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Ends and means in flame theory
12
STRUCTURE AND PROPAGATION OF LAMINAR FLAMES
2
ENDS AND MEANS IN FLAME THEORY By D. B. SPALDING 1. Purposes of this Paper Although the research chemist and the design engineer both need to predict flame phenomena theoretically, there is at present little general understanding of what can and cannot be achieved by flame theory. This paper gives a brief review of current achievements for the benefit of nonspecialists (sections 2 and 3). The whole field of theoretical flame problems which interest chemists and engineers is very large. Review shows, however, that almost all the work has been done in one corner of this field : the one-dimensional laminar flame. One result has been that the theoretical methods developed there are useful only for this flame; continued neglect of two- and three-dimensional problems is therefore encouraged. In section 4 of this paper some methods are suggested which, while being applicable to the one-dimensional flame, can also be extended to flames of more practical importance. Some of these methods are being developed in the author's laboratory. I t is hoped that this section will interest the specialist in flame theory. 2. The Aims of Flame Theory (2.1)
THE
NATURE
OF FLAME
THEORY
Three bodies of knowledge form the basis of the theory of flames: (a) Reaction kinetics: the relation of the volumetric rates of chemical change to the local gas composition and temperature in a differential volume. (b) Transport phenomena: the relations of the fluxes across the boundaries of the differential volume, of work, heat and matter, to the gradients of velocity, temperature and composition normal to those boundaries. (c) The conservation laws of matter, momentum and energy, together with the thermodynamic properties of materials. These laws relate the fluxes to the volumetric rates of change. Flame theory has the task of combining (a) (b) and (e) so as to predict what sort of flame will ensue when given reactants are caused to flow into a given combustion chamber under given conditions; the general statements about the infinitesimal volumes have to be added up to see what they demand in the behavior of the particuIar whole.
(2.2)
PURPOSES
OF FLAME
THEORY
Flame theory has two main purposes. The first is practical. Combustion chamber testing is expensive. A theory which reliably predicts the result of an experiment, without the need to e~)nduct the experiment, speeds and cheapens development. In this branch of flame theory, the reaction kinetic and transport data are regarded as known; the interest is in the effect of changing the geometry and flow conditions. Usually the assumed reaction kinetics have to he simplified. Flame theory is also used with a fundamental aim in view. Measurements on simple flames are often easier than direct measurement of reaction kinetic data. A reliable theory enables the latter to be deduced from the former. In this branch of flame theory, the geometry and flow conditions are usually as simple as possible; the interest is in the effect of changes in the assumed reaction kinetics, and full justice has to be done to complex kinetic schemes. Theoretically considered, the differences between the practical and fundamental branches of the subject lie only i.n the varying number of dependent and independent variables, and in the nature of the boundary conditions. Similarities exceed differences. 3. Present Achievements of Flame Theory The object of this section is to state what types of flame have been dealt with so far, to classify the theories of the various investigators, and to point out differences and similarities. (3.1)
THE ONE-DIMENSIONAL LAMINAR PRE-MIXED ADIABATIC F L A M E
The inner cone of the Bunsen burner has claimed the greatest attention among investigators. The problem is this: given data about the reaction kinetics and transport properties of the mixture, what will be the distribution of temperature and gas composition along a line normal to the flame, and what must be the velocity of flow along this line? Since these questions can also be answered experimentally, comparison of theory and experiment is a check on the data. In the simplest case the reaction is single-step
ENDS AND MEANS IN FLAME THEORY
and the resultant diffusivity equals thermal diffusivity, for then it suffices to consider the temperature variation--composition being linked to temperature in a known way. A single differential equation must be solved. If the diffusivities are not equal, both temperature and composition must be watched; then two simultaneous equations must be solved. Increasing complexity of the reaction increases the number of equations by one for each new reactant that is involved. Each of these differential equations has the same independent variable, viz. distance. The subject of applied mathematics is highly developed and provides many techniques which may be applied to these equations. The apparent differences between "theories" of flame propagation, which so bewilder the onlooker, are usually only differences of notation or of procedure; each investigator chooses the mathematical tool which is best suited to his own situation. These differences of method have often obscured the fact that the differences between the solutions obtained in given cases are very slight. Differences in the solutions exist because every mathematical method is to some extent approximate. In some methods approximation is inherent, for example where it is desired to present the result as an algebraic formula. Such are the methods of Zeldovieh and Frank-Kamenetsky,~ Boys add Corner, ~ Adams, 3 and yon Kgrm~n and Penner. 4 In other methods, where the result is to be presented as a set of numbers valid for particular cases, the degree of approximation can be reduced to zero merely by taking more trouble in computation. Such are the methods of Hirsehfelder, Curtiss, and Campbell, ~ who integrate the equations numerically; of Hirschfelder's associates (Klein 6 and SchatzkF), who use respectively solutions by successive quadrature and by relaxation; and of Spalding} who obtains the steady flame speed as an asymptotic solution of the unsteady state equations. All of these methods have been applied with success to problems where there are only two dependent variables, usually the ease of a singlestep reaction with unequal diffusivities. All of them become difficult to use when there are more dependent variables, as in chain reactions of the hydrogen-bromine type, to take a popular example. Solutions to these complex flame problems have been delayed by the search for the best mathematical procedures, but the certainty of success is not in question. In addition to the above-mentioned researches,
13
"theories" of the one-dimensional flame exist which do not start from consideration of the fundamentals mentioned in section 2.1, but attempt to take a short cut by assuming, in the light of intuition, that a single property or process is dominant in flame propagation. Such, for example, are the "diffusion theory" of Tanford and Pease, 9 the "excess enthalpy hypothesis" of Lewis and von Elbe, 1° and the "H-radical momentum theory" of Manson. 1' In the view of the author it is now time to put aside these "shortcut" theories. Although the insight which some of them have afforded in the past has been valuable, they now confuse rather than simplify. The effort that is still wasted in discussing the relative merits of "thermal" or "diffusional" theories could be more usefully employed. (3.2)
M U L T I - D I M E N S I O N A L STEADY L A M I N A R P R E M I X E D FLAMES
Very few problems of this type have been studied. Von Kgrmgn and Millgn'2 have considered effects near the wall when a flame propagates through gas at rest in a tube. Marble and Adamson13have studied the mixing and reaction of adjacent semi-infinite streams of equal velocity and ultimate composition but different temperature. The transient method mentioned in 3.1 has been applied to the mixing of finite jets of equal velocity.8 No other work of this type is known to the author. ~ (3.3)
T U R B U L E N T FLAMES
There are no turbulent flame theories in the sense of this paper, b Tentative extension of lanfinar results to turbulent conditions is usually made by postulating augmented transport properties. This is made necessary by present ignorance of the proper differential description of turbulence even without combustion; the relations between time-mean reaction rates and time-mean concentrations also await investigation. Most practical flames are turbulent. Satis"Karlovitz, Knapschaefer, and Wells14 discussed flames in the presence of velocity gradients in a "short-cut" theory, assuming the laminar flame speed to be a constant of the mixture, irrespective of the local flow. b "Short-cut" theories by Karlovitz, Denniston, and Wells 15 and Scurlock and Grover 16exist. They assume that turbulent flames are composed of laminar ones. Many doubt whether this is a helpful assumption, e.g., Summerfield, Reiter, Kebely, and Maseolo. x7
14
STRUCTURE AND PROPAGATION OF LAMINAR FLAMES
factory ways of writing the differential equations governing turbulence must be discovered if theory alone is to predict flame phenomena. Section 4.32 of this paper indicates how a combination of theory and model experiment makes prediction possible already. (3.4)
I D E A L I Z A T I O N S OF F L A M E S I N S T E A D Y - F L O W ENGINES
An idealized flow system which has features in common with some engine flames is the homogeneous reaction zone of Avery and Hart, 33 in which variations in both space and time are assumed absent. Longwell, Frost, and Weiss34 have shown how, for a single dependent variable, the gas condition can be calculated as a function of the steady-flow rate. Cases of two dependent variables have also been considered in connection with chain reactions8b and radiation heat transfer2 5 In eact-~ case it is found that the flame must be extinguished when the flow rate exceeds a critical value. The homogeneous reaction zone represents a simple limiting case, and future work must account for the spatial inhomogeneities arising in real combustion systems due to the finite transport rates. Spalding8~ deals with one such case: the cylindrical vortex with uniform diffusivity. This exhibits extinction phenomena also, as must all steady-flow systems whether homogeneous or not. (3.5)
