished, the appropriate flame temperature in rhe should be evaluated at q~= 1. However, the equivalence ratio in the is not evaluated at ~b = 1 since all o f the eddies are not at stoichiometric. Instead, the fraction o f stoichiometric eddies controlling flame stabilization should be proportional to the overall equivalence ratio. Thus as ~ov increases, the ignition delay time becomes smaller and the flame becomes more stable. Application o f the characteristic time model to lean blowoff is now demonstrated using experimental data obtained from the three combustors of Fig. 4. Together, variations in combustor pressure, inlet temperature, air velocity, flame-holder size and type, housing dimension, fuel type, and injector size have been examined. The effect o f fuel penetration observed with configuration C [13] is discussed in a later section.

CORRELATION OF GEOMETRY, INLET, AND FUEL VARIATIONS Characteristic times for flame stabilization are first quantified for the lean-blowoff data obtained in

70

S.L. PLEE and A. M. MELLOR

prevaporizing-premixing combustors. Specifically, extinction limits obtained by Ballal and Lefebvre [12] using gaseous propane have eliminated all possible heterogeneous effects. Using these data, appropriate scaling parameters for rsl and rhc can be determined without the influence of incomplete vaporization, mixing, and fuel penetration. Based on the experimental data of Ballal and Lefebvre [12], the ignition delay time for the characteristic time model was found to be

T e21,O00/RT the ' = 10- 4

(msec), Tin

(17)

~b

where the prime indicates that rhc' contains the ratio T/Tin from the annulus velocity. Note that both the pre-exponential (10 - 4 ) and the activation energy (E = 21,000 cal/mole) are determined empirically. An activation energy of 21,000 cal/mole gives the best correlation and compares favorably with values of 24,600 cal/mole and 24,200 cal/mole reported by Engleman et al. [31] and Edelman [32], respectively, for a single-step quasiglobal combustion reaction of hydrocarbon fuels. Following Table 2, both the temperature and equivalence ratio in (17) are evaluated at the approach flow equivalence ratio just prior to blowoff (temperature evaluated using a version of the equilibrium computer program of Gordon and McBride [33] ). Figure 5 illustrates the correlation described in (16) 1 and Table 2 for the lean-blowoff data of Ballal and Lefebvre [12]. The model collapses the experimental data to a straight line representing the lean limit with a y intercept approximately zero as the theory predicts. The correlation also indicates that one approaches the lean-blowoff limit from the stable region by either increasing the annulus velocity or decreasing the flame-holder characteristic dimension ($rsl), or decreasing the approach flow equivalence ratio (trhe'). Analysis of the turbulent mixing time predicts an optimum disk diameter for maximum stability at a given power setting that is solely a function of 1 Failure to include the ratio T/Tin from the velocity term decreases the correlation coefficient in Fig. 5 from 0.98 to 0.93.

the housing diameter: 2 maximum flame stabilization exists (%1 is a maximum) at De/X/~. Thus increasing the characteristic dimension of the flame holder leads to an increase in the range of stable operation; however, as the size of the stabilizer exceeds De/x,~3, a further increase narrows the blowoff limit ($rsl). Note that the turbulent mixing time in (16) is identical to that obtained by Tuttle et al. [7] for CO emission; however, kinetic times are quite different. For configuration B, Plee and Mellor [13] found that decreasing the fuel volatility narrowed the lean-blowoff limit in agreement with results of other investigators [34, 35]. These influences can easily be interpreted via the droplet-evaporation time [36]. According to Lefebvre et al. [36], there must be sufficient fuel vapor to exceed the gas-phase lean flammability limit in the critical region of the flow field before ignition will occur. Injector size, viscosity, and volatility are important since high boiling points and large droplets reduce the local concentration of fuel vapor. Therefore, given the same mixing time as in the purely homogeneous system, the heterogeneous combustion process must both evaporate the fuel and ignite the fuel vapor in the same time span. Thus droplets represent a perturbation on the flame-stabilization process that narrow the limit. To account for heterogeneous effects, (16) is modified to Tsl ~ Th e' 3r aTeb'

