Detonative and Deflagrative Combustion

Detonative and Deflagrative Combustion RICHARD B. MORRISON, THOMAS C. ADAMSON, JR., and ALEXANDER WEIR, JR.

Downloaded by UNIV OF MINNESOTA on May 6, 2013 | http://pubs.acs.org Publication Date: January 1, 1958 | doi: 10.1021/ba-1958-0020.ch007

University of Michigan, Ann Arbor, Mich.

The combustion of flammable gaseous hydrocarbon mixtures may be divided into two processes: deflagration, or subsonic combustion, and detonation, or supersonic combustion. Some of the experimental and theoretical aspects of propagation, stabilization, unsteady phenomena, and chemical phenomena of deflagration and detonation are reviewed. The significance of flow parameters in dealing with hydrocarbon combustion is discussed. A brief review of pertinent, important work recently performed in these fields is presented.

I he study of combustion in a flow field presents many complex problems. A t the present time, a complete description of the various mechanisms involved seems too complicated to obtain even in the case of the simplest first-order reactions in onedimensional flow. Simplifying assumptions are necessary in order to obtain either an exact solution to the approximate problem or an approximate solution to the exact problem. This paper discusses some of these assumptions and categorizes the various combustion phenomena which exist in subsonic and supersonic flow. Combustion models which consider the thickness of the reaction zone usually ac centuate either heat conduction mechanisms (thermal theory) or the diffusion mecha nisms (diffusion theory) and the models are of necessity of limited value. Simpler models in which the reaction zone or flame front is considered to be an infinitesimally thin dis continuity in the flow, while not simulating exactly the observed conditions, allow the model to be of more general utility and many combustion phenomena become easier to understand because of this simplification. It is the latter approach which is discussed first in this paper—i.e., the combustion process is regarded as a wave phenomenon.

One-Dimensional Waves Figure 1 illustrates the case wherein a planar wave is pictured to be stationary with respect to the observer and the flow, assumed to be one-dimensional, is passing left to right through the wave. If subscript 1 is used to denote conditions upstream of the STATIONARY WAVE

Figure 1. One-dimensional planar wave 69 In LITERATURE OF THE COMBUSTION OF PETROLEUM; Advances in Chemistry; American Chemical Society: Washington, DC, 1958.

70

ADVANCES IN CHEMISTRY SERIES

wave (unburned gas i n the ease of combustion processes) and subscript 2 is used to denote conditions downstream of the wave (burned gas), the following conservation equations may be written to describe the wave: Conservation of mass (1)

PiUi == P2M2

conservation of momentum Pi +

PiUi

2

-

P2 +

P2U2

(2)

2

conservation of energy h + (uiV2) + 0 = h + (u2 /2)


(3)

2

2

where Ρ is pressure; ρ is density; u is velocity; h is enthalpy per unit mass; and Q is heat added at the wave per unit mass. The above equations describe both adiabatic and nonadiabatic waves; in the former case, Q » 0. Equation 2 may be rearranged to give P1M1 2

which upon factoring

(piWi)

2

P2U2 -

-

Pi

2

Pi

and noting from Equation 1 that (pitti) =

(P2M )

2

2

2

becomes (piUi) (l/p, -

l/p )

2

2

-

P

-

2

Pi

or Piu

t

2

- (P - P,)(F! 2

V) t

«

(P2U*)

(4)

2

where F = specific volume Inasmuch as (piWi) must always be positive, it is noted that the numerator and de nominator of the right side of Equation 4 must always possess the same sign. 2

Table I. 1 2

( P - Pi) + — 2

(Vi - V ) + — 2

(Ui - U2) + —

Types of Waves Process when Q « 0 Shock Expansion (impossible)