NON-PRE-MIXED
FLAMES
Particularly simple (aerodynamic) considerations govern flame position when fuel and oxidant flow in separately. Pioneer investigations in this field by Nusselt TM and Burke and Schumann~9 have been followed by Hottel} ° Fay, 2~Spalding,22 and many others. The rate of fuel consumption is calculable without knowledge of the reaction kinetics. But few workers 23, 24 have studied the extinction of this flame type, when chemical reaction becomes controlling, yet this phenomenon sets the limit to aircraft engine performance at high altitude. (3.6) COMMENTS The research chemist has more reason to be pleased with the present achievements of flame theory than has the engineer, because the Bunsen burner is his chosen laboratory tool. Even so, serious use of theory in making reaction kinetic deductions from flame measurements has scarcely begun, partly because the right theoretical problems have not been solved, but still more because
the available solutions are cast in forms which chemists find obscure. The engineer on the other hand will look in vain for adequate theories of flame propagation in boundary layers, of the effect of velocity ratio on propagation from pilot flames, on ignition in static and moving gases, etc., to name only laminar flame phenomena of interest to him. Nor will current work on one-dimensional flames throw much light on these multi-dimensional ones; entirely different techniques appear suitable for the latter, as will be seen. The final section of this paper is intended to interest the specialist in the multi-dimensional problems by suggesting theoretical and other methods which are not restricted to the one-dimensional flame. 4. " S o u r c e - a n d - S i n k " M e t h o d s
The mathematicM theory of heat conduction with heat sources but without convection is highly developed. It is built on the Poisson equation which involves second derivatives of temperature with respect to distance, and a source term. Analytical solutions often involve the integration of Green's function for the prescribed boundary conditions (e.g., Carslaw and Jaeger2~). Numerical solution by relaxation methods is an established technique?6. 27 Analogues of various kinds have been widely used. ~ If there is only one independent variable, numerical integration is particularly easy (e.g., Hartree29). These techniques will here be given the generic name "source-andsink methods," since they all involve the source concept explicitly or implicitly. If flame problems can be expressed in terms of Poisson-like equations the above techniques become available. This is the aim of the methods to be described below. Because of the complicated nature of the source (reaction rate) terms, analytical methods are often out of the question; nevertheless, the Green's function approach may be used in iterative numerical methods, and forms the basis of the analogues. Transformations are indicated which in simple cases eliminate the convection term from the flame equations, so making them of the Poisson form. Discussion of these methods begins with the one-dimensional flame because of its familiarity. (4.1)
NUMERICAL PROCEDURES
(4.1.1) One-dimensional flames For simplicity, a flame with one dependent variable (temperature) will be considered. In
15
ENDS AND MEANS IN FLAME THEORY
terms of a dimensionless space co-ordinate y, the equation of conservation is d2r
dr
dy 2
dy
XO(r)
(1)
Terms on the left-hand side represent molecular transport and convection respectively; ¢ is a function of r and represents volumetric reaction rate; X is related to burning velocity, and represents the eigenvalue to be determined. Most workers 14 have followed Zeldovich and Frank-Kamenetsky in transforming the equation to new variables p
=-
and r, giving
The author circumvents the difficulty, which arises whatever the form of the equations, by introducing a fictitious "catalyst-plug" at the hot boundary which always supplies a small but finite amount of heat to the flame. It may be thought of as an adiabatic porous catalyst into which the gas flows and which brings about thermodynamic equilibrium at its upstream surface. Solutions for the value of X are then obtainable for any finite temperature gradient at the plug surface. The required value for zero temperature gradient cannot be obtained directly; it must be determined by extrapolation. (4.1.3) Discussion of Equation (1) in terms of
dp
p ~
-
p =
-x,(r)
I t is this transformation which, in the past, has turned flame theory away from the main stream of heat and mass transfer theory. It is advantageous for approximate methods but, since the equation is now first order, is not amenable to the "source-and-sink" techniques. It does not appear capable of extension to multidimensional ~Iames. The methods to be described do not make this transformation but operate either directly on Equation (1) or on transformations of it which are still of second order. (4.1.2) "Catalyst-plug" boundary conditions An often-discussed difficulty of flame theory arises because the analytical expression for reaction rate has a finite, though very small, value in the gas stream before it approaches the flame. This difficulty is avoided by some authors by assuming the reaction rate terms to be exactly, instead of nearly, zero below a certain temperature. Hirschfelder, Curtiss, and Campbell s have instead introduced a fictitious "cold plug" at the upstream boundary and suppose that the fame must always lose a small but finite amount of heat to this plug. A less widely known difficulty of flame theory arises at the hot boundary where, in the real flame, temperature and concentration gradients become zero. Since the reaction rate is also zero there, Equation (2) shows that
(dp)
0 indeterminate ~=,=1-x5=
sources and sinks
(2)
(3)
Special devices must therefore be used before the integration from the hot boundary can begin, e.g., expansion in series.
Solution of Equation (1) involves obtaining the temperature distribution through the flame. If the R.H.S. of Equation (1) were everywhere very small (except at the e~talyst plug surface, DIRECTION
o~
.._
~,o,,,--
,-.,,~' \
Jl
~ i
~
-i r
UPStrEAM CATALYGT PLUO
~-
- =~
I
I DISTANCE
~,
I ~ =o
l
1
I
FIG. 1. Temperature distribution in one-dimensional gas stream if X is very small (negligible reaction except at plug face). of course), this distribution would simply be an exponential decay curve upstream of the plug and a horizontal line downstream of the plug (Fig. 1). This distribution is the effect of a unit plane source of heat normal to the stream direction, situated at the plug surface. If the source were situated farther upstream, the distribution curve would merely be shifted correspondingly. Now the R.H.S. of Equation (1) is linear, so that, if two sources are present, the temperature at any point is the sum of the temperatures which would prevail there due to each source individually. In particular the temperature distribution due to a unit source at y = ~ and a unit sink at the hot plug (y = 0) is shown in Figure 2. Curves such as this are easily calculated for any value of 7. Suppose their ordinate is G(y,~). Now the R.H.S. of Equation (1) states how the heat sources are distributed albeit with respect to temperature. The foregoing discussion shows that the temperature distribution which solves the equation is the sum of the effects of all the sources in the flame region, these effects being determinable from the curves such as Figure 2
STRUCTURE AND PROPAGATION OF LAMINAR FLAMES
16
and from the function ~. In mathematical form this is stated by re-writing Equation (1), together with its boundary conditions, as an integral equation: 0
P
--
~
< y
~-(y) = e ~ + ~ / d-
¢(~-) d~
G(y,v),
where G(y,~) = 1 - e~, G(y,~)
e~-~
=
--
Lnv~y<0 e~,
Ln y ~ ~
<
0
(4)
G(y,~), the integrating kernel of Equation (3), is the Green's function of the problem.
]
DIRECTION __
oF
"-J I
~LOW ~
"_J
a first approximation, the corresponding ¢ is calculated, and Equation (4) is used to give a new approximation for T. If this is done point by point, the space having first been divided up into finite intervals, the procedure is essentially one of relaxation; if, for speed, sources are adjusted simultaneously at all points, the procedure is essentially one of successive quadrature performed by means of tables of G(y,~). This procedure, or variants of it, can be applied to the general case where there are several dependent variables. The arithmetic, though simple, is heavy; automatic devices soon become necessary. Apart from digital computers, analogue devices such as those mentioned later may be used; indeed analogues represent the most suitable means of carrying out solutions of this nature. (4.1.5) Flames in more than one dimension
~..,.t
"T"--7~,
*
*
'
'
'
t
*
l
v
t
'
~,~=0
FIG. 2. Temperature distribution G(y, ~) in gas stream caused by unit heat source at y = ~ and unit heat sink at y = 0 (plug surface). ~ = - 1 in this case. ~'DIRECTION
,
~*
-~
,
G • I-'¢ P
t
,
p
I
~
r
I I
,
i
i
VARIAE~LE 50URCE POSITION ~
1JNIT HEAT S~NI~
FIG. 3. Influence function (Green's function) G(y, ~) showing temperature rise incurred at y when a unit source is placed at ~ and a unit sink at the catalyst plug (~ = 0). y = - 1 in this case. The idea of a Green's function can perhaps be made acceptable to engineers by pointing out the similarity of Figure 2, for example, to a bending moment diagram representing the effects at all points (y) of a beam when a force is placed at a given point (7). Alternatively G(y,~) may be plotted as in Figure 3. This curve represents the effects at one point (y) for various positions (7) of the source; its engineering analogy is the "influence line" of structural theory. It is notable that if the source is upstream of y its effect at y is independent of source position.