(18)

where feb is computed from the "d 2 law" of (1) and evaluated under conditions in the shear layer (prime again denotes ratio T/Tin from velocity term). The parameter a in (18) is necessary because these times are not absolute and is chosen empirically to givethe bestfitto the experimental data [7]. Following Table 2, 3 the characteristic time 2 The optimum diameter is also a function of flameholder shape, but to a lesser extent. 3 In general, tube equivalence ratios are fuel lean; however, under some circumstances, ~ exceeds unity at blowoff. Thus the stabilization curve will have the same shape as the laminar flow blowoff curve in air [19] due to mixing of the O2-rich outer flow with the fuel-rich mixture from the tube. When blowoff occurs at Ot > 1, the stoichiometric flame temperature replaces TOt in Table 2 to account for the mixing of oxygen and subsequent reduction in Zhe.

CHARACTERISTIC TIMES IN LEAN BLOWOFF

71

- 11L-t /

9.6

8.4

STABLE 7.2

6.0

, tv

Zs£

C~sec)

~*

ro o/

/

UNSTABLE

4.8

3.6

TS~,,,

/

CI~'/0 00

2.4

/ 1.2

2.11T~c

-

.46

r - .98 72 data Cy

-

. 38

Configuration A

0 ~~0 ' 0

.8

1.6

2.4

I

I

I

3.2

4.0

4.8

~c C~ec) Fig. 5. Characteristic time correlation for configuration A.

correlation for configuration B is shown in Fig. 6. Note that here inlet-air vitiation effects are included in the' in the manner described in Plee et al. [37]. It can be seen that one approaches the blowoff limit by increasing the velocity ($rs0, increasing the initial droplet size (l'7"eb') and decreasing the tube equivalence ratio (l'rhe'). The model also predicts that droplet effects will become important at high velocities (small rsl) since at this point The' and feb' are the same order of magnitude. This has been verified experimentally [13]. The characteristic time correlation for configu-

ration C is shown in Fig. 7 using the appropriate scaling parameters of Table 2. Note, however, that this graph does not include data obtained from the jet and diesel-fuel experiments using the small disk. For the 6.99-cm-diameter flame holder, heterogeneous effects associated with fuel droplet penetration are important [13]. Specifically, Plee and Mellor [13] found that the smallest disk was the most stable when burning either jet or diesel fuels. However, this contradicts the characteristic time model that states that the optimum flameholder size (De/V~) for this particular housing is

72

S.L. PLEE and A. M. MELLOR

2.00 Lean Blowoff Limit 1.75

~~000 /

STABLE

1.50

(~ A/fl~

/

UNSTABLE

1.25 rs£ {msec)

1.00 r q

Y

= =

.90 .21

161 data

.75

.SO

i

0 A

Jet A Propane

.25 Configuration B I

0

.20

I

I

I

I

I

.40

.60

.80

1.00

1.20

T~e

+ .032 Z~b

(r~ec)

Fig. 6. Characteristic time correlation for configuration B. very near the dimension of the middle-sized disk. In addition, Plee and Mellor [13] observed that diesel fuel #2 (least volatile) was more stable than propane (most volatile). These apparent discrepancies are attributed to heterogeneous effects, specifically, the presence of a free-stream flame zone that enhances flame stabilization [18]. The inclusion of fuel-penetration effects in the model is the subject of the next section. For flames with negligible fuel-penetration effects, lean-blowoff data from all three combus-

tor configurations presented in Fig. 4 are collapsed onto a straight line in Fig. 8. The characteristic time model, although not as straightforward as conventional loading parameters, does include variations in pressure, inlet temperature, reference velocity, flame-holder geometry, housing diameter, fuel type, and injector size with a minimum amount of algebra. Knowledge of the flame-holder configuration inlet conditions and atomization determines rsl and Teb'; from these, ~'hc' can easily be computed that is a function of the inlet condi-

CHARACTERISTIC TIMES IN LEAN BLOWOFF

73

Configuration C

3.2

[7 Disc Dia = 11.43 cm, Jet A

2.8

•

Disc Dia = 8.89 cm, Jet A

ODisc

Dia = 6.99 cm, Propane

2.4

2.0

STABLE

Zs£ (reset) 1.6

•

J

Blowoff Limit

UNSTABLE

1.2

(3Z~ O0

.8

r = .90 cry = .Z2

67 data

.4 v

s£ = 1.47 ( r h c + . O l l r ; b ) + .08

0

I

I

I

I

I

I

.3

.6

.9

1.2

1.5

1.8

Z'hc + .OllTeb (msec)

Fig. 7. Characteristic time correlation for configuration C (negligible fuel-penetration effects).

tions and the blowoff equivalence ratio. The blowoff velocity is also calculated in a similar fashion if the geometry, pressure, inlet temperature, atomization, and equivalence ratio are given. In addition, the correlation can predict simple flame-holder modifications necessary to optimize flame stabilization for a given operating condition.