Process when Q g£ 0 Detonation Deflagration

There are indicated i n Table I two types of possible waves: one, which produces a pressure increase, density increase, and velocity decrease, and the other, which produces a pressure decrease, density decrease, and a velocity increase. F o r nonadiabatic waves the former are classified as detonations or explosion waves, and the latter are classed as deflagrations. Hugoniot Relations. The above classical distinction between detonative and deflagrative combustion has been known to investigators (2, 8, 24) since the latter part of the nineteenth century. Furthermore, it was usual for these investigators to introduce an equation for the conservation of energy in conjunction with those for mass and mo mentum conservation and eliminate the velocity terms from this system of equations— i.e., from Equations 1, 2, and 3. The resulting equations, called Hugoniot relations, and the graphs corresponding to these equations are a convenient and standard representation describing combustion phenomena. The Hugoniot relations, shown in Equations 5 and 6, are identical and may be con verted one to the other by noting that Λ = e + Pv }

hi - Λ -h Q - H(Pi - P )(t> + υι) 2

2

2

(5)

or ei -

e + Q = 2

-V (Pi 2

+ P«)(t>i -

»»)

In LITERATURE OF THE COMBUSTION OF PETROLEUM; Advances in Chemistry; American Chemical Society: Washington, DC, 1958.

(6)

MORRISON, ADAMSON, AND WEIR—DETONATIVE AND DEFLAGRATIVE COMBUSTION

71

where e is the internal energy per unit mass. F o r each value of Q selected and for each set value of the initial conditions, ( P i , Vi), a curve may be drawn for P , v . These curves are called Hugoniot curves and points on the Hugoniot curve are the locus of all possible solutions ( P , #2). Figure 2 illustrates such a curve. Points along the curve in region A to Β represent end states wherein P > P i and v < v hence a region of detonation. Region C to D is a region where P < P i and v > v or a region of deflagration. Solu tions are prohibited in region Β to C. The dashed line is representative of a Hugoniot curve for a higher value of Q. From Equation 4, it may be seen that a straight line from point 0 to some other point M on the Hugoniot curve defines the wave velocity. I n general, this line inter sects the Hugoniot curve at another point, N. This duality of solutions leads to a 2

2

2

2

2


2

Figure 2.

2

h

h

Hugoniot curves

further classification of combustion waves, the one whose end state is at M being called a strong detonation wave, and the one within its end state at Ν being called a weak detonation. The intersection of the tangent to the curve from point 0—i.e., point Τ— defines a Chapman-Jouguet detonation wave. It has been shown by many investigators that this latter wave possesses the particular property that the velocity of the gas i n the wake of the wave is moving at the velocity of sound relative to the front—i.e., at a M a c h number of one. Identical considerations may be applied to the deflagrative branch of the Hugoniot curve. Φ(Μ) Relations. Application of a similar analysis to the deflagrative branch of the Hugoniot curve leads one to the consideration of further fluid dynamic properties of combustion waves. Attention has been focused on the thermodynamic changes across the waves, this having been accomplished by elimination of the kinetic terms i n the conservation equations. If the kinetic terms are retained and the thermodynamic state terms eliminated, many of the obscure points concerning the Hugoniot relations become apparent. Such may be accomplished with extreme ease if, in Equations 1, 2, 3, and 4, the gas is regarded as a perfect gas. For this case the latter system of equations can be In LITERATURE OF THE COMBUSTION OF PETROLEUM; Advances in Chemistry; American Chemical Society: Washington, DC, 1958.


72

solved to Mi

(1 +

(1 +

Μ{ψ

Ί

(7)

yMW


where M, M a c h number, is velocity of gas/(speed of sound in the gas) ; C„ is specific heat at constant pressure; C , is specific heat at constant volume; and y is ratio of specific heats.

i

Ai φ (M) vs. Mach number

Figure 3.

The function on the right side of Equation 7 is often defined as AP ( l + *

(

M

:L

-=-

φ(Μ)—i.e.,

M*)

!

(1 + yMV

)

Figure 3 shows a plot of φ(Μ) vs. M a c h number.