As far as the one-dimensional flame is concerned, the above method is just one among many. It is presented here because it can be extended, in either its numerical or analogue form, to multi-dimensional flames--that is, to partial differential equations. The only essential change lies in the extended nature of the Green's function which must represent the influence of a heat source at point A on the temperature at point B, wherever these points are situated. If an analogue is used it is not necessary to evaluate these influences explicitly; they are "built-in" to the apparatus. (4.1.6) Transformations of the conservation equations Numerical integration, relaxation techniques, and analogues all deal with second-order differential equations most happily when the first derivative is absent. A number of transformations make this possible provided that, although heat and matter flow may be multi-dimensional, the general stream velocity is uniform. They will be indicated for the one-dimensional flame only. The dependent variable in Equation (1) can be transformed by putting =
t e ~/~
(5)
Then Equation (1) becomes (4.1.4.) Methods of solution Since ~b(r) in Equation (4) can be given definite values only when r is known as a function of y, iterative methods of solving Equation (4) suggest themselves. A r distribution is assumed as
d2t 1 = - t - Xe-'J/~(r) dy 2 4 --
(6)
Thus the first derivative has been eliminated at the cost of introducing a new source term and
17
ENDS AND MEANS IN FLAME THEORY
making the original source term depend upon position. I t is still amenable to the various sourceand-sink techniques, and is particularly suitable for expression in finite difference form and solution by relaxation. Computation is unnecessary at points so far upstream that chemical reaction has become negligible, for the exponential nature of the solution is known there; that y goes to infinity is not a serious disadvantage. This transformation has been used to solve analytically the heat conduction problems with moving heat sources encountered, for example, in welding, a° As an alternative, the independent variable may be transformed by putting =
e~
(7)
Then Equation (1) becomes d~r d~2
-
x
¢ (~) - ~2
(8)
This has the advantage that the independent variable only varies between 0 and 1 in the whole flame region. I t is usually preferable to Equation (6) for one-dimensional problems. A different but related transformation which is of relevance to the electric analogue follows from multiplying Equation (1) by e-v giving --
dy
e -'a
=
e-U¢(r)
(9)
In finite difference form this is easily simulated electrically, and y of course varies from 0 to --
o o
Each of these transformations has its own special features. Which to use depends on the details of the problem and the solution technique which is to be used. Source-and-sink methods are applicable in each case. None of these transformations works when the flow velocity is not uniform. (4.1.7) Other methods
applications
of
source-and-sink
The method of solving transient problems mentioned in section 3.1 involves the use of sources and sinks; it makes use of the linearity of all terms of the equation, except that for reaction rate, when distance and time are the independent variables. The distribution of chemical reaction in steady diffusion fames (section 3.4) can be studied by obvious modification of the techniques mentioned above. In this case the source and sink magnitudes depend on position in a new way, only hav-
ing finite values in a restricted region. The boundary conditions are such as to make numerical integration very difficult; relaxation or successive quadrature are possible, however, as is the use of an analogue. (4.1.8) Closing note The above formulations have not yet been tried with any very complicated flames. They are mentioned here to indicate that there are many more ways of solving flame problems theoretieally than the current literature of the subject would suggest. (4.2.)
ANALOGUE METHODS FOR
PRESCRIBED
PROBLEMS
WITH
FLOW
When the flow is not uniform, as for example in laminar boundary layer problems, the calculation of the Green's functions is a formidable task. This is avoided if a device is constructed which obeys the same differential equation as does the flame, i.e., if an analogue is used. An electric analogue for example must be so constructed that a unit current source at point A produces a voltage increase at point B which is proportional to the temperature rise at the corresponding point b in tile flame when a unit heat source is present at the corresponding point a. Since they automatically satisfy the differential equation and the boundary conditions, analogues may be regarded as having "built-in" Green's functions. Usually the simulation must be of a finite-difference nature, at least as far as the source terms are concerned. Electrical and other analogues for heat transfer and diffusion problems are well known 2s and will not be described. Here some modifications needed for solving flame problems will be mentioned. I t is assumed that the flow pattern and the transport properties are known beforehand. (4.2.1) Fully electrical analogues Electrical resistance networks with controllable current inputs at the mesh-points obey Poisson's equation in finite difference form. Transformations of the flame equations into the Poisson form have been described above. The analogue for Equation (6), for example, is a line of equal resistance in series representing the left-hand side; a set of equal resistances connecting each junction of the line to ground, representing the 1/~t term; and a controlled current input at each ]unction to represent the reaction rate term. One end of the resistor line is held at
18
STRUCTURE AND PROPAGATION OF LAMINAR FLAMES
a fixed voltage above ground; this corresponds to the hot plug. The other end, theoretically at infinity, is at ground voltage. The current sources are adiusted successively until they accord with the local voltages for an arbitrary value of h. The current taken from the terminating battery is a measure of the temperature gradient at the hot plug. For problems with many dependent variables, many such networks are required, interconnected at least through the input currents. With several independent variables, several networks are cross-connected electrically. Equation (7) can be simulated more easily: the line of resistance is finite in length, and the leaks to ground are absent. Equation (9), on the other hand, requires an infinite line of series resistances, but these decrease in value the greater the distance from the hot plug; leakage resistors are absent. This technique and others are described by Johnson and Alley. 31 Problems with nonuniform velocity cannot be described by Poisson-type equations by any transformation known to the author; the convection terms cannot be eliminated. If network analogues are to be used for these problems, they must incorporate resistances which have different values according to which end they are seen from. Suitable devices are described by Soroka; ~ they involve electronic equipment. (4.2.2) Electro-thermal analogues When simulation of convection terms is unavoidable, a thermal resistance network may be used in place of the electrical one. Mesh-points are connected by metal tubes; thermal conduction occurs along thin walls while fluid flows through them with a distribution of velocities which simulates that appropriate to the flame. Electrical resistances wound around the tubes provide concentrated heat sources, and thermocouples measure the local temperature. The external control relates the rate of heat input to the temperature, which represents the dependent variable of the flame problem. This technique has not so far been used. AIthough somewhat clumsy, it offers an inexpensive means of investigating flame phenomena in boundary layers where the velocity distribution, although complex, can be calculated without knowledge of the flame pattern. (4.3)
A N A L O G U E S FOR F L A M E S I N COMPLEX A E R O DYNAMIC F I E L D S
In engineering equipment, flow is usually furbulent and multi-dimensional, and moreover in-
volves regions of mixing and recirculating flow which aerodynamics can at present neither predict nor describe in detail sufficient as a basis for flame theory. This difficulty must be surmounted if flame theory is to serve practical ends. Two ways of doing so are here described. The first is simple in concept but difficult in practice; it has not been used. The second involves an analogue which avoids these difficulties; it has been in operation since mid-1954. (4.3.1) Experiment on models, combined with
calculation An early step in one of the theoretical methods (section 4.1) is the calculation of the Green's function relating temperature increment at one p o i n t to heat source increment at another. This is impossible for complex turbulent flow fields. However, these influence coefficients could be determined experimentally, by making a dynamically similar model of the flow system (combustion chamber) and, for example, measuring the concentration of a tracer at position A when a thin stream of tracer is supplied at B; this would be done at a sufficient number of point-pairs to characterize the flow. The next step, iterative solution of the conserration equations in integral form, then follows. Though straightforward, it demands much computation even in relatively simple cases. Thus both the initial experimental step and the followng computational step are time-consuming. (4.3.2) Electro-thermal analogue The experimental step in the above method is eliminated and the computational step shortened by use of an electro-thermal analogue. A model of the combustion chamber is constructed through which air flows simulating the reactive gas. Distributed through the model are thermocouples measuring the air temperature and electric resistance wires providing heat. External controls relate local heat input to local air temperature in a way similar to the prescribed reaction rate function. The flow in the model automatically ensures that temperature increments resulting from a heat source increment are distributed through the ftow space in the correct manner. The analogue constructed at the Cambridge University Engineering Laboratory and now operating at Imperial College3~ solves two-dimensional turbulent flow problems with a single dependent variable. The reaction=rate function used is rectangular in shape for simplicity. Control of heaters is automatic. Each solution takes
ENDS AND MEANS IN FLAME THEORY
about one minute. The heater-thermoeouple elements are permanently in position; changes in combustion chamber shape are made by interposing metal baffles. The main limitations of this type of analogue are inability to simulate the density changes consequent on reaction, and inability to account for the effects of two-phase flow (spray combustion). Separate introduction of fuel (assumed gaseous) and air is simulated by a preliminary temperature-distribution experiment with a heat source simulating fuel injection; the local reaction rate is adjusted to accord with the local fuel/air ratio. Heat losses and complex reaction schemes may in principle be simulated by simultaneous use of several analogues with cross-coupled source terms. Further development will be directed to these points, to use of three dimensions and the better simulation of reaction-rate-temperature functions.