CORRELATION

OF FUEL-PENETRATION

EFFECTS According to the equation for reb in (1), an increase in fuel viscosity or a decrease in volatility leads to an increase in the droplet-evaporation time. As a result, the correlation predicts that lean-

74

S.L. PLEE and A. M. MELLOR

9.6

0 Lean Blowoff

8.4

Configuration 7.2

O

A

Fig. 4A

[2 B

Fig. 4~

C

Fig. *C

0

6.0

0

rs£

STABLE

C~ec)

4.8

0

0 UNSTABLE

3.6

0 0

2.4

1.2

s~ : 1.sz ('r'hc ", .OlZreb) ", .04 r = .93 300 data c~ = .36 Y

O

0

.8

1.6

2.4

3.7

4.0

4.8

r ' + .Ollreb (msec) hc

Fig. 8. Complete characteristic time correlation for inlet, geometry, and fuel variations on three simplified bluff-body-stabilized flames (negligible fuel-penetration effects).

blowoff limits for heavy fuels such as DF 2 should be much narrower than for lighter fuels. Unfortunately, the characteristic time model developed in the previous section cannot be applied to flame stabilization data obtained from the 6.99-cmdiameter disk (configuration C) burning aviation or diesel fuels (Fig. 9). These data show opposite trends that appear to contradict results of other investigators [34, 35] who observed an increase in the blowoff equivalence ratio with heavier fuels.

Thus fuel property variations on the limit observed with the small disk cannot be interpreted solely as a function of feb ; instead, fuel-penetration effects are expected to be important. As droplet-evaporation times increase, the flame structure is believed to change in the manner described in Fig. 10. Liquid fuel penetrates the existing shear layer and creates a free-stream flame where the fluid mechanics in this region are characterized by rfi, a fluid-mechanic mixing time

CHARACTERISTIC TIMES IN LEAN BLOWOFF

75

3.2 Fuels

Eq. (18) Fig. 8

[~ Jet A

2.8

<>Jet A B1. ~7 JP4 A DF2

2.4

2.0

rsi

STABLE

C q.6 1.Z

mD<>w .8

W .4

ot o

UNSTABLE

Configuration C

.4

.8

1.2

"r'hc+ .OllZeb

1.6

2.0

2.4

(msec)

Fig. 9. Failure of model to correlate fuel-penetration effects.

associated with the fuel-injection process. Since more time is available for ignition (~'sl + ~'fi) when fuel penetration effects are important, the overall equivalence ratio at blowoff is much lower resulting in wider flame-stabilization limits. These results are not totally unexpected since Tuttle et al. [16] also observed slight fuel-penetration effects with liquid propane while performing detailed probing measurements inside the disk-induct combustor. Many other investigators have also noted that fuel-spray characteristics influence combustor performance [38-40]. In particular,

Isaac and Cookson [39] used the fuel-penetration concept to correlate flame-stabilization data obtained by injecting a gaseous jet perpendicular to the air flow. Assuming that fuel penetration is important to the flame stabilization process for configuration C, the mixing time in (13) is modified as follows D rsl,f p' ~

Va,T

+ arfi

(19)

to include this effect. The fluid-mechanic mixing time ~'fi, associated with the fuel-injection process

76

S.L. PLEE and A. M. MELLOR

f

Duct Wall I

- Air

=- rs~

"¢"

V.,

F

s"rac

shear layer flame

F

Negligible Fuel Penetration Effects

Air

"C

r,.

g

f

~

..iii!i:-.:.-ii~..._ Liqui .:!~ii#::" - - Spray ::ii...... i

free stream

shear layer /-- flame

.

flame

..... .......-.-.,.-..

ue \~

t~.

m

~=mmn

~

m

Fuel Penetration Effects Important Fig. 10. Burner schematic illustrating the effect of fuel penetration on flame structure (configuration C).