It is noted from liquation 8 that when:

M

- 0

φ (M)

= 0

M

=

1

Φ'Μ)

=

M

«

«

*(Ai) =

1

j — ^ y -

27

1

2

The plot of φ(Μ) represents the locus of all possible solutions for a one-dimensional wave whether the process is adiabatic or nonadiabatic. T o clarify some of the properties of this function consider a stationary combustion front possessing an incoming initial M a c h number and a value for Q/C T. Then from Equation 7, it follows that there will be a solution for the final M a c h number, i l / , as long as the value of the left side of the equation lies between 0 and 1/(2(7 4- 1)). In the range of values from (y - 1)/(2γ ) to 1/(2(7 -f 1)) there will be two solutions for M> (Figure 3). A positive value of Q/C T increases φ(Μ) from its initial value. For the branch of the φ(Μ) curve 0 < M < 1 this corresponds to an increase in Λ/ and for M > 1 to a decrease in M . Writing Equation 2 in a more suitable form for this discussion, we obtain Equation 9 P

2

2

P

I\

1 4-

yMx*

Pi

1 +

7Λ/2

2


(9)

73


It follows from Equation 9 that when .

Mi

< M

2

Pi

>

Pi

Mi

> M

2

Pi

1 to that of detonations. Point Β at which M - 1 is the Chapman-Jouguet point. It is seen from the above that detonative combustion

is a supersonic phenomenon (M > 1) and deflagrative combustion a subsonic phenomenon ( M < 1). Equation 7 may be simplified to +

CT P

φ{Μι)

ol

{

m

where T i, the stagnation temperature at state 1 (Figure 1), has replaced the static temperature, 7Ί. It would seem that φ(Μ ) could be made arbitrarily large by making Q/CpToi sufficiently large. Such is not the case, however, for the solution for M becomes imaginary when the value of φ(Μ )[\ 4- (Q/(C T i))] exceeds 1/(2(7 + 1 ) . If φ(Μ ) is replaced in Equation 10 by this maximum value of 1 /(2(φ 4- 1)) and solved for Q/C T , then 0

2

2

ι

p

0

2

p

-

-2-, ^pl ol

or in terms of

(Π)

1 ) 2

2(7 4- D(Mi*)(l + , Î\/ ι ι Ύ

oi

Ί-γ^Μιή ±

Q/C Ti P

/_SL\ \C Tjuuùùnz p

=

(Ms -

p.

2(7 4- D ( M i 2 )

If values of Q/C Ti or Q/C T„i greater than the limiting value are insisted upon, then initial conditions must change to be consistent with the conservation equations. I n combustion work, the latter is referred to as "thermal choking." The same processes traced on the Hugoniot curve may also be traced on a φ(Μ) curve. In this way the fluid dynamic aspects of each process are made more perspicuous. The various possible processes are shown in Figure 4 and in Table I I . p

p


74


Table II. Initial State A A A


Ε Ε Ε

Processes of Combustion Waves Classification Weak deflagration Chapman-Jouguet deflagration Strong deflagration Weak detonation Chapman-Jouguet detonation Strong detonation

Final State

Β

C D D C

Β

Not a l l the processes indicated in Table I I or Figure 4 are feasible. A process from Ε to A is an adiabatic process from supersonic conditions to subsonic conditions and is recognized as a shock wave. The entropy for this process increases from Ε to A, hence the reverse process from A directly to Ε entails an entropy decrease and is impossible. A strong deflagration, A to D, is therefore impossible except via C, a path involving an exothermic process from C to D, followed by an endothermic process D to E. It seems unlikely that such a combustion process would be found in nature, although it is not impossible. The process from A to Β is a weak deflagration, the type of combustion wave usually encountered. This combustion occurs at a rate which is governed to a large extent by the multiple diffusion processes within the neighborhood of the flame front. Such processes are leisurely and the propagation rates or "flame speeds ' associated with such combustion are on the order of a few feet per second. The process from A to C is a Chapman-Jouguet deflagration and while possible, it is not usually encountered as a deflagration wave. Flame speeds corresponding to point A are usually much higher than those actually observed. I n combustors employing flame stabilization devices, it is possible to reach point C (thermal choking), but this should not be confused with the case of a Chapman-Jouguet deflagration wave. The process from Ε to Β v i a A is a process made up of a shock from Ε to A and a deflagration from A to B. I n this case, the high temperatures necessary for initiating the combustion are supplied by the shock rather than by the diffusion processes previously mentioned. This explanation for detonation phenomena was advanced independently b y Chapman, and by Jouguet, around 1900. If Q/C Ti is sufficiently large to carry the process to point C, this is a Chapman-Jouguet detonation. A s a consequence of the physical means by which detonations are produced, this is the type of detonation usually encountered. The process from Ε to D, that of a weak detonation, is open to much discussion. If Chapman and Jouguet's explanation (8, 2J() of a detonation as a shock plus a deflagration is used, the process is composed of a shock from Ε to A and a strong deflagration from A v i a β to D . The strong deflagration, as mentioned earlier, is usually regarded as impossible, hence barring the existence of a weak detonation. A process from Ε directly to D, however, is permissible. Work on condensation shocks wherein heat is liberated by the condensation to a supersonic stream are at present used as examples of a weak detonation. It is possible to imagine other sets of circumstances under which this process could be induced. F o r example, a supersonic stream of photosensitive material might be triggered into releasing energy upon passing a light beam. Whether the latter examples are weak detonations depends upon the definition one wishes to choose to describe the phenomenon. A Chapman-Jouguet detonation wave may be described as an exothermic supersonic wave that propagates itself at the minimum possible velocity consistent with the con servation laws, whereas a Chapman-Jouguet deflagration is an exothermic subsonic wave that propagates itself at the maximum possible velocity consistent with the conservation laws. 1

