Quasi-steady flames on an evolving atmosphere

1 3 SEP. t984

Cranfield

College of Aeronautics Report No. 8406 February, 1984

TECHNISCHE HOGESCHOOL DELFT LUCHTVAART- EN RUIMTEVAARTTECHNIEK

BIBLIOTHEEK Kluyverweg 1 - DELFT

Quasi-Steady Flames on an Evolving Atmosphere

by

J F Clarke

College of Aeronautics Cranfield Institute of Technology

Cranfield

College of Aeronautics Report No. 8406 February, 1984

Quasi-Steady Flames on an Evolving Atmosphere

by

J F Clarke

College of Aeronautics Cranfield Institute of Technology

Cranfield, Bedford, UK.

ISBN 0 947767 02 9 £7.50

"The views expressed herein are those of the authors alone and do not necessarily represent those of the Institute."

"The views expressed herein are those of the authors alone and do not necessarily represent those of the Institute".

Theoretical studies of pre-mixed flames are complicated by the fact that the atmosphere ahead of the combustion wave is essentially in a state of disequilibrium. Various devices have been proposed to overcome this difficulty and this paper proposes, and examines, a model that admits from the outset that the basic atmosphere will be undergoing temporal changes, at least.

These are allowed to progress, and the flame is superimposed upon them in a manner that allows it to be treated as a quasi-steady combustion wave whose slow evolution is shown to be linked directly with that of the ambient atmosphere itself. The flames are shown to have structures that cover the whole spectrum of possibilities from convection-reaction balances to the conventional thermal-flame-type of wave.

1

-?.. Basic Equations

The energy, species and momentum equations for a one-dimensional unsteady flow are, respectively

where >', ƒ > "^ i "^s > *^ >'^•t ^""^ pressure, density temperature, stagnation temperature, namely

1.1

H »X,T: ; 1.2

u'»-1.4

velocity and mass fraction of species-«f ( « K , ^ for oxidant and fuel, respectively). Thermal conductivity is written as X' while ^ r and L e are the Prandtl and Lewis numbers. The specific heat at

constant pressure is C ' (assumed constant), Q ' is the combustion energy per unit mass, W ^ is the molecular weight of species-» ;

u s. P signifies the product species and

W ^ ^ W^-^W^ a

^^(\^<r)

. 1.5

By implication the combustion reaction is

r ^ K - ^ Ï ,

whose rate of progress is

The chemical time ' ^ ^ is dependent upon activation energy E j ^ > sound speed A' , etc, as follows

where R* is the Universal Gas Constant and K' is the thermal diffusivity,

' = v'//^ .

1.8

The index i in (1.7) is a small number, of order ±. \ for example.

The thermal equation of state will be used in the form

where Y^ is the (constant) ratio of frozen specific heats, and the continuity equation is

\f' ^ \ If'-')

^c .

The frozen sound speed is A . , such that

2.1

2.2

2.3

2. Ambient Atmosphere

If a combustible atmosphere is spatially uniform, so that iC'^A-K.'

is zero for all ( > , (1.1, 2 and 3) give

To f o f "fct' '• > ^ ^ o '

where subscript —o denotes the value of a quantity in this atmosphere.

Equation (1.10) shows that

fi =

constant 2.4

whence (1.9) gives

V' = ^.S'

(^^

)^^e

•

Equations (2.1 and 5) combine to give

C V ' T ^ .

- VNf.pO'^.' 2.6

and this result together with (2.2) shows that

<"T'o' -^ 0'(%^-*-^^) ^ ^>^*^-. 2.7

where C-^ is specific heat at constant volume.

Similarly (2.2, •< t x ,P ) give

2.8

where subscript-I signifies an initial value. Thus

and (2.7) can now be written as

C ^ T V (i-^<rS<Jp C_ = constant . 2.10

It will be assumed from now on that the fuel/oxidant mixture is a lean one, so that combustion ceases when e ^ , or in the particular case e ^ , is zero. Thus (2.10) can be written as

where T ^ is the final temperature in the present constant-volume process.

In terms of initial values 7^^ and ^p^ (2.10) can be

written as

C v T ^ ' -V ('Ucr^O'c^. * C ^ " ^ ^ ^6^^W<^y^ • 2.12

Thus

and (^^ is a constant for any given initial conditions.