(Ltp)

is related to a fuel-penetration length scale divided by an appropriate velocity in the free stream (Va). (Note that Va in l"fi is not evaluated at an elevated temperature since this is primarily a free-stream effect). The appropriate length scale in rfi is assumed proportional to the product of the droplet-evaporation time in (1) and the initial fuel velocity (Vf, expression given by Crowe [41]).

T/Tin

Multiplying through by the ratio as before, the characteristic time model with fuel-

penetration effects now becomes

D + aVfreb(T/Tin) v~ T(I 0 --

Tin

4e21"°: °/RT + 0.011Teb ) -

(20)

or Tsl + arfi' ~ rhc' + 0.01 lreb' (msec)

(21)

77

CHARACTERISTIC TIMES IN LEAN BLOWOFF TABLE

3

Comparison of Fuel-Penetration Effects with Configuration C Fuel

¢~bo,vlt

1"eb(msec)

Vf (m/see)

7ebV"f (cm)

D (cm)

(rfi'/~'sl)

a. Geometry variations (Tin = 700°K, P = 2.36 arm, rha = 1.5 kg/sec) Jet A Jet Aa Jet Aa

0.069 0.088 0.093

4.38 3.90 3.93

Jet A Jet A blended

0.052 0.049

5.21 6.82

C3H8a

0.128

~0 b

JP 4 DF 2

0.060 0.043

2.89 9.31

51.7 60.1 62.0

22.6 23.4 24.4

6.99 8.89 11.43

10.6 8.64 7.01

b. Fuel variations (Tin = 500° K, P = 2.36 atm, n~a = 1.5 kg/sec) 41.1 42.4

21.4 28.9

6.99 6.99

14.1 19.0

-

~0

6.99

~0

50.6 36.8

14.6 34.3

6.99 6.99

9.61 22.6

a Fuel-penetration effects have been assumed negligible. b Propane should flash vaporize on injection under these operating conditions.

(prime denotes T/Tin dependence, as before). The inclusion of Lfp(Ffreb) accounts for the lengthening of the shear-layer region as a result of fuel penetration and evaporation. Table 3 summarizes these effects for selected burner operating conditions; here, both geometry [Table 3(a)] and fuel [Table 3(b)] variations are examined. It can be seen that the ratio of fuelpenetration length to disk diameter increases with decreasing flameholder size and fuel volatility. Therefore, since data obtained from the two large stabilizers and the small disk burning propane were correlated assuming negligible heterogeneous effects (Fig. 7), it appears from Table 3 that fuelpenetration effects will be important when the ratio is greater than approximately ten. Note that for JP 4 only a slight influence is noticeable in Fig. 9; however, for DF 2, the length-scale ratio has more than doubled. All of the flame stabilization data of Table 1 are correlated in Fig. 11 using the equation rsl + 0.12rfi' = 2.12(The' + 0.01 1Zeb') + 0.095,

(22)

where characteristic times are evaluated in the same manner as discussed previously for each par-

ticular configuration. Note that Teb' in (22) is negligible for mixtures in which the fuel is completely vaporized prior to combustion. Also, the mixing time associated with the fuel-injection process is only expected to be important for the disk-in-duct combustor when fuel-penetration effects are present. For configuration C, increased viscosity and decreased volatility (~'Teb') have two effects, the latter o f which dominates the flamestabilization process. First, feb' increases on the right-hand side of (22), causing the lean-blowoff limit to become narrower; however, increasing feb' also increases the effective length scale in rfi, which provides more time for ignition to occur. The net effect is a decrease in the blowoff equivalence ratio for the heavier fuels using configuration C.