p

Finite Reaction Zone Theories Detonation. Past investigators have usually considered that a Chapman-Jouguet detonation wave is the only stable detonation wave that normally exists and have ex pended most of their analytic efforts i n describing the change i n thermodynamic properties that occurs across this wave. This is accomplished by use of the equations for conserIn LITERATURE OF THE COMBUSTION OF PETROLEUM; Advances in Chemistry; American Chemical Society: Washington, DC, 1958.



75

vation of mass and momentum together with a suitable representation for conservation of energy. The extremely high temperatures i n the wake of these waves complicated the analysis due to the lack of sufficiently exact data for high temperature gases. I n later years, with more complete information in this field, calculations have been made with good accuracy which predict detonation velocities i n very close agreement to ex perimental measurements. Such calculations are usually lengthy and tedious. Recently, efforts have been made to describe the structure of the detonation front where the front is no longer regarded as a discontinuity but as a region through which thermodynamic and hydrodynamic properties change continuously. I n most of these analyses the shock is assumed to occur without producing appreciable chemical reaction; then a deflagration is assumed to occur. Sometimes, the assumption has been made that the pressure remains constant throughout the zone of chemical reaction, but it is apparent that the pressure must decrease during this combustion process by a factor of nearly two. Other investigators have attempted to trace the process on Hugoniot curves together with Rayleigh lines. While such a model would describe processes for ex tremely thick fronts of slowly reacting mixtures, experimental evidence indicates that the reactions are quite rapid with severe temperature gradients. Rayleigh, Fanno, and Hugoniot curves can be used only to give the locus of end points of possible detonation or deflagration waves. They cannot be used to study the structure of any process where thermal, or velocity, or concentration gradients are all important. The Chapman-Jouguet wave is usually considered as the only stable detonation wave that exists, but actually the type of wave encountered depends only on the boundary conditions imposed. I n flame tubes where detonations are generated from accelerating flame fronts, Chapman-Jouguet detonations will ultimately develop when the tube is of sufficient length. If, however, a detonation is forced by a piston in the tube behind the wave or if one is stabilized i n a highly supersonic stream in a manner similar to a shock i n a supersonic wind tunnel, then, strong detonations can be realized. The latter case of a standing detonation wave has been investigated analytically by Rutkowski and Nicholls (35) and found to be feasible for strong detonations as well as for ChapmanJouguet detonations. Strong detonations can be discussed qualitatively on Hugoniot or φ(Μ) curves but an additional classification is necessary if one is to analyze these waves quantitatively. A convenient means is provided by the introduction into the conservation equations of the function (1), /

η

\

2

^ + W W ) r%" W

- i ) .

P

( 1 3 )

where M initial M a c h number, is Ui/a (Figure 1). For this analysis, assuming a perfect gas, expressions for the thermodynamic state changes may be written in terms of the initial M a c h number and Q/C T with the following results: h

x

P

X

=

r\ Bl _ Pi -

7 + 1 F_

y + 1 &-[·

+;¥i

Detonative and Deflagrative Combustion

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