Halogenated Fire Suppressants

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Theoretical Investigation of Inhibition Phenomena in Halogenated Flames SERGE GALANT and J. P. APPLETON Mechanical Engineering Department, Massachusetts Institute of Technology, Cambridge, Mass. 02139

Literature reviews on chemical inhibition (1),(2),(3) reveal that a great deal of experimental work has been carried out in an attempt to gain a better understanding of inhibition phenomena in flames. Since strong experimental evidence (2),(3),(4) supports the view that a basic understanding of inhibition phenomena can be obtained by studying inhibition effects in premixed laminar flames, researchers have studied inhibition effects in such systems. Yet, two somewhat different approaches have prevailed. One crude but efficient way of screening potential inhibiting agents has been to measure the decrease in nominal laminar flame speed as a function of the inhibitor concentration. (5),(6),(7), (8),(9),(10). Another related type of experiment has been concerned with the modifications of the ignition limits of a given mixture when inhibiting agents are introduced. (3),(6),(9). On the basis of such experimental evidence, a plethora of speculative kinetic models have been proposed (for exhaustive listings, see References (1),(2),(3)). A feature common to almost all these proposals is their lack of conclusive agreements when it comes to making "ab initio" predictions of the reduction of the laminar flame speed. On the other hand, it can be argued that a better insight into inhibition can be gained by looking at flames on a molecular level (2). In particular, spectroscopic measurements (11),(12), (13),(14) have yielded a wealth of information by providing actual nonequilibrium concentration and temperature profiles. Most important, speculations have been quantified into models which identified some of the critical kinetic steps. However, the uncertainty in those measurements and the lack of detailed knowledge about the various nonlinear couplings between transport phenomena and chemical reactions which occur in flames, have impaired any conclusive description of the actual inhibiting effects. Our theoretical approach is aimed at reconciling these two experimental approaches: using the rate coefficients and diffusion parameters as obtained from "molecular experiments" (11), 406 Gann; Halogenated Fire Suppressants ACS Symposium Series; American Chemical Society: Washington, DC, 1975.

15.

Inhibition Theory

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(12),(15),(16),(17), we intend to use a new numerical technique to solve the general one dimensional unsteady flame equations and, i n turn, to predict the local and global behavior of i n h i bited flames, v i z . temperature and concentration profiles but also, flame velocities and ignition limits. Such an approach i s not new per se. Day et_ a l . (6) solved approximations to the flame equations to study inhibition of rich hydrogen-air flames. In doing so, they were able to pinpoint the fundamental kinetic steps needed to account for experimental observations. Yet a proper adjustment of unknown reaction rates was required to match to the experimentally observed decrease of the laminar flame speed. In the present investigation, a computing procedure which had been successfully applied to the ozone-oxygen flame (18) has been improved and generalized to be used for an automatic computer code which describes both ignition and steady propagation of deflagrations i n planar and spherical coordinates. The calculations reported here have been obtained with a mathematical model which includes a l l effects that are believed to be important, i.e., molecular diffusion, heat conduction and thermal diffusion, together with a general reaction kinetic scheme and f u l l y r e a l i s t i c thermodynamic data. A short overview of the equations and of the computational method to solve them i s given in the f i r s t section of this paper. As noted earlier, a f i r s t logical approach i s to model i n h i bition phenomena i n flames for which the kinetic, transport and thermodynamic data are f a i r l y well established. The archetype i s evidently the hydrogen-oxygen (or hydrogen-air) flame which is abundantly documented i n the literature. In the second section we consider inhibition effects for a low pressure rich H2-O2 flame (19) supported on a porous plug burner. In so doing, we hope to demonstrate the capability of the numerical method to describe actual experiments. These calculations are the f i r s t i n a series of simulations dealing with inhibition phenomena i n hydrogen-oxygen a i r flames. Further work on ignition limits and flame velocity computations w i l l be reported later. Theory The Mathematical Model The equations which describe the unsteady one-dimensional propagation of a premixed flame i n a frame of reference fixed with respect to the moving flame front are (20) mass continuity: species continuity:

+ r ^ n

p^ j

(r ^ pv) = 0

pv

=

r"^3 (r^pG,)

W

j

where

Gann; Halogenated Fire Suppressants ACS Symposium Series; American Chemical Society: Washington, DC, 1975.

