An investigation of the diffusion flame around a porous cylinder under conditions of natural convection

COMB USTION AND FLAME 36: 233-244 (1979)

233

An Investigation of the Diffusion Flame Around a Porous Cylinder Under Conditions of Natural Convection TAKEO SAITOH

Department of Mechanical Engineering II, Tohoku University, Sendal, 980, Japan

A numerical analysis was performed for the natural convectivediffusion flame around a porous horizontal cylinder from which fuel gas was ejected into a quiescent oxidant atmosphere. Two dimensionless parameters were utilized to describe the characteristics of the present flame. As a consequence, the phenomena of thermal quenching, which arises when the gas ejection velocity diminishes, and flame extinction, which is caused by chemical limitation, were predicted. Moreover, various features like flame structure, compressibility effect, heat-transfer characteristics, and geometric difference were clarified in detail.

INTRODUCTION Relatively few investigations have been devoted to diffusion flames under conditions of natural convection despite their broad applications to droplet combustion and many other important problems such as spontaneous ignition [1]. Elucidation of these problems is important for estimation of buoyancy effects on forced convective diffusion flames. Tsuji and Yamaoka [2, 3] have proposed a counterflow diffusion flame stabilized in the forward stagnation region of a porous horizontal cylinder as a fundamental flame model for flame structural analysis. They pointed out that a flame model of this type can be used to examine basic flame structural characteristics such as the distribution of temperature, concentrations, mass-flux fraction, net reaction rate of various species, and heat-release rate. Thereafter, Milne et al. [4] utilized the same model for the study of inhibition effectiveness of gaseous and powdered agents. El Wakil [5] has also performed an experiment employing Tsuji and Yamaoka's model and has revealed flame structures by means of inteferometry and gaschromatography. In the present study the same configuration as that of Tsuji and Yamaoka is employed, except Copyright © 1979 by The Combustion Institute Published by Elsevier North Holland, Inc.

that there exists no forced flow field. Under a natural convection condition the key parameters involved are the diameter of the cylinder and the gas-ejection velocity, that is, the Damk6hler number and the ejection parameter when expressed in dimensionless forms. Two limiting phenomena appear when these two parameters decrease beneath the certain critical values. Thermal quenching takes place in the case of low ejection velocities. Under such conditions the heat loss from the flame to the cylinder surface increases, followed by a decrease in the flame temperature, thereby eventually causing flame extinction. The other limiting phenomenon is due to chemical limitation caused by decreasing the Damktihler number, namely, the diameter of the cylinder. On first thought it would seem that the latter phenomenon is not inherent in the flames established in a natural-convection flow, but consideration similar to that for the extinction of a counterflow diffusion flame [6, 7] leads to the same extinction mechanism based on the chemical limitation associated with a characteristic convective flow. It can be shown that the effective Damktihler number using the reference velocity gradient of natural convection flow can be defined as a characteristic flow time.

0010-2180/79/090233+12501.75

234 The present paper reports the numerical results of the natural convective diffusion flames that were analyzed by studying only the vicinity of the stagnation point and adopting a laminar boundary-layer approximation. Compressibility effects were also accounted for by introducing Howarth-type transformation [8]. Some features of the present study were compared with the work of Aldred et al. [9] in which flame structures were examined for the combustion of a porous ceramic sphere wetted with nheptane fuel, although the flame-supporting mechanism at the cylinder surface is not the same as in the present case. The distribution of temperature, concentrations, and velocities of the present flame were numerically clarified in detail. Moreover, variations of the heat-transfer coefficient and maximum flame temperature versus the ejection parameter are revealed using the cylinder diameter as a parameter. Since the present flame model has a quite simple configuration, it may serve as a more fundamental means for flame structural analysis than the counterflow diffusion flames because no forced-flow equipment is needed.

T. SAITOH

/f

Fig. 1. Schematic model and coordinate system for diffusion flame under natural-convection conditions. step Arrhenius second-order equation; and (6) viscous dissipation and the Dufour-Soret effect are minor. Under these assumptions the governing equations are as follows [6] :

a(puro~)

-

GOVERNING EQUATIONS

The flame model and coordinate system for a freeconvection diffusion flame are schematically shown in Fig. 1. Fuel gas issues from the surface of a porous horizontal cylinder of outer radius R and reacts with oncoming oxidant flow that is induced by the buoyancy force produced by the flame itself. The flame is then located at a certain position that is determined by temperature, concentrations, and flow fields. A curvilinear coordinate system is employed to express the basic equations. The use of this system and the boundary-layer assumption, which is introduced later, confines the cylinder to a relatively large size. For the analysis, the following simplifying assumptions and restrictions are made: (1) flow is steady laminar and the boundary-layer approximation is valid; (2) the Schmidt and Prandtl numbers are constant and equivalent; (3) the products pp, pX, and p2D i take constant values throughout the field, (4) radiation effect from the flame is negligible; (5) reaction obeys the single-