5

-3. Derivation of an Equation for the Temperature

Since T Q and c^^^ are functions of b ' only, (2.1) can be

subtracted from (1.1), and (2.2) from (1.2), to give the following

equations for the differences

\~\

and c ^ - e . ^ :

f r 1 V *^

i

^' Co it'

Prandtl number has been given the value ^ and Lewis number the value

unity; these assumptions make the analysis simpler without impeding

seriously the proper modelling of the events whose progress is to be

followed. At this point the proposition will be adopted that these events

are of a quasi-steady character so that, in a first approximation at least,

the partial time derivatives of T,'-7^' and <-^—<=-B<.C can be

neglected relative to the convective, w-'^C^/Vx.' , derivatives of these

quantities. By the same token (1.10) simplifies to the extent that

» V

^

^

^'/tO

^ f^"^^

3.3

where vw*-' is a quasi-steady, or slowly-varying mass-flux and

f j ,^^

are the values of density and flow velocity where ^'

-*'

— •* •

In terms of the variable ? , where

^ = [/-'C:^7V)^^' , 3.4

the quasi-steady versions of (3.1 and 2) become

4t

M^

"•*

Equations (1.5 and 8) have been used in the derivation of these results ,

which show that

3.7

,' — '

3.8

3.9

since T-^T^ and c^ -^ < ^ where

X

—V- — •« «by hypothesis .

With the lean-mixture assumption (1.6 and 7) can be used to write

where

»' - E / / f i . ' .

But (3.7) makes

o^^-yp's. * o-<rY:)'^ -^^i-^J-X) ^ i^:^ 3.10

or, in view of the fact that the reaction will cease when

<z.

vanishes and

T j takes the value

Tj^ ,

where

T'

is given by

'n!(\L''^^')^

0-^-)Q'<.^^ .^i-:^ , 3.11

one can write this relation (3.10) in the form

Equation (2.11) shows that

7

-so that (3.8) with (3.12 and 13) becomes

^ . U ^ l . ft'/T'

. ^^;W,i^;"''t-)-f^t^^"f.^-•'>l

s.

_{' %im\^^-)-if0im)^'^^^"}

and we note the appearance of two dimensionless groups namely (<*.'./(*• ^ » of which more will be said shortly, and

The final form of (3.16) follows from the fact that V / ^ , , is equal to

( T ' / T O ' \ \ o < o!^ ^ \ , in many circumstances. Little accuracy in the modelling for present purposes will be sacrificed if we choose the simplest situation,

UJ 4- 4 »-o

and this will be adopted from now on.

The local frozen Mach number can be written as

u'.*- ^^ ^ ^.A^rJ^ 3.17 3.18 where ..I

-'.

' ^

< X,

3.19

The subscript ^-vx values are ambient - intitial values and M is the ambient initial Mach number of the flow. Since 9 is usually a large temperature

(typically Z « lo'' K. ) it is clear that the dominant behaviour of the right-hand side of (3.15) is encapsulated in the exponentials ^ - « ^ f ^ — 8 Y T ' ) and «jc^(—BV"^') in the { ^ term, together with the differences ("^^-Tf,')

and C'^^(.' ~ T(^' ) • The coefficient of { ^ has an important role to play as a determinant of magnitude of the right-hand side of (3.15), but

its variations with % throughout the T -domain ( — •«<. ? < f ««3

that come from changes in w. and T (see (3.18), are less significant and so (3.15) will be simplified to read

where

3.21

and all temperatures T^ , T., , T^ ^ T* are now measured in units of

the intial ambient value T ^ . The quantity TH. in (3.21a) is equal to M ^ at the inital instant, but must be allowed to vary (slowly) with time

thereafter; that is to say one must take into account the time-modulations

implicit in (3.18) but the spatial changes in U . * - / T are neglected.

The dimensionless stagnation temperature is evidently given by

T ^ = T 4- ^ ( v ^ ~ i ) M * - v < ^ 3.22

in view of (1.11), ( S > & ) 8< ('j.is) .

The proposition that M is very small now reduces (3.20) to a single equation for T , given that T ^ is known from the work of §2, as follows '.

Identification of the terms here as C (convection), b (diffusion) and (2, (reaction) will be useful later on.