HALOGENATED FIRE SUPPRESSANTS

408

m

η = Σ

η

energy equation:

equation of s t a t e :

ρ = P/(RTn)

o v e r a l l mass c o n s e r v a t i o n :

m Σ w.m. 3=1 3

= 0

3

m

Σ j=l m

Σ J-l

n m 3

m.G. J

= 1

3

- 0

3

Here t i s the time, r i s the coordinate along the dimension of i n t e r e s t , ν i s the v e l o c i t y component i n the r d i r e c t i o n , ρ i s the l o c a l d e n s i t y of the f l o w i n g gas, n^ i s the c o n c e n t r a t i o n of the j - t h species i n moles per gram of mixture, mj i t s molecular weight,C$+ i t s molar heat c a p a c i t y a t constant pressure, Hj i t s molar enthalpy, Dj i t s molecular d i f f u s i o n c o e f f i c i e n t , D.T i t s thermal d i f f u s i o n c o e f f i c i e n t and WJ i s the net mass production (or d e s t r u c t i o n ) r a t e of the j - t h s p e c i e s per u n i t volume. I t has been t a c i t l y assumed t h a t s p e c i e s d i f f u s e according to F i c k ' s law, that heat t r a n s f e r by conduction w i t h i n the system obeys F o u r i e r ' s law, λ being the heat conduction c o e f f i c i e n t , and that the mixture c o n s i s t s of i d e a l gases a t constant pressure P. F i n a l l y , by s e t t i n g γ = 0, 1 or 2 the equations can be obtained i n terms of c a r t e s i a n , p o l a r or s p h e r i c a l coordinates respectively. Together w i t h the set of equations (1) there must be a s e t of adequate i n i t i a l and boundary c o n d i t i o n s which complete the d e s c r i p t i o n of the system. Two types of i n i t i a l c o n d i t i o n s have been d e a l t with i n the past. For steady s t a t e flame s t r u c t u r e c a l c u l a t i o n s , guessed Sshaped p r o f i l e s are assumed to d e s c r i b e the v a r i a t i o n s of the dependent v a r i a b l e s ( c o n c e n t r a t i o n , temperature) through the flame r e g i o n i n i t i a l l y , I f enough energy i s s t o r e d i n these i n i t i a l p r o f i l e s , a flame w i l l propagate and the p r o f i l e s u l t i mately assume t h e i r steady-state shapes. P h y s i c a l l y , t h i s means that steady s t a t e flame propagation does not depend upon the way i n which the system was i g n i t e d . Yet, such a r b i t r a r y i n i t i a l c o n d i t i o n s have no p h y s i c a l b a s i s . In order to consider the i g n i t i o n phase, i t would be p r e f e r a b l e to use i n i t i a l c o n d i t i o n s


15.

Inhibition Theory

GALANT AND APPLETON

409

which were compatible w i t h a more p h y s i c a l s i t u a t i o n such as e x posing " p o c k e t s " of combustion products of d i f f e r e n t dimensions to a l a r g e body of combustible mixture ( 1 8 ) . More g e n e r a l l y , i n i t i a l c o n d i t i o n s which a r e square (discontinuous) waves i n concentrations and/or temperature might be thought of as r e p r e s e n t i n g the e a r l y phases of i g n i t i o n ( e x t i n c t i o n ) phenomena: i n deed, they a r e approximate models of the l a r g e c o n c e n t r a t i o n and temperature g r a d i e n t s which i n i t i a l l y do e x i s t i n , s a y , spark o r l a s e r i g n i t e d systems. Boundary c o n d i t i o n s f o r steady flame propagation have been d i s c u s s e d a t l e n g t h . The problem i s to s p e c i f y i n t e r n a l l y c o n s i s t e n t boundary c o n d i t i o n s f o r an a c t u a l n o n - a d i a b a t i c m u l t i - d i m e n s i o n a l flame. For the c a l c u l a t i o n s presented i n the second p a r t , we use the concept of a porous p l u g burner (20),(21) which i s v a l i d o n l y f o r p l a n a r c o o r d i n a t e s , i . e . , Y=0. A t the flameholder a temperature g r a d i e n t i s assumed t o e x i s t which tends t o s t a b i l i z e the p o s i t i o n of the flame. Be cause of d i f f u s i o n of the product s p e c i e s from the flame a l l the way back t o the f l a m e h o l d e r , the chemical composition of the gas at the flameholder i s not s p e c i f i e d . However, s i n c e the mass f l u x of the j - t h s p e c i e s must be continuous a t the c o l d boundary (r=0), we o b t a i n