AIR

Y

-

ax

a(pVrok) + -

-

ay

- o,

(J)

pu -ax - + pV-~y = -Pgx - -ax - + - - ay

oc,,u a--~+oc,,v-fig

ay \

t / u~y

/ ,

(2)

ay /

n

+ ]~ H?~,

(3)

i=l

pu

aY~+

ax

+ pv ~ y = - ~ y

i~y]

+ (P2 i -- Pl i)mt ~,

(4)

Y~+ e = pR°T ÷ 2~

i=1 mt

(5)

The parameter k in Eq. 1 indicates the type of flow: k = 0 for planar flow and k = 1 for axisymmetric flow. The reaction term in Eqs. 3 and 4 is

DIFFUSION FLAME WITH NATURAL CONVECTION expressed by the one-step Arrhenius second-order equation written as

235

where L [ ] designates an operator defined

d 2~

d~ +Scf-dT?2 d1?

L[~,n] = ~ v = K p)2 y °m+ Y fo+exp ( m

f

E/ RO_+

.

(6)

The above overall equation serves as a good approximation in many situations [10]. The first and second terms in Eq. 2 yields, from the condition at infinity, ~P --Pgx

-- --

ax

----

(Pe - - P)gx.

T=

mCp

pvro k = -- -

ay'

m

Oti-

PePe

~X

f ( ~ ) G r l / 4 x k + 1,

(9)

--Pe p dy.

(10)

The conventional Howarth type transformation is adopted to include compressibility effects [8]. The governing equations are rewritten by virtue of the above variables as

f"'+ff"+

k+i

2

re

T +,

h~

vli'mi) ,

(14)

Yz"= aiYi +,

m =

v~

so=-,

Oi

kx/T-~lD

Grll4

T"

£plimi i

a,-

=

Ep2imi ' i

gO3 12e 2 '

(8)

- ,

Restricting the attention only to the stagnation region of a porous cylinder, it is possible to obtain similar solutions by use of the following variables:

~7= ~

l/V2 i __

Gaf

c~

D1 =

=

H~

T+=

a~

puro k = - -

--

Primes denote the differentiation with repect to r/. In the above equations, use has been made of the following nondimensional variables:

(7)

This term contributes as a driving force for a natural convective flow. Next we introduce the customary stream function ff satisfying Eq 1.

a~

+nScDIY°Yfexp

(f')2

=0,

(11)

L[T. 1] =0.

(12)

L [ Y , - 1] =0.

(13)

KPeVl ° v1 fD 2

ve Grl/2(k + 1)m

KPeP1 °v I f m(k + 1 ) ~

.

(15)

In the above equations D1 denotes the DamkShler number involving the square root of the ratio of the cylinder diameter to acceleration of gravity. The term ~ has the dimension [1/sec] and cotresponds to the velocity gradient in the case of the counterflow diffusion flame. The Damk551er number plays an important role in the understanding of the extinction mechanism of this flame. Boundary conditions for Eqs 11-13 are as follows:

D Gr-114 T e x/-ff71Ve Tw

7? 0; f = fw

f ' = 0,

T=T~

(17)

Y/(O)

Yfw = af - - Sc f,~ '

(18)

Yo'(0) Yow

--

(16)

-

-

Se fw

r/=oo; i f = O ,

(19)

,

T=Te,

Yf=O,

Yo=Yoe.

(20)

236

T. SAITOH

Equations 18 and 19 are derived by considering the mass balances at the cylinder surface. It is assumed that only fuel gas penetrates through the cylinder wall. The value of f a t the surface, fw, appears as an important parameter to prescribe the ejection velocity (Henceforth it is referred to as the e/ection parameter) [2].

-- (1 "FSchfi(n~'Ti+l(n+l)-~ /

.

= h 2 Sc D 1 Yf(n)Yo (n)

NUMERICAL PROCEDURE The present basic equations resemble those used for ignition and extinction analyses in forced flows [6]. However, the boundary conditions contain the derivatives of concentrations, so that many difficulties arise in the numerical computations of the simultaneous ordinary differential equations. For this reason an implicit-type finitedifference method was employed. The momentum equation (Eq. 11) was solved as a second-order equation by putting p = f ' . A nonlinear term (f')2 was then expanded in a Taylor series and a quasilinearization technique was applied to construct the finite-difference equation. The pure implicit finite-difference formulations for Eqs 11-13 are as follows:

X exp

T(n).

