Thermodynamics of combustion 293

Gibbs energy than the reactants; these are called the products (usually carbon dioxide, water, etc). In the case of the spark-ignition engine the ionisation energy is provided by the spark, which ignites a small kernel of the charge from which the flame spreads. In the diesel engine part of the mixture produced in the cylinder cannot exist in the metastable state: it will spontaneously ignite. Such a mixture is termed a hypergolic mixture.

9 2 x e r3

Q) \ Metastable

\ Stable equilibrium

State Fig. 15.1 Energy states associated with combustion

Some mixtures are unstable at room temperature, and their constituents spontaneously ignite. An example of such a mixture is hydrogen and fluorine. This concept of spontaneous ignition will be returned to later.

All of these thermodynamic processes take place somewhere in the combustion zone, and many of them occur in the flame. Before passing on to the detailed discussion of flames it is worthwhile introducing some definitions and concepts.

15.2.1 REACTION ORDER

The overall order of a reaction is defined as n = cy=, v,, summed over the q species in the reactants.

First-order reactions

These are reactions in which there is spontaneous disintegration of the reactants. These reactions do not usually occur, except in the presence of an ‘inert’ molecule.

Second-order reactions

These are the most common reactions because they have the highest likelihood of a successful collision occurring.

Third-order reactions

These are less likely to occur then second-order ones but can be important in combustion. An example is when OH and H combine to produce an H,O molecule. This H,O molecule

294 Combustion and James

will tend to dissociate almost immediately unless it can pass on its excess energy - usually to a nitrogen molecule in the form of increased thermal energy.

15.2.2 PROCESSES OCCURRING IN COMBUSTION

The thermodynamics of combustion generally relates to the gas in isolation from its surroundings. However, the surroundings, and the interaction of the gas with the walls of a container, etc, can have a major effect on the combustion process. The mechanisms by which the combusting gas interacts with its container are:

(i) transport of gaseous reactant to the surface - diffusion; (ii) adsorption of gas molecules on surface; (iii) reaction of adsorbed molecules with the surface; (iv) desorption of gas molecules from the surface; (v) transport of products from the surface back into the gas stream - diffusion.

These effects might occur in a simple container, say an engine cylinder, or in a catalytic converter. In the first case the interaction might stop the reaction, while in the second one it might enhance the reactions.

' 15.3 Explosion limits

The kinetics of reactions was introduced previously, and it was stated that reactions would, in general, only occur when the atoms or ions of two constituents collided. The reaction rates were derived from this approach, and the Arrhenius equations were introduced. The tendency for a mixture spontaneously to explode is affected by the conditions in which it is stored. A mixture of hydrogen and oxygen at 1 bar and 500°C will remain in a metastable state, and will only explode if ignited. However, if the pressure of that mixture is reduced to around 10 mm Hg (about 0.01 bar) there will be a spontaneous explosion. Likewise, if the pressure increased to about 2 bar there would also be an explosion. It is interesting to examine the mechanisms which make the mixture become hypergolic. The variation of explosion limits with state for the hydrogen-oxygen mixture is shown in Fig 15.2.

lOUU0

h

r" E E , IO0 v

B 2 2 a

I U

I

3911 .rin 430 45(1 471 490 510 530 550 570 590

Temperature / ("C)

Fig. 15.2 Explosion limits for a mixture of hydrogen and oxygen (from Lewis and von Elbe, 1961)

Explosion limits 295

The kinetic processes involved in the H,-0, reaction are:

H+O,-OH

which leads to the further branching steps

0 + H 2 4 OH + H OH + H , 4 H 2 0 + H

(15.1)

The first two of the three steps are called branching steps and produce two radicals (highly reactive ions) for each one consumed. The third step does not increase the number of radicals. Since all steps are necessary for the reaction to occur, the multiplication factor (i.e. the number of radicals produced by the chain) is between 1 and 2. The first step is highly endothermic (it requires energy to be supplied to achieve the reaction), and will be slow at low temperatures. This means that an H atom can survive a lot of collisions without reacting, and can be destroyed at the wall of the container. Hence, H2-02 mixtures can exist in the metastable state at room temperatures, and explosions will only occur at high temperatures, where the first step proceeds more rapidly.