(n

j P

V - PG^I^Q = p° u°

η.

0

where the s u p e r s c r i p t denotes c o n d i t i o n s i n the mixing chamber, and r = 0 a t the c o l d boundary. Using the d e f i n i t i o n of G . i n (1) we get 0

3

Τ

> j - Ψ- I V ë>- ' νν ' {u

+

ν (

} r = 0 c >0

A l s o , the temperature a t the flameholder i s supposed t o remain constant w i t h t i m e . F i n a l l y , a t the hot boundary ( r - * » ) , i t i s assumed that the s p e c i e s d i f f u s i o n f l u x e s and heat d i f f u s i o n f l u x a r e z e r o . Therefore, the s e t of boundary c o n d i t i o n s r e a d s : Τ

r

=

0

:

ί ΐ

-

ψ

i f -

+

^

Τ (0,t) = Τ ο r + ~ ^ = 0 ,

g}

+

P

O

U

O

(

n

j

0 _

n

j

)

.

o,

t>0

, t > 0 g

=0, t >

0

Before s e a r c h i n g f o r s o l u t i o n s of s e t ( 1 ) , f u r t h e r i n f o r m a t i o n i s needed about the f u n c t i o n a l form of the d i f f u s i o n c o e f f i c i e n t s , heat c a p a c i t i e s , e n t h a l p i e s , and chemical p r o d u c t i o n terms. These m o d e l l i n g questions are considered i n the Appendix


410


Computational Method. It i s only recently that numerical techniques to describe one-dimensional premixed laminar flames have become available. Spalding (21) and Dixon-Lewis (22) dev eloped similar relaxation methods where the unsteady flow equations are solved for arbitrary i n i t i a l conditions. Although computation times seem quite reasonable, t r i a l and error must be used to determine an economical stable step size, because the equations are linearized locally i n time. Alternatively, the steady state equations and their corresponding eigenvalue—the adiabatic flame velocity—may be solved via shooting methods quasi-linearization or f i n i t e difference techniques (23), (24). The latter method can become very time consuming, whereas, the former two present d i f f i c u l t i n s t a b i l i t y problems i n both the backward and forward integration because of the sensitivity of the numerical solutions to the guessed boundary conditions (19). From this brief glance at already existing methods to solve the flame equations, i t can be recognized that the choice of a numerical technique i s governed by trade-offs between computation time and problems of s t a b i l i t y , convergence and accuracy. Since one of our objectives was to develop a generalized computer pro gram to provide the user with a fast and reliable technique which could analyze a wide range of physical situations, i t was our belief that none of the methods discussed above f u l f i l l e d a l l the requirements for a general user package. The technique presented now i s the so-called "method of l i n e s " which was f i r s t applied to one dimensional flame propagation by Bledjian (18). Basically, instead of approximating the p a r t i a l derivatives with respect to a l l the independent variables by f i n i t e different expressions, one can convert the original set of parabolic par t i a l d i f f e r e n t i a l equations (1) to a system of ordinary d i f f e r ential equations by discretizing only the space variable, the time being l e f t continuous. In our case, for a system of m reacting species, the system of m parabolic nonlinear differen t i a l equations, m-1 species continuity equations and 1 energy equation , i s transformed into a set of m Ν nonlinear d i f f e r t i a l equations where Ν i s the number of grid points in the spatial direction: the i n i t i a l conditions to solve this system are provided by the i n i t i a l conditions applied to the original partial d i f f e r e n t i a l equations. Basic numerical advantages characterize this method • once the problem i s reduced to an i n i t i a l value problem, very efficient numerical methods of integration are available, with a computation cost proportional to the dimension of the system (mN) • unsteady problems can be dealt with i n a general fashion in particular, the modelling of ignition and/or extinction pheno mena becomes feasible. • flames i n spherical or c y l i n d r i c a l coordinates can be treated with no major inherent d i f f i c u l t y . The immediate


15.