19
ternational) on Combustion, p. 190. Baltimore, The Williams & Wilkins Co., 1953. 6. KLEIN, G. : University of Wisconsin SQUID--1
(1955).
7. SCHATZKI,T. K. : University of Wisconsin CM 853 (1955). 8. SPALDING,D. B. : (a) Aircraft Engineering, 25, 264 (1953) ; (b) Phil. Trans. Roy. Soc., A249, 1 (1956). 9. TANFORD, C., AND PEASE, R. N.: J. Chem. Phys., 15, 433 (1947). 10. LEwis, B., AND YON ELBE, G.: Combustion, Flames and Explosions. New York, Academic Press, 1951. 11. MANSON, N.: Rev. Inst. Fr. Pdtrole, ~, 338 (1949). 12. yON KXRMXN, T., AND MILLAN, G.: Fourth Symposium (International) on Combustion, p. 173. Baltimore, The Williams & Wilkins Co., 1953. 13. MARBLE, F. E., AND ADAMSON,T. C.: Selected Combustion Problems, p. 111. London, Butterworth's, 1954. 5. Conclusions 14. KARLOVITZ, B., KNAPSCHAEFER, D. H., AND WELLS, F. E.: Fourth Symposium (InternaThe theoretical flame problems dealt with by tional) on Combustion, p. 613. Baltimore, the design engineer and research chemist have The Williams & Wilkins Co., 1953. much in common, differing only in the relative 15. KARLOVITZ, B., DENNISTON, D. W., AND number of dependent and independent variables. WELLS, F. E.: J. chem. Phys., 19, 541 (1951). Methods are becoming available for solving 16. SCURLOCK,A. C., AND GROVER, J. H. : Selected most of these problems. The analogue methods Combustion Problems, p. 215. London, Butwhich are most suitable for complex flow patterns terworth's, 1954. are similar in spirit to numerical methods which 17. SUMMERFIELD,M., REITER, S. H., KEBELY, K., may be used for solving the more familiar probANDMASCOLO,R. W. : Jet Propulsion, 25, 377 (1955). lems of the one-dimensional laminar flame. 18. NUSSELT, W.: Z.V.D.I., 60, 102 (1916). 6. Acknowledgment 19. BURKE, S., AND SCHUMANN, T.: Ind. Eng. Chem., 20, 998 (1928). The electro-thermal analogue described in 20. HOTTEL, H. C.: Fourth Symposium (Internasection 4.3.2 has been developed as part of an tional) on Combustion, p. 97. Baltimore, The extra-departmental research sponsored by the Williams & Wilkins Co., 1953. Mechanical Engineering Research Laboratory 21. FAY, J. A.: J. Aero. Sci., 21,681 (1954). of the Department of Scientific Research. 22. SPAI~DING, D. B.: Proc. I. Mech. E., 168, 545 (1954). REFERENCES 23. ZELDOVICH,Y. B. : J. Tech. Phys. Moscow, 19, 1. ZELDOVICH,Y. B., AND FRANK-KAMENETSKY, 1199 (1949); trans, as NACA Tech. Memo D. A.: J. Phys. Chem. Moscow, 12, 100, 1296 (1951). 24. SPALDING,I). B. : Fuel, 33, 253 (1954). (1938). 2. BOYS, S. F., AND CORNER, J. : Proc. Roy. Soc., 25. CARSLAW,H. S., AND JAEGER, J. C.: ConducAI97, 90, (1949); CORNER, J.: Proc. Roy. tion of Heat in Solids. New York, Oxford Soc., AI98, 388 (1949). Univ. Press, 1947. 3. ADAMS,E. N.: See I-IENKEL,M. J., SPAULDING, 26. EMMONS, H. W.: Trans. A.S.M.E., 65, 607 W. P., ANDHIRSCnFELDER,J. O. : Third Sym(1943). posium on Combustion, p. 21. Baltimore, 27. DUSINBERRE, G. M.: Numerical Analysis of The Williams & Wilkins Co., 1949. Heat Flow. New York, McGraw-Hill, 1949. 4. VON ]~(.~RM.~N, W., AND PENNER, S.: Selected 28. SOROKA,W. W. : Analog Methods in Computation and Simulation. New York, McGrawCombustion Problems, p. 5. London, ButterHill, 1954. worth's 1954. 5. HIRSCHFELDER, J. O., CURTISS, C. F., AND 29. HARTREE, D. R.: Numerical Analysis, New CAMPBELL, D. E.: Fourth Symposium (InYork, Oxford Univ. Press, 1952.
20
STRUCTURE AND PROPAGATION OF LAMINAR FLAMES
30. I{,OSENTHAL, D.: Trans. A.S.M.E., 68, 849 (1946). 31. JOHNSON, W. C., AND ALLEY, •.
33. AVERY, W. If.. AND HART, t{. W.: Ind. Eng. Chem. 45 1634 (1953). 34. LONGWELL, J. P., FROST, E. E., AND WEISS, M. A.: Ind. Eng. Chem., 1628 (1953). 35. SPALDING,D. B., AND TALL, ]3. S. : Aero Quarterly, 5, 195 (1954).
E. : Rep. 3.
ONR. Contract N6 105, Task Order VI June 1948. 32. SPALDING,D. ]~.: Proc. I. Mech, E., to be published.
3
LIMITS O F I N F L A M M A B I L I T Y
By J. W. L I N N E T T AND C. J. S. M. SIMPSON Introduction Many experimental determinations have been made of limits of inflammability, but less has been done on the theory of the problem, and, at the present time, there is no agreed explanation of why composition limits are observed. Recent work has been concerned mainly with suggesting
BURNING VELOC I TY
COMPOSITION FIG. 1. Probable theoretical variation of burning velocity with composition (diagrammatic). features that might account for the existence of a finite limit of composition; there has been little attempt to see whether the known facts support the suggestions. Why is there this difficulty regarding the existence of limits? The recent steady-state treatments of flame propagation are thorough. 1 However, they include no factor that would account for the existence of limits. Theoretically steady-state
solutions (see also LayzerD can exist over the whole composition range, so that present theories of this type would probably predict a burning velocity curve of the form shown in Figure 1. (Figure 1 is not to scale.) Theory will predict that the burning velocity will be close to zero for a range of compositions near each end of the curve (i.e., the "tails" will be much closer to the composition axis than is drawn). Because of this situation it has been necessary to try to account for limits by additional suggestions. One of these has been to consider the effect of perturbations on the steady-state condition2 It has been shown that for some mixtures a perturbation would die away while for others it would grow and the flame would fail to return to its steady-state condition. Before examining these ideas it is important to consider the question: Does experimental evidence indicate whether an infinite horizontal plane flame moving through a quiescent homogeneous mixture would or would not show a composition limit of inflammability? The answer is not certain for no such flames have been studied. The nearest approach is provided by the circular flat flame burner of Powling,4 Egerton and Thabet 5 and Badami and Egerton? But this is only about 3 in. across so that convection between the hot products from the flame and the surrounding gases may be important. If the answer to the above question were "Yes, such flames would show a limit," a second may need to be asked, namely: Are the limits usually determined close to or far from the limits that might exist for such infinite undisturbed plane flames?