-

Before starting numerical computations, the initial distributions of f,, T, Yi should be given in appropriate functional forms chosen in a trial and error fashion. It is emphasized here that these initial profiles are important to obtain meaningful convergent solutions.

NUMERICAL RESULTS AND DISCUSSION Numerical calculations were performed for a planar stagnation flow (k = 0) using propane as fuel. The result for a sphere is shown only for comparison. Physical properties and various constants for propane are given below. The values of activation energy, specific heat at constant pressure, and frequency factor were obtained from the paper of Ablow and Wise [11]. Other properties such as heat of combustion were obtained from Fristrom and Westenberg [12]. Cp = 0.409 kcal/kg °C,

XPi+lln+l)=k+ 1

he = 11120 kcal/kg,

Te (21)

-

I 1 Sc ht',( n)q "-'~ I v ' . . . ( n + l ) 2

j-r.~-,

(22)

Te + = Tw + = 300°K,

Yoe+ = 0.232 (for air), 0% = 1.275,

E = 20.9 kcal/mole,

af = 4.6363,

m = 452.2.

A typical structure of the natural-convective diffusion flame is depicted in Fig. 2 when the ejection parameter, - f w , takes a relatively small value of 0.12. This condition corresponds to vw = 0.2 cm/sec and D = 3 cm if use was made of the constants listed before. In this case the flame is located very near to the cylinder surface because of a low ejection velocity. Hence the maximum flame temperature decreases considerably and the reaction zone broadens. In particular, it is noted that oxy-

DIFFUSION FLAME WITH NATURAL CONVECTION I

237

I

FUEL C3Hs ambient.-, air

41-

04

D,: 2.71 xlOr -f,¢0.12

.~000

0.3 2 T ÷

f

f

vo

T ~-

I

2

i Vf

J 02 I

I000 0

I

I

-I 0

0

I

2

3

4

-~O.I

olo

Fig. 2. Typical flame structure for a relatively small ejection parameter ( - ] w = 0.12, D 1 = 2.71 × 107).

gen concentration has a nonzero value at the cylinder surface. The heat transfer from the flame to the cylinder is intensified due to the large temperature gradient at the cylinder wall, and eventually, a phenomenon of flame quenching will be brought about, as is shown later. The dimensionless radial velocity, expressed as vD p f(rl) -

Gr -1/4

(24)

Pe Pe

is also shown in Fig. 2. Another flame structure for a large ejection parameter ( - f w = 1, hence vw = 1.7 cm/sec) is plotted in Fig. 3, which clearly shows a thinflame condition. The position of maximum temperature is located far from the cylinder and the temperature profde has a sharp summit. The location of the flame in this case is calculated to be y* = 5.6 mm, with y* as the distance measured from the cylinder surface. Again, previous constants have been used in calculations. Shown in Fig. 4 are the distributions of radial and tangential velocities, u and v, respectively. The same ejection parameter and Damk6hler number are employed as in Fig. 3 for the sake of compar-

ison. It is found that the radial velocity varies in a quite complicated manner because of the existence of the maximum temperature region. Also shown in the figure is the Aldred et at. [9] experimental data measured for a ceramic porous cylinder wetted with n-heptane liquid film. Although the situations vary considerably between the two investigations, it is seen that the qualitative agreement is moderately good. Figure 5 plots the variations of temperature profiles with various ejection parameters in the case of D = 1 cm. The position of maximum temperature approaches the cylinder surface as the ejection parameter decreases. At the same time the maximum temperature descends gradually. Figure 6 indicates one of the important two characteristics of the present diffusion flame. There exists a critical ejection velocity at which the maximum flame temperature goes down steeply to a frozen-state value. It is thought that this aspect implies the so-called thermal quenching phenomenon that was pointed out by Tsuji and Yamaoka [2, 3] in their experiment on the counterflow diffusion flame around a porous cylinder. As explained in their paper, this may be caused by the heat loss from the flame to the cylinder.

238

T. S A I T O H

i 4-

I

i

FUEL C3H8

4

-f,--I D, = 2.71 x 107

1

3 -20( ) 0 ,

f

I

Tt

- 3

f 2- T

1 0.3

Yf

I -

2

Yo 0.2

I0( )0 I

° f 0

I I

0

~ 3

2

4

0

0.1

0

17

Fig. 3. Typical flame structure for relatively large ejection parameter ( - f w = 1.0, D 1 = 2.71 × 107).