15.3.1 EXPLODE The effect of the multiplication factor can be examined in the following way. Assume a straight chain reaction has lo* collisions/s, and there is 1 chain particle/cm3, with 10'' molecules/cm3. Then all the molecules will be consumed in 10" seconds, which is approximately 30 years. However, if the multiplication factor is now 2 then 2" 10". giving N=62. This means that all the reactants' molecules will be consumed in 62 generations of collisions, giving a total reaction time of 62 x O-' seconds, or 0.62 ps: an extremely fast reaction! If the multiplication factor is only 1.01 then the total reaction time is still only 10 ms. Hence, the speed of the reaction is very dependent on the multiplication factor of the reactions, but the overall multiplication factors do not have to be very high to achieve rapid combustion.

THE EFFECT OF MULTIPLICATION FACTOR ON THE TENDENCY TO

A general branched chain reaction may be written

M - R initiation k l

k 2 R + M - a R + M chain branching, a > 1

k3 R + M - P product formation removes radical

chain termination I R 'L, destruction

R A destruction

wall

gas

(15.2)

where M is a molecule, R is a radical, and P is a product. a is the multiplication factor. The value of a necessary to achieve an explosion can be evaluated. The rate of formation of the product, P , is given by

-- d [ P 1 - k , [ R ] [ M ] dt

(15.3)

296 Combustion and flames

The steady state condition for the formation of radicals is

-- d[RI - 0 = k l [ M l + k 2 ( a - l ) [ R ] [ M ] - k 3 [ R ] [ M ] - k 4 [ R ] - k , [ R ] dt

Solving eqn (15.4) for R and substituting into eqn (15 .3) gives

(15 .4)

(15.5)

The rate of production of the product P becomes infinite when the denominator is zero, giving

(15.6)

Thus, if amact > aCnt the reaction is explosive; if the axact < a,fit then the combustion is non-explosive and progresses at a finite rate. A few explosion limits, together with flammability limits are listed in Table 15.1. This text will concentrate on non-explosive mixtures from now on.

Table 15.1 Flammability and explosion limits (mixtures defined in % volume) at ambient temperature and pressure

Lean Rich

Mixture Flammability Explosion Flammability Explosion Stoichiometric ~~~~~~

H, - air 4 18 74 59 29.8 co-0, 16 38 94 90 66.7 CO-air 12.5 74 29.8 NH3-02 15 25 79 75 36.4 c3Hfl-02 2 3 55 37 16.6 CH.-air 5.3 15 9.5 1 C,H, -air 3.0 12.5 5.66 C3H, -air 2.2 9.5 4.03 C,H,, - ax 1.9 8.5 3.13

15.4 Flames

A flame is the usual mechanism by which combustion of hydrocarbons takes place in air. It is the region where the initial breakdown of the fuel molecules occurs. There are two different types of flame, as described above: premixed flames and diffusion flames. Premixed flames will be dealt with first because it is easier to understand their mechanism.

15.4.1 PREMIXED FLAMES

Premixed flames occur in any homogeneous mixture where the fuel and the oxidant are mixed prior to the reaction. Examples are the Bunsen burner flame and the flame in most spark-ignited engines. Premixed flames can progress either as deflagration or detonation

Flames 297

processes. This text will consider only deflagration processes, in which the flame progresses subsonically. Detonation processes do occur in some premixed, spark-ignited engines, when the ‘end gas’ explodes spontaneously making both an audible knock and causing damage to the combustion chamber components.

When considering laminar flame speed, it is useful to start with a qualitative analysis of a Bunsen burner flame, such as depicted in Fig 15.3. If the flow velocity at the exit of the tube is low then the flow of mixture in the pipe will be laminar. The resulting flame speed will be the laminar flame speed. While most flames are not laminar, the laminar flame speed is a good indication of the velocity of combustion under other circums- tances. It can be seen from Fig 15.3(b) that the shape (or angle) of the inner luminous cone is defined by the ratio of the laminar flame speed (or burning velocity), ut, to the flow velocity of the mixture. In fact, the laminar flame speed ut= ug sin a. While this is a relatively simple procedure to perform, it is not a very accurate method of measuring laminar flame speed because of the difficulty of achieving a straight-sided cone, and also defining the edge of the luminous region. Other methods are used to measure the laminar flame speed, including the rate of propagation of a flame along a horizontal tube and flat burners.