STRUCTURE AND PROPAGATION OF LAMINAR FLAMES
2
ENDS AND MEANS IN FLAME THEORY By D. B. SPALDING 1. Purposes of this Paper Although the research chemist and the design engineer both need to predict flame phenomena theoretically, there is at present little general understanding of what can and cannot be achieved by flame theory. This paper gives a brief review of current achievements for the benefit of nonspecialists (sections 2 and 3). The whole field of theoretical flame problems which interest chemists and engineers is very large. Review shows, however, that almost all the work has been done in one corner of this field : the one-dimensional laminar flame. One result has been that the theoretical methods developed there are useful only for this flame; continued neglect of two- and three-dimensional problems is therefore encouraged. In section 4 of this paper some methods are suggested which, while being applicable to the one-dimensional flame, can also be extended to flames of more practical importance. Some of these methods are being developed in the author's laboratory. I t is hoped that this section will interest the specialist in flame theory. 2. The Aims of Flame Theory (2.1)
THE
NATURE
OF FLAME
THEORY
Three bodies of knowledge form the basis of the theory of flames: (a) Reaction kinetics: the relation of the volumetric rates of chemical change to the local gas composition and temperature in a differential volume. (b) Transport phenomena: the relations of the fluxes across the boundaries of the differential volume, of work, heat and matter, to the gradients of velocity, temperature and composition normal to those boundaries. (c) The conservation laws of matter, momentum and energy, together with the thermodynamic properties of materials. These laws relate the fluxes to the volumetric rates of change. Flame theory has the task of combining (a) (b) and (e) so as to predict what sort of flame will ensue when given reactants are caused to flow into a given combustion chamber under given conditions; the general statements about the infinitesimal volumes have to be added up to see what they demand in the behavior of the particuIar whole.
(2.2)
PURPOSES
OF FLAME
THEORY
Flame theory has two main purposes. The first is practical. Combustion chamber testing is expensive. A theory which reliably predicts the result of an experiment, without the need to e~)nduct the experiment, speeds and cheapens development. In this branch of flame theory, the reaction kinetic and transport data are regarded as known; the interest is in the effect of changing the geometry and flow conditions. Usually the assumed reaction kinetics have to he simplified. Flame theory is also used with a fundamental aim in view. Measurements on simple flames are often easier than direct measurement of reaction kinetic data. A reliable theory enables the latter to be deduced from the former. In this branch of flame theory, the geometry and flow conditions are usually as simple as possible; the interest is in the effect of changes in the assumed reaction kinetics, and full justice has to be done to complex kinetic schemes. Theoretically considered, the differences between the practical and fundamental branches of the subject lie only i.n the varying number of dependent and independent variables, and in the nature of the boundary conditions. Similarities exceed differences. 3. Present Achievements of Flame Theory The object of this section is to state what types of flame have been dealt with so far, to classify the theories of the various investigators, and to point out differences and similarities. (3.1)
THE ONE-DIMENSIONAL LAMINAR PRE-MIXED ADIABATIC F L A M E
The inner cone of the Bunsen burner has claimed the greatest attention among investigators. The problem is this: given data about the reaction kinetics and transport properties of the mixture, what will be the distribution of temperature and gas composition along a line normal to the flame, and what must be the velocity of flow along this line? Since these questions can also be answered experimentally, comparison of theory and experiment is a check on the data. In the simplest case the reaction is single-step
ENDS AND MEANS IN FLAME THEORY
and the resultant diffusivity equals thermal diffusivity, for then it suffices to consider the temperature variation--composition being linked to temperature in a known way. A single differential equation must be solved. If the diffusivities are not equal, both temperature and composition must be watched; then two simultaneous equations must be solved. Increasing complexity of the reaction increases the number of equations by one for each new reactant that is involved. Each of these differential equations has the same independent variable, viz. distance. The subject of applied mathematics is highly developed and provides many techniques which may be applied to these equations. The apparent differences between "theories" of flame propagation, which so bewilder the onlooker, are usually only differences of notation or of procedure; each investigator chooses the mathematical tool which is best suited to his own situation. These differences of method have often obscured the fact that the differences between the solutions obtained in given cases are very slight. Differences in the solutions exist because every mathematical method is to some extent approximate. In some methods approximation is inherent, for example where it is desired to present the result as an algebraic formula. Such are the methods of Zeldovieh and Frank-Kamenetsky,~ Boys add Corner, ~ Adams, 3 and yon Kgrm~n and Penner. 4 In other methods, where the result is to be presented as a set of numbers valid for particular cases, the degree of approximation can be reduced to zero merely by taking more trouble in computation. Such are the methods of Hirsehfelder, Curtiss, and Campbell, ~ who integrate the equations numerically; of Hirschfelder's associates (Klein 6 and SchatzkF), who use respectively solutions by successive quadrature and by relaxation; and of Spalding} who obtains the steady flame speed as an asymptotic solution of the unsteady state equations. All of these methods have been applied with success to problems where there are only two dependent variables, usually the ease of a singlestep reaction with unequal diffusivities. All of them become difficult to use when there are more dependent variables, as in chain reactions of the hydrogen-bromine type, to take a popular example. Solutions to these complex flame problems have been delayed by the search for the best mathematical procedures, but the certainty of success is not in question. In addition to the above-mentioned researches,
13
"theories" of the one-dimensional flame exist which do not start from consideration of the fundamentals mentioned in section 2.1, but attempt to take a short cut by assuming, in the light of intuition, that a single property or process is dominant in flame propagation. Such, for example, are the "diffusion theory" of Tanford and Pease, 9 the "excess enthalpy hypothesis" of Lewis and von Elbe, 1° and the "H-radical momentum theory" of Manson. 1' In the view of the author it is now time to put aside these "shortcut" theories. Although the insight which some of them have afforded in the past has been valuable, they now confuse rather than simplify. The effort that is still wasted in discussing the relative merits of "thermal" or "diffusional" theories could be more usefully employed. (3.2)
M U L T I - D I M E N S I O N A L STEADY L A M I N A R P R E M I X E D FLAMES
Very few problems of this type have been studied. Von Kgrmgn and Millgn'2 have considered effects near the wall when a flame propagates through gas at rest in a tube. Marble and Adamson13have studied the mixing and reaction of adjacent semi-infinite streams of equal velocity and ultimate composition but different temperature. The transient method mentioned in 3.1 has been applied to the mixing of finite jets of equal velocity.8 No other work of this type is known to the author. ~ (3.3)
T U R B U L E N T FLAMES
There are no turbulent flame theories in the sense of this paper, b Tentative extension of lanfinar results to turbulent conditions is usually made by postulating augmented transport properties. This is made necessary by present ignorance of the proper differential description of turbulence even without combustion; the relations between time-mean reaction rates and time-mean concentrations also await investigation. Most practical flames are turbulent. Satis"Karlovitz, Knapschaefer, and Wells14 discussed flames in the presence of velocity gradients in a "short-cut" theory, assuming the laminar flame speed to be a constant of the mixture, irrespective of the local flow. b "Short-cut" theories by Karlovitz, Denniston, and Wells 15 and Scurlock and Grover 16exist. They assume that turbulent flames are composed of laminar ones. Many doubt whether this is a helpful assumption, e.g., Summerfield, Reiter, Kebely, and Maseolo. x7
14
STRUCTURE AND PROPAGATION OF LAMINAR FLAMES
factory ways of writing the differential equations governing turbulence must be discovered if theory alone is to predict flame phenomena. Section 4.32 of this paper indicates how a combination of theory and model experiment makes prediction possible already. (3.4)
I D E A L I Z A T I O N S OF F L A M E S I N S T E A D Y - F L O W ENGINES
An idealized flow system which has features in common with some engine flames is the homogeneous reaction zone of Avery and Hart, 33 in which variations in both space and time are assumed absent. Longwell, Frost, and Weiss34 have shown how, for a single dependent variable, the gas condition can be calculated as a function of the steady-flow rate. Cases of two dependent variables have also been considered in connection with chain reactions8b and radiation heat transfer2 5 In eact-~ case it is found that the flame must be extinguished when the flow rate exceeds a critical value. The homogeneous reaction zone represents a simple limiting case, and future work must account for the spatial inhomogeneities arising in real combustion systems due to the finite transport rates. Spalding8~ deals with one such case: the cylindrical vortex with uniform diffusivity. This exhibits extinction phenomena also, as must all steady-flow systems whether homogeneous or not. (3.5)