0.2

0 '

I

0.4

Y

0.6

I

'

I

'

0.8 i

I

cm I

FUEL C3H a -fw= I

15

O,: 2.71 x I07 •

:

2

•

•

POSITIONOF Tm

%./i ~ • AIdredetol?s )(~ \ experimentaldata I ~~for n-Heptone

sJ 24

I0

(3 °

a~

16~

." I

~

1

2

0

3 -2

Fig. 4. Distribution of velocities u and v. Also shown are experimental data of Aldred et al. [9] (indicated by dots).

DIFFUSION FLAME WITH NATURAL CONVECTION

239

I

2000

I

FUEL C3H 8 D, = 1.56 x 107 -fw 1.5

1500

I

T -~

0.6 0.4

I 0 O0

500 30C

0

I

I

2

5

'9

4

Fig. 5. Variation of temperature distributions with ejection parameter for D = 1 cm.

K i 2000:

I

I

FUEL

'

C3He

IOOC

300 O

0

I

I

05

I -f.:

I

1.5

~-0-~Gr~uw

Fig. 6. Maximum flame temperature vs ejection parameter showing thermal quenching condition.

240

T. SAITOH K

2000

J

FUEL C3 He

-f~=l

I000

300

/j _ _

o

i

Io'

i

,o

o''crit i

io'

J

Io'

i

Io

i

Io

Io '°

Oi

Fig. 7. Maximum flame temperature plotted against Damk6hler number, showing flame extinction due to chemical limitation. In Fig. 7 an S-shaped curve appears, which has been used for iginition and extinction arguments in forced-flow situations. The maximum flame temperature decreases abruptly at a critical Damk6hler number. Only the upper branch is designated in the figure (the solid line) because of the difficulties in numerical computations of middle branches by the finite-difference method. If the ejection velocity is assumed to be Vw = 2 cm/sec, the corresponding critical cylinder diameter becomes D = 0.01 cm, which is too small for the present boundary-layer approximation to hold true. However, according to a separate quasisteady analysis [13] for the droplet combustion and assuming spherical symmetry, a similar aspect arises when the droplet diameter is decreased. Therefore, the present results are deemed to be valid, at least qualitatively. To examine the extinction mechanism of the free convection flame, a particular effective Damk6hler number using a maximum velocity gradient amax at the stagnation point given by

KPePl °Pl f Deaf-

( k + 1)mamax

D1

(26)

(f')max"

It is of interest that flame extinction due to chemical limitation may occur also in natural convection condition. Figure 8 plots the maximum temperature vs effective Damk6hler number. It is apparent that the mechanism of extinction of this flame can be explained by analogy with the flame under forced convection. The fact that the dimension of the cylinder plays an important role in describing the flame characteristics may be the most noteworthy point of the present flame. The relation between the Nusselt number and - f w is given in Fig. 9. The Nusselt number is defined by hD

Nu = --,

(27)

X and the ordinate is related with the next equation.

Ve G r 112

amax

02

! (f)max

can be defined as follows:

(25) Nu 4 ( r m cr1

-

= -

-a-Tw an

(28)

DIFFUSION FLAME WITH NATURAL CONVECTION

K 2000

I

I

J \

1000

241

\

FUEL

C3H.

PLOT OF T,~ VERgUS EFFECTIVE DAMI~HLER NUMBER D e ft

% % % % \

3OO

ol

I

I

10 5

106

I

1

IO7

I0'

O,tf Fig. 8. Maximum flame temperature vs effective bamk6hler number, which shows an S-shaped curve similar to that generally used for extinction phenomena.

0.2[-

0

FUEL C3H8

0.5

I

1.5

-f.

Fig. 9. Heat-transfer rate and ejection parameter.

It is seen from Fig. 9 that the thermal quenching mechanism is due to heat loss to the cylinder. The critical ejection parameter, --fw,erit, is plotted against the cylinder diameter in Fig. 10. The values of the critical ejection parameter become

small with increasing cylinder size. The effect of compressibility is indicated in Fig. 11, from which it is seen that the compressibility has only a minor influence on the thermal quenching characteristics. The difference in the geometry of a

242

T. SAITOH

JI 0.1

0.01

J IC)~

I I(] 2

i I01

i I

J I0 cm

D

I I0 2

i I0 3

I0"

Fig. 10. Plot of critical ejection parameter and cylinder diameter.

K 2000

I

incompressible

F U E L C3 H8 D, = 2.71 x 107

~

compressible

I000

300

0

0.5

-f.