CONCLUSIONS The characteristic time model originally developed for correlating gaseous emissions has now been extended to include flame stabilization for three different bluff-body flame holders. Neglecting heterogeneous effects, the model states that the ratio of a fluid mechanic to a chemical time (rsl/rhe) evaluated in the shear layer is constant

78

S.L. PLEE and A. M. MELLOR

STABLE 9.6 O

Lean Blowoff Limit

8.4

Configuration 7.2

O

A Fig. 4A

[] B Fig. 4B ,% C Fig. 4C

6.0

"C

UNSTABLE

4.8 O

.~w3.6 b, ¢,,I

0

+

0

O

0

~m2.4

T,z + . n~ h

1.2

~)

o

0

- z. 12 cT~c ÷ • ° n : ; b ) + . 095

r=

.93

~y = .43

397 data 0

.8

1.6

2.4

3.2

4.0

4.8

T' + .011 Z' Cmsec) hc eb

Fig. 11. Complete characteristic time correlation for blowoff limit.

for the limit in agreement with Zukoski and Marble [2]. Since this particular ratio is proportional to a Damk/Shler similarity group, this is not an entirely new concpt. However, this model does include variations in combustor pressure, inlet temperature, reference velocity, and geometry, as well as heterogeneous effects associated with changes in fuel type and injector size in a quantitative manner for the first time. In addition, lean-btowoff data obtained from simplified prevaporizing-premixing

and conventional gas-turbine-type flames are correlated simultaneously. Heterogeneous effects are included through characteristic times associated with the fuelinjection (r~i) and evaporation (feb) process. However, in practical gas-turbine systems, strong fuel penetration effects will probably be minimized by the introduction of high-momentum quench air through discrete air-addition holes. As as result, r~i in (22) will be zero, and the charac-

CHARACTERISTIC TIMES IN LEAN BLOWOFF teristic time model reduces to ~'sl = 2.12(rh¢' + 0.01 l r e b ' ) + 0.095.

(23)

79 8. 9. 10. 11.

MeUor,A. M.,AIAA J. Energy 1:244-249(1977). Mellor,A. M.,AIAA J. Energy 1:257-262(1977). Hammond,D. C.,AIAA J. Energy 1:250-256(1977). Putnam, A. A., and Jensen, R. A., Third Symposium

on Combustion and Flame and Explosion PhenomThis equation predicts that increased droplet evaporation times will narrow the lean limit in agreement with Lefebvre et al. [36]. Plee [22] and Plee et al. [42] have recently used the characteristic time model to correlate a limited amount of lean-blowoff data obtained from actual gasturbine combustors.

12. 13. 14.

15.

The authors wouM like to thank Professors A. H. Lefebvre, C. Ferguson, and C. Smith o f Purdue University and Dr. M. B. Colket o f United Technologies Research Center for their valuable suggestions during the course o f this investigation. Thanks are also extended to Mr. D. Ramsey, who aided in the data analysis. This research was sponsored in part by the Air Force Office o f Scientific Research (11. T. Wolfson, technical monitor), the Army Tank-Automotive Research and Development Command (P. Machala and A. Jaeger, technical monitors), and the Army Research Office (J. Murray, technical monitor). The U.S. government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon.

REFERENCES 1. Caretto, L. S., in Progressin Energy and Combustion Science (N. A. Chigier, Ed.), Pergamon, Oxford, 1976, Vol. 1, pp, 47-71. 2. Zukoski, E. E., and Marble, F. E., Proceedingsof Gas

16

17.

18.

19.

20.

21. 22. 23. 24.

Dynamics Symposium on Aerothermochemistry, Northwestern Univ., 1956, pp. 205-210. 3. Swithenbank, J., Poll, I., Vincent, M. W., and Wright,

25.

D. D., Fourteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh,

26.

1973, pp. 627-638. 4. Vranos, A., Combust. Flame 22:253-258(1974). 5. Mellor, A. M., in Progressin Energy and Combustion Science, (N. A. Chigier, Ed.), Pergamon, Oxford, 1976, Vol. 1, pp. 111-133. 6. Damk~ihler,G., Z. Elektrochem. 46:601(1940). 7. Turtle, J. H., Colket, M. B., Bilger, R. W., and MeUor, A. M., Sixteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1977, pp. 209-219.

27.

28.