NON-PRE-MIXED
FLAMES
Particularly simple (aerodynamic) considerations govern flame position when fuel and oxidant flow in separately. Pioneer investigations in this field by Nusselt TM and Burke and Schumann~9 have been followed by Hottel} ° Fay, 2~Spalding,22 and many others. The rate of fuel consumption is calculable without knowledge of the reaction kinetics. But few workers 23, 24 have studied the extinction of this flame type, when chemical reaction becomes controlling, yet this phenomenon sets the limit to aircraft engine performance at high altitude. (3.6) COMMENTS The research chemist has more reason to be pleased with the present achievements of flame theory than has the engineer, because the Bunsen burner is his chosen laboratory tool. Even so, serious use of theory in making reaction kinetic deductions from flame measurements has scarcely begun, partly because the right theoretical problems have not been solved, but still more because
the available solutions are cast in forms which chemists find obscure. The engineer on the other hand will look in vain for adequate theories of flame propagation in boundary layers, of the effect of velocity ratio on propagation from pilot flames, on ignition in static and moving gases, etc., to name only laminar flame phenomena of interest to him. Nor will current work on one-dimensional flames throw much light on these multi-dimensional ones; entirely different techniques appear suitable for the latter, as will be seen. The final section of this paper is intended to interest the specialist in the multi-dimensional problems by suggesting theoretical and other methods which are not restricted to the one-dimensional flame. 4. " S o u r c e - a n d - S i n k " M e t h o d s
The mathematicM theory of heat conduction with heat sources but without convection is highly developed. It is built on the Poisson equation which involves second derivatives of temperature with respect to distance, and a source term. Analytical solutions often involve the integration of Green's function for the prescribed boundary conditions (e.g., Carslaw and Jaeger2~). Numerical solution by relaxation methods is an established technique?6. 27 Analogues of various kinds have been widely used. ~ If there is only one independent variable, numerical integration is particularly easy (e.g., Hartree29). These techniques will here be given the generic name "source-andsink methods," since they all involve the source concept explicitly or implicitly. If flame problems can be expressed in terms of Poisson-like equations the above techniques become available. This is the aim of the methods to be described below. Because of the complicated nature of the source (reaction rate) terms, analytical methods are often out of the question; nevertheless, the Green's function approach may be used in iterative numerical methods, and forms the basis of the analogues. Transformations are indicated which in simple cases eliminate the convection term from the flame equations, so making them of the Poisson form. Discussion of these methods begins with the one-dimensional flame because of its familiarity. (4.1)
NUMERICAL PROCEDURES
(4.1.1) One-dimensional flames For simplicity, a flame with one dependent variable (temperature) will be considered. In
15
ENDS AND MEANS IN FLAME THEORY
terms of a dimensionless space co-ordinate y, the equation of conservation is d2r
dr
dy 2
dy
XO(r)
(1)
Terms on the left-hand side represent molecular transport and convection respectively; ¢ is a function of r and represents volumetric reaction rate; X is related to burning velocity, and represents the eigenvalue to be determined. Most workers 14 have followed Zeldovich and Frank-Kamenetsky in transforming the equation to new variables p
=-
and r, giving
The author circumvents the difficulty, which arises whatever the form of the equations, by introducing a fictitious "catalyst-plug" at the hot boundary which always supplies a small but finite amount of heat to the flame. It may be thought of as an adiabatic porous catalyst into which the gas flows and which brings about thermodynamic equilibrium at its upstream surface. Solutions for the value of X are then obtainable for any finite temperature gradient at the plug surface. The required value for zero temperature gradient cannot be obtained directly; it must be determined by extrapolation. (4.1.3) Discussion of Equation (1) in terms of
dp
p ~
-
p =
-x,(r)
I t is this transformation which, in the past, has turned flame theory away from the main stream of heat and mass transfer theory. It is advantageous for approximate methods but, since the equation is now first order, is not amenable to the "source-and-sink" techniques. It does not appear capable of extension to multidimensional ~Iames. The methods to be described do not make this transformation but operate either directly on Equation (1) or on transformations of it which are still of second order. (4.1.2) "Catalyst-plug" boundary conditions An often-discussed difficulty of flame theory arises because the analytical expression for reaction rate has a finite, though very small, value in the gas stream before it approaches the flame. This difficulty is avoided by some authors by assuming the reaction rate terms to be exactly, instead of nearly, zero below a certain temperature. Hirschfelder, Curtiss, and Campbell s have instead introduced a fictitious "cold plug" at the upstream boundary and suppose that the fame must always lose a small but finite amount of heat to this plug. A less widely known difficulty of flame theory arises at the hot boundary where, in the real flame, temperature and concentration gradients become zero. Since the reaction rate is also zero there, Equation (2) shows that
(dp)
0 indeterminate ~=,=1-x5=
sources and sinks
(2)
(3)
Special devices must therefore be used before the integration from the hot boundary can begin, e.g., expansion in series.
Solution of Equation (1) involves obtaining the temperature distribution through the flame. If the R.H.S. of Equation (1) were everywhere very small (except at the e~talyst plug surface, DIRECTION
o~
.._
~,o,,,--
,-.,,~' \
Jl
~ i
~
-i r
UPStrEAM CATALYGT PLUO
~-
- =~
I
I DISTANCE
~,
I ~ =o
l
1
I
FIG. 1. Temperature distribution in one-dimensional gas stream if X is very small (negligible reaction except at plug face). of course), this distribution would simply be an exponential decay curve upstream of the plug and a horizontal line downstream of the plug (Fig. 1). This distribution is the effect of a unit plane source of heat normal to the stream direction, situated at the plug surface. If the source were situated farther upstream, the distribution curve would merely be shifted correspondingly. Now the R.H.S. of Equation (1) is linear, so that, if two sources are present, the temperature at any point is the sum of the temperatures which would prevail there due to each source individually. In particular the temperature distribution due to a unit source at y = ~ and a unit sink at the hot plug (y = 0) is shown in Figure 2. Curves such as this are easily calculated for any value of 7. Suppose their ordinate is G(y,~). Now the R.H.S. of Equation (1) states how the heat sources are distributed albeit with respect to temperature. The foregoing discussion shows that the temperature distribution which solves the equation is the sum of the effects of all the sources in the flame region, these effects being determinable from the curves such as Figure 2
STRUCTURE AND PROPAGATION OF LAMINAR FLAMES
16
and from the function ~. In mathematical form this is stated by re-writing Equation (1), together with its boundary conditions, as an integral equation: 0
P
--
~
< y
~-(y) = e ~ + ~ / d-
¢(~-) d~
G(y,v),
where G(y,~) = 1 - e~, G(y,~)
e~-~
=
--
Lnv~y<0 e~,
Ln y ~ ~
<
0
(4)
G(y,~), the integrating kernel of Equation (3), is the Green's function of the problem.
]
DIRECTION __
oF
"-J I
~LOW ~
"_J
a first approximation, the corresponding ¢ is calculated, and Equation (4) is used to give a new approximation for T. If this is done point by point, the space having first been divided up into finite intervals, the procedure is essentially one of relaxation; if, for speed, sources are adjusted simultaneously at all points, the procedure is essentially one of successive quadrature performed by means of tables of G(y,~). This procedure, or variants of it, can be applied to the general case where there are several dependent variables. The arithmetic, though simple, is heavy; automatic devices soon become necessary. Apart from digital computers, analogue devices such as those mentioned later may be used; indeed analogues represent the most suitable means of carrying out solutions of this nature. (4.1.5) Flames in more than one dimension
~..,.t
"T"--7~,
*
*
'
'
'
t
*
l
v
t
'
~,~=0
FIG. 2. Temperature distribution G(y, ~) in gas stream caused by unit heat source at y = ~ and unit heat sink at y = 0 (plug surface). ~ = - 1 in this case. ~'DIRECTION
,
~*
-~
,
G • I-'¢ P
t
,
p
I
~
r
I I
,
i
i
VARIAE~LE 50URCE POSITION ~
1JNIT HEAT S~NI~
FIG. 3. Influence function (Green's function) G(y, ~) showing temperature rise incurred at y when a unit source is placed at ~ and a unit sink at the catalyst plug (~ = 0). y = - 1 in this case. The idea of a Green's function can perhaps be made acceptable to engineers by pointing out the similarity of Figure 2, for example, to a bending moment diagram representing the effects at all points (y) of a beam when a force is placed at a given point (7). Alternatively G(y,~) may be plotted as in Figure 3. This curve represents the effects at one point (y) for various positions (7) of the source; its engineering analogy is the "influence line" of structural theory. It is notable that if the source is upstream of y its effect at y is independent of source position.