I

I

I

1~5

Fig. 11. Effect of compressibility (-fw = 1). porous body is dearly seen in Fig. 12. Thermal quenching for the axisymmetric flow occurs at a larger critical ejection parameter than for the planar flow.

around a porous cylinder under conditions of-natural convection: 1.

CONCLUDING REMARKS The following conclusions can be drawn from the present numerical analysis for a diffusion flame

2.

This combustion phenomenon can be rigorously described by two key parameters, that is, by the Damk6hler number and the ejection parameter. Thermal quenching occurs when the ejection parameter is decreased beneath a

DIFFUSION FLAME WITH NATURAL CONVECTION K 2OOO!

150(

CYLINDER

/ ~

S

P

H

E

R

243

E FUEL

I00(

CsH a

D,= 2.71X IOr

500

300 I

0

0.5

I

i

I

1.5

" fw

Fig. 12. Comparisonof geometry of porous bodies.

.

certain value. By examining the heat-transfer rate at the cylinder surface, this phenomenon can be attributed to heat losses. There exists flame extinction for the natural-convection diffusion flames as well. A plot of maximum temperature versus the Damkt~hler number shows the so-called Sshaped aspect, and using an effective Damk6hler number, it was shown that the extinction mechanism is essentially the same as that under forced flows,

In closing, it would be of particular interest to make a simulated analysis for the combustion of a droplet with phase-change conditions at the liquid sphere surface.

NOMENCLATURE Cp D Di D1 Deff E f fw gx Gr h Hi ° n c

he

specific heat at constant pressure diameter of cylinder diffusion coefficient of species i Damk/Shler number effective Damkohler number activation energy defined by Eq. 9 ejection parameter acceleration due to gravity; x denotes x component Grashof number mesh length standard heat of formation per mole of species i at temperature T° heat of combustion of fuel per mole heat of combustion of fuel per unit mass

The author extends his sincere thanks to Prolessor 1-1. Tsu/i, Institute o f Space and Aeronautical Sciences, University o f Tokyo, for many invaluable suggestions received in the course o f this research, He is also indebted to Mr. H. Arai for the assistance in the numerical computations and to the Computer Center at Tohoku University for the use o f ACOS 6 SYSTEM 7 time-sharing systems.

k K mi Nu Pr P Pi

flow parameter; k = 0 for planar flow, k = 1 for axisymmetric flow frequency factor molecular weight of species i Nusselt number Prandtl number pressure = f'

244 ro R° Sc

T T+ T° Tm u, v x,y

~b k /a v vl i, ~z i p

T. SAITOH radius of body curvature universal gas constant Schmidt number dimensionless temperature temperature reference temperature maximum flame temperature radial and tangential velocities, respectively coordinates shown in Fig. 1 chemical reaction rate similarity variable thermal conductivity viscosity kinematic viscosity stoichiometric coefficients of reactant i and product ], respectively density stream function

Sub~'ipts and Super~ripts e f o w

value at infinity fuel oxygen cylinder surface

REFERENCES 1. Kim, J, S., Ris, de J.~ and Kroesser, F. W., Internat. J. Heat Mass Transfi, 17:439 (1974). 2. Tsuji, H., and Yamaoka, I., Eleventh S~mposiurn (International) on Combustion, The Combustion Institute, Pittsburgh, 1967, p. 979. 3. Tsuji, H., and Yamaoka, 1., Inst. Space Aero. Sci., University of Tokyo, Report No. 404, 1966, p. 95. 4. Milne, T, A., Green, C.L., and Benson, D. K., Cornbust. Flame 15:255 (1970). 5. El Wakfl, M. M., in Combustion Measurements (R. Goulard, Ed.), Hemisphere Publishing Corporation, 1976, p. 225. 6. Saitoh, T., Internat. J. Heat Mass Transl., 17, 1063 (1974). 7. Saitoh, T., and Otsuka, Y., Transact. JSME 40-333: 1412 (1974). 8, Lees, L.,JetI~opul. 26:259 (1956). 9. Aldred, J. W., Patel, J. C., and Williams, A., Cornbust. Flame 17:139 (1971). 10. Saitoh, T.,and Otsuka, Y., Combust, $ci. Technol. 12:135 (1976). 11. Ablow, C. M., and Wise, H., Combust. Flame 22:23 (1974). 12. Fristrom, R. M., and Westenberg, A. A., Flame Structure, McGraw-Hill, New York, 1965. 13. Saitoh, T., Technol. Rep., Tohoku Univ. 43:47 (1978). Received 18 December 19 78