GALANT AND APPLETON

411

Inhibition Theory

shortcomings of the technique stem from the particular nature of the resulting set of d i f f e r e n t i a l equations. • Since rapid changes within the flame occur over a very limited spatial domain, a large number of discretized points Ν i s needed to accurately describe those changes. In case of an insufficient number of grid points, strong i n s t a b i l i t i e s which most often yield negative concentrations appear readily. • On the other hand, i t has been shown with the help of simple linear parabolic equations (25) that, as the number of grid points Ν i s increased, the "stiffness" of the d i f f e r e n t i a l equations becomes very c r i t i c a l , i . e . , smaller and smaller time step sizes are needed to obtain stable and accurate results. • Moreover, since the original equations are already " s t i f f " due to the presence of nonlinear chemical source terms, this immediately suggests the use of an implicit " s t i f f " method of integration such as Gear's code (26). But then the cost of computation i s no longer proportional to the dimension of the system, m Ν: i t varies as (mN) or even (mN) because the Jacobian matrix of the nonlinear system must be inverted when a loss of accuracy i s detected. Most of the numerical problems inherent to the solution of the flame equations i n their original form were overcome by transforming both the dependent and independent variable. The use of the von Mises transformation (21) together with a mapping (27) bringing the i n f i n i t e or semi-infinite domains ]-°°, + °°[, [0,°°] into [0,1] allowed for a non-uniform grid size i n the r direction and improved the s t a b i l i t y characteristics of the re sulting set of equations. The stiffness of the equations was further reduced by using the new dependent variables 2

n. = log n. 3 3 f

3

i=l,...m

• log Τ

which also eliminates the problem of dealing with negative con centrations. The discretization of the spatial derivatives was accomplished by using a non-central five-point difference scheme (28) . Such a scheme allows for a sensible reduction i n the num ber of spatial grid points needed to follow the flame history within a given accuracy. Finally, a slightly modified version of the variable step size variable order integration method devised by Krogh (29) was implemented: i t leads to stable, ac curate and relatively inexpensive calculations for the set of equations considered above. Numerical Simulations Motivations. The flame which was chosen i s a low pressure, rich, H2-O2 flame, for which temperature and concentration measurements exist (19). Although such a flame i s not a very


412


good approximation to the burning of inhibited polymers i n an atmospheric environment, i t was selected to i ) demonstrate the a b i l i t y of the computational scheme to reliably describe an actual burner type experiment, i i ) pinpoint eventual improvements to be made i n the model l i n g (kinetics, transport, properties,...)* i i i ) obtain a general description of inhibition phenomena when the chosen flame i s seeded with halogenated compounds (HBr, Br. 2 •··), iv) check the influence of thermal diffusion which had been considered as a secondary effect but which i s claimed to be im portant under some experimental conditions (14). Within this context, no attempt was made to c r i t i c a l l y evaluate the parameters which are needed to describe both unin hibited and inhibited flames. Rather, the results shown here should be considered as a check for the v a l i d i t y of our flame model and a f i r s t step toward a basic understanding of i n h i b i tion phenomena i n gaseous mixtures. Results Ife-(>2 flame. The present predictions have been obtained using a set of rate constants l i s t e d i n Table I (from there on, referred to as Flame 1 : 75% H , 25% 0 ) . 2

2

TABLE I Reaction Mechanism and Reaction Rate Data (a) Reaction Mechanism and Forward Rate Constants Reaction Rates are expressed as Κ = AT exp (-E/RT) with k i n cm mole" sec" (cm mole"" sec" for three body reactions), Τ i n Κ and Ε i n kilocalories C