As far as the one-dimensional flame is concerned, the above method is just one among many. It is presented here because it can be extended, in either its numerical or analogue form, to multi-dimensional flames--that is, to partial differential equations. The only essential change lies in the extended nature of the Green's function which must represent the influence of a heat source at point A on the temperature at point B, wherever these points are situated. If an analogue is used it is not necessary to evaluate these influences explicitly; they are "built-in" to the apparatus. (4.1.6) Transformations of the conservation equations Numerical integration, relaxation techniques, and analogues all deal with second-order differential equations most happily when the first derivative is absent. A number of transformations make this possible provided that, although heat and matter flow may be multi-dimensional, the general stream velocity is uniform. They will be indicated for the one-dimensional flame only. The dependent variable in Equation (1) can be transformed by putting =
t e ~/~
(5)
Then Equation (1) becomes (4.1.4.) Methods of solution Since ~b(r) in Equation (4) can be given definite values only when r is known as a function of y, iterative methods of solving Equation (4) suggest themselves. A r distribution is assumed as
d2t 1 = - t - Xe-'J/~(r) dy 2 4 --
(6)
Thus the first derivative has been eliminated at the cost of introducing a new source term and
17
ENDS AND MEANS IN FLAME THEORY
making the original source term depend upon position. I t is still amenable to the various sourceand-sink techniques, and is particularly suitable for expression in finite difference form and solution by relaxation. Computation is unnecessary at points so far upstream that chemical reaction has become negligible, for the exponential nature of the solution is known there; that y goes to infinity is not a serious disadvantage. This transformation has been used to solve analytically the heat conduction problems with moving heat sources encountered, for example, in welding, a° As an alternative, the independent variable may be transformed by putting =
e~
(7)
Then Equation (1) becomes d~r d~2
-
x
¢ (~) - ~2
(8)
This has the advantage that the independent variable only varies between 0 and 1 in the whole flame region. I t is usually preferable to Equation (6) for one-dimensional problems. A different but related transformation which is of relevance to the electric analogue follows from multiplying Equation (1) by e-v giving --
dy
e -'a
=
e-U¢(r)
(9)
In finite difference form this is easily simulated electrically, and y of course varies from 0 to --
o o
Each of these transformations has its own special features. Which to use depends on the details of the problem and the solution technique which is to be used. Source-and-sink methods are applicable in each case. None of these transformations works when the flow velocity is not uniform. (4.1.7) Other methods
applications
of
source-and-sink
The method of solving transient problems mentioned in section 3.1 involves the use of sources and sinks; it makes use of the linearity of all terms of the equation, except that for reaction rate, when distance and time are the independent variables. The distribution of chemical reaction in steady diffusion fames (section 3.4) can be studied by obvious modification of the techniques mentioned above. In this case the source and sink magnitudes depend on position in a new way, only hav-
ing finite values in a restricted region. The boundary conditions are such as to make numerical integration very difficult; relaxation or successive quadrature are possible, however, as is the use of an analogue. (4.1.8) Closing note The above formulations have not yet been tried with any very complicated flames. They are mentioned here to indicate that there are many more ways of solving flame problems theoretieally than the current literature of the subject would suggest. (4.2.)
ANALOGUE METHODS FOR
PRESCRIBED
PROBLEMS
WITH
FLOW
When the flow is not uniform, as for example in laminar boundary layer problems, the calculation of the Green's functions is a formidable task. This is avoided if a device is constructed which obeys the same differential equation as does the flame, i.e., if an analogue is used. An electric analogue for example must be so constructed that a unit current source at point A produces a voltage increase at point B which is proportional to the temperature rise at the corresponding point b in tile flame when a unit heat source is present at the corresponding point a. Since they automatically satisfy the differential equation and the boundary conditions, analogues may be regarded as having "built-in" Green's functions. Usually the simulation must be of a finite-difference nature, at least as far as the source terms are concerned. Electrical and other analogues for heat transfer and diffusion problems are well known 2s and will not be described. Here some modifications needed for solving flame problems will be mentioned. I t is assumed that the flow pattern and the transport properties are known beforehand. (4.2.1) Fully electrical analogues Electrical resistance networks with controllable current inputs at the mesh-points obey Poisson's equation in finite difference form. Transformations of the flame equations into the Poisson form have been described above. The analogue for Equation (6), for example, is a line of equal resistance in series representing the left-hand side; a set of equal resistances connecting each junction of the line to ground, representing the 1/~t term; and a controlled current input at each ]unction to represent the reaction rate term. One end of the resistor line is held at
18
STRUCTURE AND PROPAGATION OF LAMINAR FLAMES
a fixed voltage above ground; this corresponds to the hot plug. The other end, theoretically at infinity, is at ground voltage. The current sources are adiusted successively until they accord with the local voltages for an arbitrary value of h. The current taken from the terminating battery is a measure of the temperature gradient at the hot plug. For problems with many dependent variables, many such networks are required, interconnected at least through the input currents. With several independent variables, several networks are cross-connected electrically. Equation (7) can be simulated more easily: the line of resistance is finite in length, and the leaks to ground are absent. Equation (9), on the other hand, requires an infinite line of series resistances, but these decrease in value the greater the distance from the hot plug; leakage resistors are absent. This technique and others are described by Johnson and Alley. 31 Problems with nonuniform velocity cannot be described by Poisson-type equations by any transformation known to the author; the convection terms cannot be eliminated. If network analogues are to be used for these problems, they must incorporate resistances which have different values according to which end they are seen from. Suitable devices are described by Soroka; ~ they involve electronic equipment. (4.2.2) Electro-thermal analogues When simulation of convection terms is unavoidable, a thermal resistance network may be used in place of the electrical one. Mesh-points are connected by metal tubes; thermal conduction occurs along thin walls while fluid flows through them with a distribution of velocities which simulates that appropriate to the flame. Electrical resistances wound around the tubes provide concentrated heat sources, and thermocouples measure the local temperature. The external control relates the rate of heat input to the temperature, which represents the dependent variable of the flame problem. This technique has not so far been used. AIthough somewhat clumsy, it offers an inexpensive means of investigating flame phenomena in boundary layers where the velocity distribution, although complex, can be calculated without knowledge of the flame pattern. (4.3)
A N A L O G U E S FOR F L A M E S I N COMPLEX A E R O DYNAMIC F I E L D S
In engineering equipment, flow is usually furbulent and multi-dimensional, and moreover in-
volves regions of mixing and recirculating flow which aerodynamics can at present neither predict nor describe in detail sufficient as a basis for flame theory. This difficulty must be surmounted if flame theory is to serve practical ends. Two ways of doing so are here described. The first is simple in concept but difficult in practice; it has not been used. The second involves an analogue which avoids these difficulties; it has been in operation since mid-1954. (4.3.1) Experiment on models, combined with
calculation An early step in one of the theoretical methods (section 4.1) is the calculation of the Green's function relating temperature increment at one p o i n t to heat source increment at another. This is impossible for complex turbulent flow fields. However, these influence coefficients could be determined experimentally, by making a dynamically similar model of the flow system (combustion chamber) and, for example, measuring the concentration of a tracer at position A when a thin stream of tracer is supplied at B; this would be done at a sufficient number of point-pairs to characterize the flow. The next step, iterative solution of the conserration equations in integral form, then follows. Though straightforward, it demands much computation even in relatively simple cases. Thus both the initial experimental step and the followng computational step are time-consuming. (4.3.2) Electro-thermal analogue The experimental step in the above method is eliminated and the computational step shortened by use of an electro-thermal analogue. A model of the combustion chamber is constructed through which air flows simulating the reactive gas. Distributed through the model are thermocouples measuring the air temperature and electric resistance wires providing heat. External controls relate local heat input to local air temperature in a way similar to the prescribed reaction rate function. The flow in the model automatically ensures that temperature increments resulting from a heat source increment are distributed through the ftow space in the correct manner. The analogue constructed at the Cambridge University Engineering Laboratory and now operating at Imperial College3~ solves two-dimensional turbulent flow problems with a single dependent variable. The reaction=rate function used is rectangular in shape for simplicity. Control of heaters is automatic. Each solution takes
ENDS AND MEANS IN FLAME THEORY
about one minute. The heater-thermoeouple elements are permanently in position; changes in combustion chamber shape are made by interposing metal baffles. The main limitations of this type of analogue are inability to simulate the density changes consequent on reaction, and inability to account for the effects of two-phase flow (spray combustion). Separate introduction of fuel (assumed gaseous) and air is simulated by a preliminary temperature-distribution experiment with a heat source simulating fuel injection; the local reaction rate is adjusted to accord with the local fuel/air ratio. Heat losses and complex reaction schemes may in principle be simulated by simultaneous use of several analogues with cross-coupled source terms. Further development will be directed to these points, to use of three dimensions and the better simulation of reaction-rate-temperature functions.