. . :f

/

Premixed gas and air

(a) (b)

Fig. 15.3 Schematic diagram of Bunsen burner flame: (a) general arrangement; (b) velocity vectors

15.4.2 LAMINAR FLAME SPEED

There are a number of theories relating to laminar flame speed. These can be classified as:

0 thermal theories 0 diffusion theories 0 comprehensive theories.

The original theory for laminar flame speed was developed by Mallard and le Chatelier (1883), based on a thermal model. This has been replaced by the Zel’dovitch and Frank- Kamenetsky (1938), and Zel’dovitch and Semenov (1940) model which includes both

298 Combustion and flames

r -

I I I

____+ +--.' 3 .+-- +--a +-- +--%+-- Mass flow

I I I + - - lndicates thermal energy flow I , I I I

I I I

I I I

I I I Tb : Concentration I I I Temperature of reactants- I - - - - - -

I - \ I I I \

\ I I

I \ I I ',

Fig. 15.4 Schematic diagram of interactions in plane combustion flame

thermal and species diffusions across the flame. The details of these models will not be discussed but the results will simply be presented.

A plane flame in a tube may be shown schematically as in Fig 15.4. It can be seen that the thermal diffusion goes from right to left in this diagram, i.e. against the direction of flow. The flame also attempts to travel from right to left, but in this case the gas flow velocity is equal to the flame speed. If the gas had been stationary then the flame would have travelled to the left at the laminar flame speed.

Considering the physical phenomena occurring in the tube: heat flows, by conduction, from the burned products zone (b) towards the unburned reactants zone (u), while the gas flows from u to b. A mass element passing from left to right at first receives more heat by conduction from the downstream products than it loses by conduction to the reactants, and hence its temperature increases. At temperature T, the mass element now loses more heat to the upstream elements than it receives from the products, but its temperature continues to increase because of the exothermic reaction taking place within the element. At the end of the reaction, defined by Tbr the chemical reaction is complete and there is no further change in temperature.

The Zel'dovitch et al. (1938, 1940) analysis results in the following equation for laminar flame speed

(15.7)

In obtaining eqn (15.7) the assumption had been made that the Lewis number

L e = k / p c , D = l

where k = thermal conductivity, p = density, cp = specific heat at constant pressure, and D =mass diffusivity Hence, Le is the ratio between thermal and mass diffusivities, and

Flames 299

this obviously has a major effect on the transport of properties through the reaction zone. The assumption Le = 1 can be removed to give the following results for first- and second- order reactions.

For first-order reactions

and for second-order reactions

(15.8a)

(15.8b)

where 2' is the pre-exponential term in the Arrhenius equation and c, is the initial volumetric concentration of reactants.

Equations (15.8a) and (15.8b) can be simplified to

(15.9)

Hence the laminar flame speed is proportional to the square root of the product of thermal diffusivity, a , and the rate of reaction, R. Glassman (1986) shows that the flame speed can be written as

(1 5.10)

which is essentially the same as eqn (15.9), where R=Ze-EtRTb. Obviously the laminar flame speed is very dependent on the temperature of the products, Tb, which appears in the rate equation. This means that the laminar flame speed, ul, will be higher if the reactants temperature is high, because the products temperature will also be higher. It can also be shown that u1 = P ( ~ - ' ) / ' , where n is the order of the reaction, and n = 2 for a reaction of hydrocarbon with oxygen. This means that the effect of pressure on u1 is small. Figure 15.5 (from Lewis and von Elbe, 1961) shows the variation of u1 with reactant and mixture strength for a number of fundamental 'fuels'. It can be seen that, in general, the maximum value of uI occurs at close to the stoichiometric ratio, except for hydrogen and carbon monoxide which have slightly more complex reaction kinetics. It is also apparent that the laminar flame speed is a function both of the reactant and the mixture strength. The effect of the reactant comes through its molecular weight, m,. This appears in more than one term In eqn (15.10) because density and thermal conductivity are both functions of m,. The net effect is that ul= l/mw. This explains the ranking order of flame speeds shown in Fig 15.5, with the laminar flame speed for hydrogen being much higher than the others shown. While the molecular weight is a guide to the flame speed of a fuel, other more complex matters, such as the reaction rates, included as Z' in these equations, also have a big influence on the results obtained. Figure 15.6, from Metgalchi and Keck (1980, 1982) shows a similar curve, but for fuels which are more typical of those used in spark-ignition engines.