29.

ena, Williams and Wilkins, Baltimore, 1949, pp. 89-98. BaUal,D. R., and Lefebvre, A. H., ASME paper No. 78-GT-144, 1978. Plee, S. L,, and Mellor, A. M., AIAA paper No. 78-1038, 1978. Godsave, G. A. E.,Fourth Symposium (International) on Combustion, Williams and Wilkins, Baltimore, 1953, pp. 818-830. Hunter, S. C., Johnson, K. M., Mongia, H. C., and Wood, M. P., USAAMRDL technical report No. 74-3A, 1974. Turtle, J. H., Shisler, R. A., Bilger, R. W., and MeUor, A. M., report No. PURDU-CL-75-04,School of Mechanical Engineering, Purdue Univ., Lafayette, Ind., 1975. Turtle, J. H., Colket, M. B., and Mellor, A. M., report No. PURDU-CL-76-05, School of Mechanical Engineering, Purdue Univ., Lafayette, Ind., 1976. Colket, M. B., Stefucza, J. M., Peters, J. E., and Mellot, A. M., USATACOMtechnical report No. 12163, Part II, 1977. Lewis, B., and von Elbe, G., Combustion, Flames and Explosions of Gases, 2nd ed., Academic Press, New York, 1961. Khitrin, L. N., Moin, P. B., Smirnov, D. B. and Shevchuk, V. U., Tenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1965, pp. 1285-1291. Cooke, D. G., and Williams, A., Combust. Flame 24:245-256 (1975). Plee, S. L. Ph.D. thesis, School of Mechanical Engineering, Purdue Univ., Lafayette, Ind., 1978. Grumer, J., Harris, M. E., and Rowe, V. R., Bureau of Mines report No. RI-5225, 1956. Altenkirch, R. A., and Mellor, A. M., report No. PURDU-CL-73-05, School of Mechanical Engineering, Purdue Univ., Lafayette, Ind., 1973. Plee, S. L., and Mellor, A. M., Combust. Flame 32:193-203(1978). Longwell, J. P., Frost, E. E., and Weiss, M. A., Ind. Eng. Chem. 45:1629-1633(1953). Altenkirch, R. A., and Mellor, A. M., Fifteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1975, pp. 11811189. Gosman, A. D., Pun, W. M., Runchal, A. K., Spalding, D. B., and Wolfstein, M., Heat and Mass Transfer in Recirculating Flows, Academic Press, New York, 1969. Blazowski, W. S., and Henderson, R. E., AFAPL-TR77-41, 1977.

80 30. Spalding, D. B., Thirteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1971, pp. 649-657. 31. Engleman, V. S., Bartok, W., Longwell, J. P., and Edelman, R. B., Fourteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1973, pp. 755-765. 32. Edelman, R. B., personal communication, SAI, 1977. 33. Gordon, S., and McBride, B. J., NASA SP-273, 1971. 34. Moses, C. A., paper presented at Joint Spring Meeting of Western and Central States Sections, The Combustion Institute, Pittsburgh, 1975. 35. Marchionna, N., and Opdyke, G., Jr., USATACOM technical report No. 12191, 1976. 36. Lefebvre, A. H., Mellor, A. M., and Peters, J. E., paper prepared for Project SQUID Workshop on Alternate Hydrocarbon Fuels for Engines: Combustion and Chemical Kinetics, September 7-9, 1977, Columbia, Maryland.

S.L. PLEE and A. M. MELLOR 37. Plee, S. L., Schmidt, D. A., and Mellor, A. M., report No. PURDU-CL-77-07, School of Mechanical Engineering, Purdue Univ., Lafayette, Ind., 1977. 38. Lefebvre,A. H., CoA Note Aero report No. 163, The College of Aeronautics, Department of Propulsion, Cranfield Institute of Technology, 1966. 39. Isaac, J. J., and Cookson, R. A., Combust. Flame 30:187-192(1977). 40. Komiyama, K., Flagan, R. C., and Heywood, J. B., Sixteenth Symposium [International) on Combustion, The Combustion Institute, Pittsburgh, 1977, pp. 549-560. 41. Crowe, C. T., WSS/CI paper No. 74-23, The Combustion Institute, Pittsburgh, 1974. 42. Plee, S. L., Clark, J. A., Peters, J. E., Schmidt, D. A., Stefucza, J. M., Ferguson, C. R., and Mellor, A. M., USATARADCOM technical report No. 12349, 1978.

Received 18 July 1978; revised 23 November 1978