Effect of plate thermal resistance on boundary layer ignition

COMBUSTIONAND

121

FLAME 43: 121-129 (1981)

Effect of Plate Thermal Resistance on Boundary Layer Ignition &SAR Department

of Fluid

TREVIltO

and MIHIR SEN

and Thermal Sciences, Division of Mechanical and Electrical Engineering, National University of Mexico(U.N.A.M.), Mexico City, Mexico

Recent studies of the flat plate boundary layer ignition problem have neglected paper a reacting flow above a thin flat plate is considered, the temperature maintained constant. Calculations were made using the parabolized governing the 6 + 0 limit, where 6 is proportional to the local flame velocity relative to indicate that the effect of the transverse plate thermal resistance is to move the while displacing the upper part of the flame front upstream

1. INTRODUCTION The process of steady-state ignition of a combustible gaseous mixture flowing along a flat plate kept at a high temperature has been the object of many studies. Dooley [l] employed an iterative technique for the analysis of a reacting flow over a catalytic wall. A nonfractional reaction order was assumed and the parabolic governing equations generated by using the Prandtl boundary layer approximation were solved through an iterative process, Ignition was taken to start at the adiabatic point on the wall. Toong [2] studied a reacting flow over a noncatalytic plate assuming a second-order chemical reaction. Through a series expansion technique, the governing parabolic equations were solved. He also described an experiment utilizing an ethanol-air mixture and found good qualitative agreement with theoretical results. Using numerical methods, Sharma and Sirignano [3] studied the ignition of a premixed fuel by a hot projectile. The adiabatic point was used by them as an ignition criterion and calculated for different flow parameters. Asymptotic analyses for the highactivation energy limit were made by Berman and Ryazantsev [4] and by Law and Law [S]. In both cases analytic expressions were obtained for the ignition distance. The finite thermal conductivity of the plate has not been considered for the ignition Copyright 0 1981 by The Combustion Institute Published by Elsevier North Holland, Inc. 52 Vanderbilt Avenue, New York, NY 10017

Faculty

of Engineering,

the thermal resistance of the plate. In this of the lower surface of the plate being equations which are shown to be valid at the local fluid velocity. Numerical results zero wall heat transfer point downstream,

problem though various studies for nonreacting thermal boundary layers have included this effect, [6], [7], and [8] among others. The present paper takes into account the effect of transverse thermal resistance of the plate on steady-state ignition, the lower surface of the plate being kept at an uniformly high temperature.

2. ANALYSIS Figure 1 shows schematically the physical model analyzed. A stoichiometric combustible gaseous mixture of a fuel and an oxidant flows along a noncatalytic, nonpermeable, flat plate of finite thickness t and of finite thermal conductivity 1, The chemical reaction in the mixture is taken to be one-step and irreversible, with the rate of generation by chemical reaction of any species given by the Arrhenius law. The free-stream conditions correspond to a velocity urn, temperature T,, and fuel concentration Y mc.The lower surface of the plate is kept at a constant temperature T,,. The gas phase governing equations are well known (see [9], for example). We can introduce a stream function * (x, y) to satisfy

a* p”=5’

ati pv=--ax’ OOlO-2180/81/110121+09$02.50

(2.1)

122

Cl&AR TREVfiO

and MIHIR SEN

I// -c::i;:.,l ll

-

/ Y A ,‘/A’ /

Y4

---

l

X

(

-t

-L-

Fig. 1. Schematic of physical model.