19
ternational) on Combustion, p. 190. Baltimore, The Williams & Wilkins Co., 1953. 6. KLEIN, G. : University of Wisconsin SQUID--1
(1955).
7. SCHATZKI,T. K. : University of Wisconsin CM 853 (1955). 8. SPALDING,D. B. : (a) Aircraft Engineering, 25, 264 (1953) ; (b) Phil. Trans. Roy. Soc., A249, 1 (1956). 9. TANFORD, C., AND PEASE, R. N.: J. Chem. Phys., 15, 433 (1947). 10. LEwis, B., AND YON ELBE, G.: Combustion, Flames and Explosions. New York, Academic Press, 1951. 11. MANSON, N.: Rev. Inst. Fr. Pdtrole, ~, 338 (1949). 12. yON KXRMXN, T., AND MILLAN, G.: Fourth Symposium (International) on Combustion, p. 173. Baltimore, The Williams & Wilkins Co., 1953. 13. MARBLE, F. E., AND ADAMSON,T. C.: Selected Combustion Problems, p. 111. London, Butterworth's, 1954. 5. Conclusions 14. KARLOVITZ, B., KNAPSCHAEFER, D. H., AND WELLS, F. E.: Fourth Symposium (InternaThe theoretical flame problems dealt with by tional) on Combustion, p. 613. Baltimore, the design engineer and research chemist have The Williams & Wilkins Co., 1953. much in common, differing only in the relative 15. KARLOVITZ, B., DENNISTON, D. W., AND number of dependent and independent variables. WELLS, F. E.: J. chem. Phys., 19, 541 (1951). Methods are becoming available for solving 16. SCURLOCK,A. C., AND GROVER, J. H. : Selected most of these problems. The analogue methods Combustion Problems, p. 215. London, Butwhich are most suitable for complex flow patterns terworth's, 1954. are similar in spirit to numerical methods which 17. SUMMERFIELD,M., REITER, S. H., KEBELY, K., may be used for solving the more familiar probANDMASCOLO,R. W. : Jet Propulsion, 25, 377 (1955). lems of the one-dimensional laminar flame. 18. NUSSELT, W.: Z.V.D.I., 60, 102 (1916). 6. Acknowledgment 19. BURKE, S., AND SCHUMANN, T.: Ind. Eng. Chem., 20, 998 (1928). The electro-thermal analogue described in 20. HOTTEL, H. C.: Fourth Symposium (Internasection 4.3.2 has been developed as part of an tional) on Combustion, p. 97. Baltimore, The extra-departmental research sponsored by the Williams & Wilkins Co., 1953. Mechanical Engineering Research Laboratory 21. FAY, J. A.: J. Aero. Sci., 21,681 (1954). of the Department of Scientific Research. 22. SPAI~DING, D. B.: Proc. I. Mech. E., 168, 545 (1954). REFERENCES 23. ZELDOVICH,Y. B. : J. Tech. Phys. Moscow, 19, 1. ZELDOVICH,Y. B., AND FRANK-KAMENETSKY, 1199 (1949); trans, as NACA Tech. Memo D. A.: J. Phys. Chem. Moscow, 12, 100, 1296 (1951). 24. SPALDING,I). B. : Fuel, 33, 253 (1954). (1938). 2. BOYS, S. F., AND CORNER, J. : Proc. Roy. Soc., 25. CARSLAW,H. S., AND JAEGER, J. C.: ConducAI97, 90, (1949); CORNER, J.: Proc. Roy. tion of Heat in Solids. New York, Oxford Soc., AI98, 388 (1949). Univ. Press, 1947. 3. ADAMS,E. N.: See I-IENKEL,M. J., SPAULDING, 26. EMMONS, H. W.: Trans. A.S.M.E., 65, 607 W. P., ANDHIRSCnFELDER,J. O. : Third Sym(1943). posium on Combustion, p. 21. Baltimore, 27. DUSINBERRE, G. M.: Numerical Analysis of The Williams & Wilkins Co., 1949. Heat Flow. New York, McGraw-Hill, 1949. 4. VON ]~(.~RM.~N, W., AND PENNER, S.: Selected 28. SOROKA,W. W. : Analog Methods in Computation and Simulation. New York, McGrawCombustion Problems, p. 5. London, ButterHill, 1954. worth's 1954. 5. HIRSCHFELDER, J. O., CURTISS, C. F., AND 29. HARTREE, D. R.: Numerical Analysis, New CAMPBELL, D. E.: Fourth Symposium (InYork, Oxford Univ. Press, 1952.
20
STRUCTURE AND PROPAGATION OF LAMINAR FLAMES
30. I{,OSENTHAL, D.: Trans. A.S.M.E., 68, 849 (1946). 31. JOHNSON, W. C., AND ALLEY, •.
33. AVERY, W. If.. AND HART, t{. W.: Ind. Eng. Chem. 45 1634 (1953). 34. LONGWELL, J. P., FROST, E. E., AND WEISS, M. A.: Ind. Eng. Chem., 1628 (1953). 35. SPALDING,D. B., AND TALL, ]3. S. : Aero Quarterly, 5, 195 (1954).
E. : Rep. 3.
ONR. Contract N6 105, Task Order VI June 1948. 32. SPALDING,D. ]~.: Proc. I. Mech, E., to be published.
3
LIMITS O F I N F L A M M A B I L I T Y
By J. W. L I N N E T T AND C. J. S. M. SIMPSON Introduction Many experimental determinations have been made of limits of inflammability, but less has been done on the theory of the problem, and, at the present time, there is no agreed explanation of why composition limits are observed. Recent work has been concerned mainly with suggesting
BURNING VELOC I TY
COMPOSITION FIG. 1. Probable theoretical variation of burning velocity with composition (diagrammatic). features that might account for the existence of a finite limit of composition; there has been little attempt to see whether the known facts support the suggestions. Why is there this difficulty regarding the existence of limits? The recent steady-state treatments of flame propagation are thorough. 1 However, they include no factor that would account for the existence of limits. Theoretically steady-state
solutions (see also LayzerD can exist over the whole composition range, so that present theories of this type would probably predict a burning velocity curve of the form shown in Figure 1. (Figure 1 is not to scale.) Theory will predict that the burning velocity will be close to zero for a range of compositions near each end of the curve (i.e., the "tails" will be much closer to the composition axis than is drawn). Because of this situation it has been necessary to try to account for limits by additional suggestions. One of these has been to consider the effect of perturbations on the steady-state condition2 It has been shown that for some mixtures a perturbation would die away while for others it would grow and the flame would fail to return to its steady-state condition. Before examining these ideas it is important to consider the question: Does experimental evidence indicate whether an infinite horizontal plane flame moving through a quiescent homogeneous mixture would or would not show a composition limit of inflammability? The answer is not certain for no such flames have been studied. The nearest approach is provided by the circular flat flame burner of Powling,4 Egerton and Thabet 5 and Badami and Egerton? But this is only about 3 in. across so that convection between the hot products from the flame and the surrounding gases may be important. If the answer to the above question were "Yes, such flames would show a limit," a second may need to be asked, namely: Are the limits usually determined close to or far from the limits that might exist for such infinite undisturbed plane flames?