9

-Reverting to dimensional forms for a moment, observe that (2.11) and (3.10) combine to give

The final term here is of order M in dimensionless variables and must be dropped to be consistent with (3.23). The temperature T|^'> or T , is then evidently given by

Thus { "^ in (3.23) vanishes when T & T ^ , as it should. It will also vanish when

3.24

T. - T .r ±

.^v!^-|.l(-..-T.)

Y « I T -r. { ^ "'- *• -' 3.25

and it is not difficult to see that this can only happen again if X is very nearly equal to T . , given that Tj^ is, say, at least 3 or 4 times larger than Tj, . The fact that T ^ will be of this sort of size, by hypothesis, guarantees the requisite T|^ behaviour, as can be seen from (3.24).

The right-hand side of (3.25) is essentially positive, hence the burnt temperature, as defined by (3.25), will be slightly less than Xj^ This will be recognized by stating from now on that { | in (3.23) vanishes when T = T, and T ^ T. where the latter is found from (3.25).

Since, from (3.24),

and X^^ is a constant, as shown by (2.13), it follows that Xj^

and hence also X^_ both increase (slowly, by hypothesis) with time as T does likewise.

4. Solutions

Equation (3.23) describes the behaviour of a disturbance to the unbounded ambient atmosphere that is not limited in amplitude (that is to say, it is not a small disturbance). The disturbance is hypothesised to be travelling at a nearly constant speed w-^

First note that with

(3.23) is a first-order non-linear autonomous differential equation for V = V ^ X ) . namely

AX " V

where

Singular points of (4.2) occur when X » X ^ 6«r X , and V is zero.

Using (3.24), h can be re-written as follows

and near to X «= X ^ this behaves like

where V^^ is defined here for convenience. Thus the integral curves of (4.2) have their behaviour prescribed by

11

-in the neighbourhood of X e s X ^ > V «* O • The characteristic equation of (4.6) or, better, of the pair

4 y - V - jU\t, (x--n > , > I = V , 4.7

has roots equal to ^ ± ( :!^ — - A - K ^ ) . Thus these roots are real, positive, and unequal since we may take it that -^-A-WLj is less than one. It follows (e.g. from Chapter 4, Part B, of the book by Hurewicz) that the point ( X ^ , O ^ is a nodal point, that is approached only

as 5 "^ — *• . Furthermore all the integral curves emanate from ( X , , o ) as tangents to the line

V -

(

i - C l J - ^ O ' ^ ^ ^ - ^ ' - ^ O

»

4.8

except for the single curve that leaves ( T ^ , 6^ along

In the neighbourhood of ^ T ^ . i 0 ^ (4.2) behaves like

Ax V

whose characteristic equation has roots ^ ^ ( i^ *•• - ^ •***-> r ^ i ' ^ i - N ) These two roots are real, unequal and of different signs, so that ( T L ^ , O )

is a saddle point. The integral curve that enters ( T ^ _ , G ) ^S f _•- «6 does so along the path that corresponds to the negative root; the magnitude of this root is strongly dependent on the size of -A_ *-i..^ ^ — 8 / * T ^ _ ) and from the definition of -A- in (3.12a) it is clear that this is

determined by the value of ItU for any given combustible mixture (i.e. any given Ö and X ^ ,in particular).

It is important to observe that even when -A. *^«-f ( ^ ^ / T w - )

is large the quantity J^^^ (see (4.5) for VC^ ) will usually be ^<èTy small as a result of the dominant influence of 4 X ^ ( - 9 / T . ) in the latter when &• is large.

Fig. 1 is a sketch of the integral curve behaviour, albeit for a very special case, as will be explained below. At this stage one should regard Fig. 1 as simply an illustration of the local behaviour in the

neighbourhood of the node at (X, , o ) , and of the saddle point at { T I , o ) ; for the present one should ignore the thick line that connects these two

singular points.

The general features of the integral curve behaviour exhibited in Fig. 1 at locations other than the singular points are validated as follows.