We can also take the following nondimensional variables:

Y’

m,= 7. YCC

(2.2)

A new set of independent variables can be defined by the Howarth-Dorodnitzyn transformation [9] of the form

where [ is the first Damkiihler maximum value cm is given by

number and its

oc+ao-l

LfBT,“‘Ti,=’

leading edge of the plate. The following simplifying assumptions are also made: negligible buoyancy forces, radiation effects, and Eckert number; mixture composed of ideal gases with constant specific heats equal for all chemical species within the mixture; mass diffusion given by Fick’s law; constant mean molecular weight of the mixture; coefficient of viscosity of the mixture proportional to its absolute temperature; and constant Prandtl and Schmidt numbers. Under these conditions the nondimensional governing equations take the form

u,( wp-

‘( wO)oo (2.4)

We assume the Prandtl boundary layer approximation for the momentum equation while retaining the longitudinal diffusion terms in the energy and species balance equations. This approximation applies, of course, away from the

(2.6)

123

BOUNDARY LAYER IGNITION 1 8’rn c+fB

Scv

Defining the nondimensional dinate within the plate as

all

transverse coor-

(2.10) Eq. (2.9) becomes j$ e,(i, o=o.

(2.11)

The nondimensional form of the lower surface boundary condition is where $is the second Damkohler number given by

ex,

- 53 = e,,

(2.12)

where Em represents the transverse thermal resistance of the plate given by (2.13)

and The solution to Eq. (2.11) is

(2.8)

s=u,+a,.

y=a,-a,-a,+l,

The nondimensional boundary conditions for the above equations are

where e,,(i) represents the temperature of the upper surface of the plate. An additional condition is imposed at the gassolid interface on considering the heat interchange between the wall and the combustible mixture. In nondimensional form we have (2.14)

f/+03:

af =m =e=i . aq c

On neglecting longitudinal heat transfer in the plate, the steady-state energy balance equation is given by -!? T,(x, y)=O. w

(2.9)

The local nondimensional parameter 6 has an important effect on the stabilization of the flame. For large values of 6 the equilibrium theory determines the stabilization mechanism [lo]. On the other hand, when 6 tends to zero the continuous ignition mechanism governs the flame stabilization [ 111. Furthermore, for vanishingly small values of 6, Eq. (2.6) and (2.7) become parabolic in the {direction and marching methods can be used for their numerical solution. Under this condition the

124

CESAR TREVIRO and MIHIR SEN

Prandtl boundary layer approximation is valid for all the governing equations, even close to the flame. Considering thermal diffusion to be a dominant factor in the determination of the flame velocity, it can be shown that 6 --. U/

Sharma and Sirignano [3-J.At the wall we have the interface condition given by Eq. (2.14). The derivative on the left-hand side of this equation is replaced by its finite difference equivalent of the form

a9 = W, 2) - W, 1) aqg=. 4 ’

(3.4)

U

The value of 6 is largest at the lower edge of the flame front, and this point is thus critical for the validity of the small 6 approximation. Physically, as the free-stream velocity increases, the value of 6 decreases and the flame inclines toward the wall in the downstream direction. In the vanishing 6 limit, longitudinal temperature and concentration gradients become negligible as compared to the transverse ones.

The momentum balance equation (2.5) is decoupled from the energy balance equation (2.6) and from the species (fuel) balance equation (2.7). For this reason, Eq. (2.5) can be reduced to the classical Blasius problem represented by (3.1)

f”‘+$“d),

where f is a function of q only. Using the vanishing 6 approximation, the energy balance and fuel balance equations (2.6) and (2.7) become

i a% -7+/E pr all if

1g -@ym,s exp - 3 , ( 11

(3.2)

1 a2mc scp- +f$ =2[

, am 2+fh;‘exp If ai

(

8, - e

1+Jl=iSh 5, (

0(i, 1)--B@, 2)

>

The calculations were carried out for the flow of a stoichiometric propane-air mixture. Parameters corresponding to this mixture are the following:

3. NUMERICAL SOLUTION AND DISCU!SSION

=25

The first index i represents the [ coordinate of a point; the second index 1 represents a point on the wall, and the second index 2 a point Au away from the wall. Equation (2.14) then gives

>I

.

(3.3)

Equations (3.2) and (3.3) were solved using a quasi-linearization method similar to that used by

8,= 28.55,

e= 8.84,

Pr = 0.75,

SC= 1.32,

u,=a,=l,

a,=O.