Fig. 155

Stoichiornetnc Stoichiornemc Stoichiometric ' H,,CO i

250 h m --.

E 200

3 r= 50 .-

5 4

0 0 10 20 30 40 50 60 70

% gas in air

Variation of laminar flame speed with reactants and mixture strength. p = 1 bar; Tu

50

45

2 25 f

E .-

15

1

=298 K

0.8 1 .o 1.2 1.4

Fuel-air equivale~ce ratio, B, Fig. 15.6 Variation of laminar flame speed with mixture strength for typical fuels (based on 1 a m ,

300 K)

Flames 301

It can be seen that the laminar flame speed is dependent on mixture strength, and this has a major influence on the design of engines operating with lean mixtures. The other feature to notice is that the laminar flame speed would remain approximately constant in an engine operating over a speed range of probably 800 to 6000 rev/min. If the combustion process depended on laminar burning then the combustion period in terms of crank angle would change by a factor of more than 7 : 1. Consider an engine operating with methane (CH,,) at stoichiometric conditions: the laminar flame speed is about 50 cm/s (0.5 m/s). If the engine bore is 100 mm then the combustion period will be 0.1 s. At 800 rev/min this is equivalent to 480" crankangle - longer than the compression and expansion periods! Obviously the data in Figs 15.5 and 15.6 are not directly applicable to an engine; this is for two reasons.

First, the initial conditions of the reactants, at temperatures of 298 K or 300 K, are much cooler than in an operating engine. Kuehl (1962) derived an expression for the combustion of propane in air, which gave

(15.11)

where

p = pressure (bar) T = temperature (K) u, = laminar flame speed (m/s)

It can be seen from eqn (15.1 1) that the effect of pressure on flame speed is very small, as suggested above. Figure 15.7 shows how the laminar flame speed increases with reactants temperature. It has been assumed that the adiabatic temperature rise remains constant at 2000 K, which is approximately correct for a stoichimetric mixture. It can be seen that the speed increases rapidly, and reaches a value of around 4 m/s when the reactants temperature

4.5 '1

300 400 500 600 700 800 900 1000

Reactants temperature / (K)

Fig. 15.7 Variation of laminar flame speed with reactants' temperature (predicted by Kuehl's equation with p = 1 bar)

302 Combustion andflames

is IO00 K. This is an increase of about a factor of 8 on the previous value which would reduce the combustion duration to about 60" crankangle. This is still quite a long combustion duration, especially since it has been evaluated at only 800 rev/min, some other feature must operate on the combustion process to speed it up. This, second, parameter is turbulence, which enhances the laminar flame speed as described below.

15.5.3 TURBULENT FLAME SPEED

It was shown previously that the laminar flame speed is too low to enable engines to operate efficiently, particularly if they are required to work over a broad speed range. The laminar flame speed must be enhanced in some way, and turbulence in the flow can do this. A popular model describing how turbulence increases the flame speed is the wrinkled laminarflame model, which is shown diagrammatically in Fig 15.8.

Burned gas

Unburned

Lamina flame

Turbulent (wrinkled) flame

Fig. 15.8 Comparison of laminar and turbulent flames

The effect of turbulence on the flame is threefold:

0 the turbulent flow distorts the flame so that the surface area is increased; 0 the turbulence may increase the transport of heat and active species; 0 the turbulence may mix the burned and unburned gases more rapidly.

The theory of turbulent flames was initiated by Damkohler (1940) who showed that the ratio of turbulent to laminar flame speeds, based on large scale eddies in the flow, is

( 1 5.1 2)

where

E = eddy diffusivity Y = kinematic viscosity of the unburned gas

Heikal et al. (1979) applied a similar approach to engine calculations, and defined the ratio of turbulent to laminar flame speed, often called the flame speed factor, as

(1 5.13)

Flammability limits 303

where a = molecular thermal diffusivity P, = Prandtl number vp = mean piston speed a , b , c , dare all empirical constants

Experience shows that in spark ignition engines the flame speed factor is a strong function of engine speed but is not greatly affected by load. Experiments by Lancaster et al. (1976) have also shown that the level of turbulence intensity in an engine cylinder increases with engine speed, but is not quite proportional to it. This means that while turbulence increases the flame speed significantly, the length of the burning period increases as engine speed is increased, which explains why the ignition timing has to be advanced.