First note that the dotted line is the locus of ^ V / A X equal to zero, namely V = V j , where

(see (4.2)). In the present context T^ is to be regarded at a constant ;

Vf. ,& and X,^ are essentially so. Thus the shape of V^ is

determined by the function ( T ^ ~ T ) rfi-»-f ^^ 8-fx) ; it can be shown that this has a single maximum for T^< T < T ^ _ lying at T^_ - T^* / 8 to leading order in ft~' when & is large; furthermore it can be shown that the singular integral curve of negative slope that enters ( T ^ _ , O ) must do so from the domain 0<NJ-*-\f, T < T | ^ . as indicated in Fig. 1. The singular integral curve that follows (4.8) must lie above V^ , which behaves like

« A - ^ o , in the neighbourhood of (jt^^o) ; hence the shape of the integral curves in this part of Fig. 1.

Any integral curve that crosses V*-0 must do so with infinite slope at any locations other than (f^o) and ( X ^ _ , o ) > in other words the points

t i A

-V-O

5. Discussion of the Solutions

The general behaviour of the integral curves is unaffected by changes in parameter values, in particular the value of JV- insofar as it depends upon the Mach n u m b e r W o f the flow (see (3.21a)). It must be recalled that Tfl is necessarily small enough to make the approximation X , ï: X tenable (see (3.22)).

For the purposes of the present discussion assume that

7»! • ft^ it *- 5.1

where both ft and v<«c are of order unity, so that

* » 5 2

It can now be seen that the integral curve that enters k from within o ^ v < V j , -r,<.T.c7^_ herein~after called the solution curve, will do so with a slope that is small like ^** for M < 0 . The maximum value of V^ will likewise be small and there must therefore be an integral curve (the solution curve) that connects A with t> . for which the exit from A is along the line of very small slope (4.8). In other words the solution curve will be one of the class of curves sketched as A t in Fig. 1, except that c will coincide with h . When Tit is relatively large the slope <^\///T of the solution curve will be small everywhere. Since the diffusive, or h

term in (3.23) is given by

V AV

in present notation it follows that b will only be comparable with the

convective term C when « S ^ V / X T is of order unity. The relatively-large-7»l solutions just des^ibed are evidently dominated by the balance between C *< R. in (3.23) and, as such, they fall into the category of convected-explosion fast flames (Clarke, 1983).

14

-As "WL diminishes the V^-locus extends upwards into larger-V values, most especially in the neighbourhood of Xj^_ . The solution curves therefore begin to approach the curve A L , which is shown on Fig. 1 as emerging from the nodal point ^ along the particular

curve (4.9) and thence connecting with t» . Any further reductions of 9)t would mean that curves like A ^ would connect with ^ (i.e. «^ would coincide with t ) ; the behaviour of such curves near to A means that they are not physically acceptable solutions. Thus the Mach number that leads to the A^-type of solution is a minimum value below which the postulated quasi-steady propagation of a combustion wave is not possible. From the nature of the relation (4.9) it is clear that <A.Vf«(T in the neighbourhood of A. is around unity. Thus low-speed combustion waves are predominantly those with a simple C b balance at their head. This is

the structure of the thermal flame, and one notes that, since «^V/XT will be quite large in the neighbourhood of k when W l is small enough to make

«A, .t-*^-^ (~h (^r^^_) behave like ^ * ^ , N > 0 the b - p r o c e s s will balance with R. to the virtual exclusion of d in this domain; the rear of the flame will therefore be a bR. or 'flame-sheet' region.

In conclusion, all of the combustion-wave structures that have been

described for the steady-state half-space "flame-holder" model (Clarke, 1983, 1984) are found in the present quasi-steady unbounded-space geometry. It is

interesting to follow through an asymptotic, large - Ö , analysis for the slowest (thermal) flame in the present situation, which is governed by (2.23). The pre-heat domain, essentially the part of solution-curve A.1^ in Fig. 1 which has positive slope, is given by

'^-•^° '

C\.--^')'-^

5.3

and the flame Mach numberTW is given by

It is noteworthy that T ^ . increases slowly with time (cf(3.26)) while T ^ - T e > and hence ' \ _ ~ T p » descreases, as can be seen from (3.24) and the fact

that Xflf. is fixed while X ^ increases. Thus (5.4) shows that the thermal flame on an unbounded constant-density atmosphere must be accelerating, although necessarily slowly according to the present theory.

References

Clarke, J.F. 1983 Combustion and Flame, 50, 125-138

Clarke, J.F. 1984 "Lectures on Fast Flames"