Figures 2 and 3 show constant temperature and constant fuel concentration curves, respectively, in the ({, q) plane for a plate lower surface temperature 8,,=3.5 and corresponding to a plate thermal resistance <,=O. Strong temperature and fuel concentration gradients indicate the position of the flame front. The lower edge of this front is not well defined due to thermal quenching effects near the wall. Figure 4 shows the effect of the plate thermal resistance on the position of the zero heat transfer point on the wall corresponding to 8,, = 3.5 and 4.0. This position has been taken to indicate the ignition point. In this figure, [, represents the distance to the zero wall heat transfer point for a thermal plate resistance 5, ,and [,, its corresponding value for &,,=O. The curves for 6,,=3.5 and 0,, = 4.0 are very close to each other and the plate thermal resistance is found to have a strong influence on the position of the ignition point for

125

BOUNDARY LAYER IGNITION

3-

9 2-

l-

o

5

10 CXld3

Fig. 2. Constant temperature curves for B,l=

15

20

3.5 and trn = 0.

.9 3

:! .4 2 d

92

1

cx10-3 5

10

l5

20

Fig. 3. Constant fuel concentration curves for ~~1 = 3.5 and trn = 0.

126

&AR

TREVIfiO and MIHIR SEN

2

dd6,\=4) 2.

21996

thd&,,=3.5)=3460

Latm

1

Fig. 4. Influence of plate thermal resistance on the zero wall heat transfer position for owl = 3.5 and for &,I = 4.0.

Fig. 5. Influence of plate thermal resistance on the wall temperature distribution for owl = 4.0.

(0,)

127

BOUNDARY LAYER IGNITION c&,,>1. The temperature profile near the leading edge of the plate for tm= 0 is qualitatively different from that when 5,-O. This means that [, for 5,-O is greater than [, for <,=O. This effect is more pronounced in the graph for the 13,~=4.0 line since here the reaction zone is closer to the leading edge and is thus more affected by conditions there. Figure 5 shows the temperature distribution on the upper surface of the plate BWUfor various values of the plate thermal resistance with 0,,=4.0. For &= 1, 8,” is almost equal to B,,, except near the leading edge. For increasing values of 4, the zero wall heat transfer point moves in the downstream direction, as is also indicated in Fig. 4. At the same time, however, the temperature behind this point reaches higher asymptotic values as t,,, increases, this being due to larger insulation provided at the wall. For all values of &,, there exists an adiabatic point on the wall. In the neighborhood of this point,

the assumption of negligible longitudinal heat conduction in the plate as compared to the transverse one is not justified. However, consideration of the full elliptic energy balance equation for the plate implies a considerable increase in the use of computer time since nonmarching methods would have to be used in the i-direction. For this reason, and also because the main conclusions are not qualitatively affected, the full elliptic equations were not used in the numerical solution. Figure 6 shows constant temperature curves for 0 = 8,, = 4.0 in the (q, [) and (Y, [) planes. Here we define

y?dE----, s 'IL(% 0

e drl

0

Y 125 -

Fig. 6. Constant temperature resistance for owl = 4.0.

(f3 = 4.0) curves for different values of the plate thermal

128

CESAR TREVfiO

where qL is the value of v] for the constant temperature at a particular [ position. The variable Y represents the distance to the wall, nondimensionalized by its value corresponding to [,=O at the same [ position. These constant temperature curves give a good idea of the location of the flame front as can be verified in Fig. 2. It is seen that at large c position behind the zero wall heat transfer point, increasing values of 5, give higher Y values for the flame front, which can be interpreted as a forward displacement of the upper part of the flame under these conditions. The higher plate thermal resistance impedes the flow of heat from the flame to the plate lower surface which is maintained at a constant temperature. Thus, in spite of the fact that we have taken d-+0 (implying zero longitudinal heat flux in the gas), the flame is found upstream of its position corresponding to a zero plate thermal resistance.