15.5 Flammability limits

Flammability limits were introduced in Table 15.1, where they were listed with the explosion limits. The flammability limit of a mixture is defined as the mixture strength beyond which, lean or rich, it is not possible to sustain a flame. The flammability limit, in practice, is related to the situation in which the flame is found. If the flame is moving in a confined space it will be extinguished more easily because of the increase in the interaction of the molecules with the walls; this is known as quenching. It is also possible for the flame to be extinguished if the level of turbulence is too high, when the flame is stretched until it breaks. The lean flammability limit is approximately 50% of stoichiome- tric, while the rich limit is around three times stoichiometric fuel-air ratio.

Bradley et al. (1987, 1992) investigated the way in which turbulence ‘stretches’ the flame and causes it to be extinguished. The results are summarised in Fig 15.9, which is a graph of the ratio of turbulent to laminar flame speed, i.e. the flame speed factor defined in eqn (15.13), against the ratio of turbulence intensity to laminar flame speed. The abscissa is closely related to the Karlovitz number, K , which is a measure of the flame stretch:

(1 5.14)

where F, = area of laminar flame; z, = transit time for flow through laminar flame; u‘ = turbulence intensity; 1, = Taylor microscale; 6, = laminar flame thickness; u, = laminar flame speed.

Also shown in Fig 15.9 are lines of constant KLe, which is the product of Karlovitz and Lewis numbers, and lines of constant Re/Le2, which is the ratio of Reynolds number to the square of Lewis number. If the value of KLe is low (e.g. 0.005) then the flame is a wrinkled laminar one, while if the value of KLe is high ( e g 6 ) the flame is stretched

304 Combustion and flames

sufficiently to quench it. This shows that the design of a combustion chamber must be a compromise between a high enough value turbulence intensity to get a satisfactory flame speed, and one which does not cause extinction of the flame.

2 0 -1

0

5 - c

.- 0 I

0

e:

I 8

16

14

12

10

8

6

4

2

0 63

Disrupted, partial quench region I

Wrinkled 6 laminar region

Quench region

- Constant KLe

- r o n s t a n t Re/Le2

0 I

0 2 4 6 8 10 12 14 16 18 20

Ratio ( turbulence intens i ty / laminar f l a m e speed)

I ' l ' l ' l ~ l ' l ' l ' l ' l ' l

Fig. 15.9 The effect of turbulence on the turbulent flame speed and tendency to quench for a premixed charge

It can be seen from Table 15.1 that the spread of flammability for hydrogen is much higher than the other fuels This makes it an attractive fuel for homogeneous charge engines because it might be possible to control the load over a wide range of operation by qualitative governing rather than throttling. This would enable the hydrogen powered engine to achieve brake thermal efficiencies similar to those of the diesel engine. The restricted range of flammability limits for hydrocarbon fuels limits the amount of power reduction that can be achieved by lean-burn running; it also restricts the ability of operating the engine lean to control NO,. Honda quote one of their engines operating as lean as 24: 1 air-fuel ratio, which enables both the engine power to be reduced and the emissions of NO, to be controlled. Such lean-bum operation is achieved through careful design of the intake system and the combustion chamber. In practice, in a car engine the lean limit is set by the driveability of the vehicle and the tendency to misfire. A small percentage of misfires from the engine will make the uHC emissions unacceptable - these misfires might not be perceptible to the average driver.

The rich and lean flammability limits come closer together as the quantity of inert gas added to a mixture is increased. Fortunately, the rich limit of flammability is more affected than the lean one, and basically the lean limit, which is usually the important one for engine operation, is not much changed.

15.6 Ignition

Tne ignition process is an extremely important one in the homogeneous charge engine because it has to be initiated by an external source of energy - usually a spark-plug. It can be shown that the minimum energy for ignition, based on supplying sufficient energy to the volume of mixture in the vicinity of the spark-gap to cause a stable flame, is