3

1

1

6

2

1

Flame 1 Reaction No. Reaction 1 0H+H = Η + H 0 2 H+0 = 0H+ 0 3 H +0 = Η + OH 4 0 +H20 = 0H+ OH 5 H+OH-m = E43+M 6 H+H +M = H + M 7 H+0 +M = 0H+ M 8 H2+O2 = 0H+ OH 9 H+0 2 4M = HO2+M 10 Η +H0 = OH+OH 11 HO 2 +0 = OH+O2 12 OH +H0 = O2+H2O 13 Η +H0 = H2+O2 2

2

2

2

2

2

2

2

A ~ 10 ,9 χ 74x 10 1 3 74x 1 0 75x 1 0 0 χ 10 1 5 2.04X10 4.6 xlO 15 2.1 xlO Ik 1.5 x l O 13 7. xlO 1. xlO 1 3 1. xlO 13 6.3 xlO 12 1 3

1 3

1 3

16

1 5

C

0. 0. 0. 0. 0. -0.31 -1.75 0. 0. 0. 0. 0. 0.

ε 5.T55 16.815 9.459 18.016 -1.49 0.0 -0.777 58.133 -0.993 0. 0. 0. 0.


Ref.

w (30) (30) (30) (31) (31) (31) (30) (31) (30) (30) (30) (30)

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GALANT

AND APPLETON

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Inhibition Theory

(b) Third Body Efficiencies. For Reactions (6) and (7) a l l third bodies are taken to be equally e f f i c i e n t . For reactions (5) and (9), the following relative efficiencies are used, a l l others being set equal to 1.

Reaction No. 5 9

H0

H

5. 16.25

1. 2.5

2

References (30),(31) (30),(31)

The species considered were U2,02, H2O, Η, 0, OH, HO2. H 0 was set aside because of i t s numerically proven minor importance, and the lack of thermodynamic data above 1500°K. The computation of the transport coefficients i s detailed i n Appendix A. The pre dicted and experimental values of temperature are compared i n Figure l a whereas i n Figure lb the heat release rate along the burner i s plotted. The agreement i s f a i r at low temperature: the deviations from experimental values might be due to modelling i n accuracies when computing the thermal conductivity and the ther mal diffusion coefficients. The heat loss by conduction to the burner i s equal to 0.062 cal.cm s"" , whereas, Eberius et a l . found i t equal to 0.15 aal.cm"" s " (19). At higher tempera ture the agreement i s very poor; this can be explained by experi mental heat losses along the burner which tend to flatten the temperature p r o f i l e . However, since this i s a low pressure flame, the recombination of radicals i n the hotter regions i s extremely slow: hence, the high temperature profiles which are displayed have not reached their steady state values (19). In Fig. 2, the predicted and measured concentrations profiles for H2,02,H 0, Η and OH are compared, together with the theoreti cal values displayed i n Ref. 19 for a model without thermal diffusion effects. The agreement i s improved and emphasis should be put on the influence of thermal diffusion over most of the reaction zone. (See Figure 4a—dotted l i n e s — t o compare the relative magnitudes of the various Η atom fluxes.) The importance of those effects i s illustrated by the f a i r l y good agreement between experimental and predicted values for the Η and OH concentration at low temperature. This also demonstrates that the contribution from thermal diffusion effects cannot be correctly estimated on the basis of steady-state flux calculations alone, as i t has been done before (22). In fact, the preflame region i s characterized by strongly nonlinear feedback effects which are hardly predictable without an adequate numerical treat ment. Finally, i t should also be mentioned that polarity effects of the water vapor molecule, which have not been considered i n those calculations, can account for the discrepancy i n the concentra tion profiles over the cold part of the flame. They may change, 2

-2

2

1

1

2


2


1300 FLAME 0

2

1100 Lu

cc

£ 900

FLAME

< cc

1

£ 700 500 3οα

Ό

HEIGHT

A B O V E B U R N E R , CM

Figure la. Temperature distribution above the burner Δ Experimental points taken from Ref. 19 Solution of the flame equations taken from Ref. 19 Solution of the flame equations for Flames 1 and 2, including thermal diffusion

Lu

Halogenated Fire Suppressants

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