4. LIMITATIONS

OF THE ANALYSIS

A parabolic form of the governing equations for the combustible mixture can be justified for vanishing values of 6, the only case for which numerical calculations have been made. It is to be noted that as the plate thermal resistance increases, so does 6. Due to insulating effect of the wall, the temperature in the boundary layer behind the flame is higher. This causes the flame velocity to increase, and since the flame lower edge is closer to the wall, the fluid velocity there is smaller. Both these effects contribute to a higher value of 6. In any case, consideration of nonvanishing 6 effects would mean the inclusion of an upstream heat transfer flux from the flame in the reacting mixture. This would imply that the flame be located upstream of its position calculated using a 6+0 approximation. The other main quantitative error arises from neglecting the longitudinal heat flux in the plate near the adiabatic point on the wall. Because of the nature of the problem this heat flux is directed toward the leading edge of the plate. Inclusion of this flux in the governing equations would imply an upstream displacement of the adiabatic point on the wall as compared to its position calculated in this paper.

and MIHIR SEN

NOMENCLATURE

a, aT

B CP .r” L “N P Pr R SC t T u, v Uf W& w x3 Y Y” Y

reaction order with respect to the specieso! temperature exponent frequency factor specific heat at constant pressure for the mixture binary mass dilfusion coefficient nondimensional stream function length of the plate nondimensional fuel concentration number of chemical species taking part in the reaction pressure Prandtl number universal gas constant Schmidt number plate thickness temperature Cartesian components of the velocity flame velocity molecular weight of the species mean molecular weight of the reaction mixture Cartesian coordinates mass concentration of the species nondimensional distance to wall

Greek Letters nondimensional parameter defined in (2.8) second Damkiihler number defined in (2.8)

nondimensional distance from plate leading edge to adiabatic point on the wall value of [, corresponding to 5, =0 nondimensional coordinates in the gas phase nondimensional temperature nondimensional activation temperature coefficient of thermal conductivity viscosity coefficient stoichiometric coefficient for the species c1 in the chemical reaction nondimensional transverse coordinate in the plate nondimensional plate thermal resistance density stream function

BOUNDARY LAYER IGNITION Indices C 0 W

wu WI

cc

refers to refers to refers to refers to refers to refers to

the fuel the oxidant the wall the plate upper surface the plate lower surface free-stream conditions

One of the authors (C. T.) acknowledges partial support from the Consejo National de Ciencia y Tecnologh of Mexico for this research. REFERENCES 1. Dooley, D. A., Proceedings, Heat Transfer and Fluid Mechanics Institute, Stanford University Press, Stanford, 1957, pp. 321-342. 2. Toong, T. Y., Sixth Symposium (International) on Combustion, Reinhold, New York, 1957, pp. 532540.

129 Sci. 3. Sharma, 0. P., and Sirignano, W. A., Combust. Technol. 1:95-104 (1970). 4. Berman, V. S., and Ryazantsev, Yu. S., Fluid Dynam. (English translation), 12(5):758-764 (1978). 5. Law, C. K., and Law, H. K., J. Fluid Mech. 92(l): 97-108 (1979). 6. Wang, R. C. C., Chung, B. T. F., and Thomas, L. C., J. Heat Transf 99:513-519 (1977). 7. Sakakibara, M., Mori, S., and Tanimoto, A., Kagaku Kogaku 37:281 (1973). 8. Davis. E. J., and Gill, W. N., Int. J. Heat Mass Transf. 13:459 (1970). 9. Williams, F. A., Combustion Theory, Addison-Wesley, Reading, Mass, 1965. 10. Ziemer, R. W., and Cambel, A. B., Jet Propul. 28(9): 592-599 (1958). 11. Zukoski, E. E., and Marble, F. E., Proceedings, Gas Dynamics (1955),

Symposium

Multicopy

on

Aerothermochemistry

Corp., Evanston, Ill., 1956, pp.

205-210.

Received

7 December

I9 79; revised

8 September

1980