which start the chain, involve collision with fuel by oxygen molecule and, as the radical pool is populated, by radicals. FU + O_{2} = FR + HO_{2} and FU + X *=* FR + XH, in which FU, FR, and X represent the fuel molecule, fuel radical and radical pool (O, OH, H, HO_{2}, etc.), respectively, fall into this class. If the temperature is sufficiently high, the cracking of either CH bonds or CC bonds in the fuel molecule becomes also important.

Standard Article

# 1 Combustion Fundamentals

Part 1. Fundamentals and Safety

Published Online: 15 MAR 2010

DOI: 10.1002/9783527628148.hoc001

Copyright © 2010 Wiley-VCH Verlag GmbH & Co. KGaA. All rights reserved.

Book Title

## Handbook of Combustion

Additional Information

#### How to Cite

Janbozorgi, M., Far, K. E. and Metghalchi, H. 2010. Combustion Fundamentals. Handbook of Combustion. 1:1:0.

#### Publication History

- Published Online: 15 MAR 2010

### 1.1 Introduction

- Top of page
- Introduction
- Combustion Thermodynamics
- Chemical Kinetics
- Laminar Premixed Flames
- Diffusion Flames
- Conclusions
- References

Combustion is a complex subject in chemical physics. A deep understanding of combustion science requires a solid grasp of a wide spectrum of scientific disciplines, such as quantum mechanics, thermodynamics, chemical kinetics, and fluid dynamics. On the application level, combustion phenomena can be classified based on interactions between exothermic chemical reactions and fluid mechanics. Such an interaction depends heavily on the relative order of magnitude of the time and spatial scales of each individual phenomenon, leading to different forms of combustion. Premixed combustion occurs when the fluid mixing is sufficiently fast as to create a near-uniform distribution of fuel/air mixture in the reactor. Depending on the thermodynamic conditions, premixed combustion can also be either strictly kinetically controlled (e.g., *autoignition*), or convection/reaction/diffusion controlled (e.g., *premixed flames*). The former condition underlies the operation of homogeneous charged-compression-ignition (HCCI) engines, diesel engines and rapid compression machines (RCMs), and is essential in understanding the engine knock. The latter introduces a fundamental physico-chemical property for any premixed mixture, that is, *laminar burning speed*. Knowledge of this property is crucially important in spark-ignition engines, partly to prevent autoignition. In the case of slow mixing and fast reaction, nonpremixed or diffusion flames will be observed. This chapter is devoted to an analysis of the above-mentioned modes of combustion, with special emphasis placed on laminar burning speeds and flame structures of different hydrocarbons at high pressures, and the experimental methods to measure them. Such data are extremely important for the validation of any reliable chemical kinetic mechanism, and will be especially useful for internal combustion engine designers.

### 1.2 Combustion Thermodynamics

- Top of page
- Introduction
- Combustion Thermodynamics
- Chemical Kinetics
- Laminar Premixed Flames
- Diffusion Flames
- Conclusions
- References

Combustion is defined as an energy-evolving (exothermic) chemical transformation [1]. While strictly involving time-dependent chemical reactions, the final yield of combustion, and how much energy can be extracted from a fuel/air mixture under a specified process, are restricted by the laws of thermodynamics. A *stoichiometric mixture* of fuel and air is defined as a mixture containing just enough oxygen to theoretically burn the hydrocarbon fuel to water and carbon dioxide (only hydrocarbon fuels are considered in this chapter). The *equivalence ratio* is commonly used to indicate quantitatively whether a fuel/oxidizer mixture is rich, lean, or stoichiometric [2]:

- (1.1)

Fuel-rich, fuel-lean, and stoichiometric mixtures are defined by , , and , respectively. In Equation 1.1, *A* is the mass of air, *F* the mass of fuel, and the “stoic” and “act” subscripts represent the stoichiometric and actual mixtures, respectively.

#### 1.2.1 Enthalpy of Reaction

The enthalpy of reaction or *enthalpy of combustion* is defined as the net change of enthalpy due to a chemical reaction. This quantity takes a positive value for an endothermic reaction, and a negative value for an exothermic reaction. This means that in the former reaction the energy is absorbed by the reacting system, whereas in the latter reaction it evolves as a result of the reaction. Considering the global stoichiometric combustion chemical reaction of a generic nonoxygenated hydrocarbon with air:

- (1.2)

this statement translates to

- (1.3)

where *h* is the enthalpy and *T* is the temperature. Depending on the thermodynamic process that the system undergoes, the products' temperature – also called the flame temperature () – may be different from the temperature of reactants (). The *heat of combustion* represents the amount of energy released or absorbed by the mixture during an isothermal chemical conversion. For example,

means that 41.16 kJ of energy will be released to the surroundings if 1 mole of carbon monoxide reacts completely with 1 mole of water vapor at constant pressure to produce 1 mole of carbon dioxide and 1 mole of hydrogen molecule [1]. The total enthalpy of a species *A* is defined as:

- (1.4)

The first term in Equation 1.4 represents the *enthalpy of formation*, and is defined as the net change in enthalpy associated with breaking the chemical bonds of the *standard state elements* and forming new bonds to create the compound of interest [2]. Here, the standard state elements are taken to be the most stable state of that element at the temperature of interest and the pressure of 1 atmosphere. For the common elements of combustion interest at , these states are carbon (C) as graphite, molecular hydrogen (H_{2}), oxygen (O_{2}), nitrogen (N_{2}) as ideal gases, and atomic sulfur (S) as solid [1]. The natural consequence of this definition is that the enthalpy of formation of, for example, an oxygen atom (O) is half of the bond dissociation energy of the oxygen molecule. The second term in Equation 1.4 represents the *sensible enthalpy change* and is defined as:

- (1.5)

Clearly, any departures from the standard state enthalpy are reflected in this term. Values of the specific heat at constant pressure, , are tabulated for many species in the CHEMKIN [3] database.

#### 1.2.2 Flame Temperature

Flame temperature is the temperature reached at the state of chemical equilibrium in a reacting system. The energy balance, , implies that, for a fixed type of work interaction with the surrounding environment, an adiabatic process, , has the highest flame temperature. Furthermore, depending on the relative order of magnitude of the chemical energy release time scale, , and that of boundary work interaction through acoustic wave propagation, , this adiabatic temperature falls between two extremes:

, resulting in . This further results in and therefore , known as constant energy–constant volume flame temperature, . Here,

*p*is the pressure and*V*represents volume., resulting in . This further translates to , known as constant enthalpy–constant pressure flame temperature, .

Since in the latter case part of the energy is used to do work against the surrounding environment, it results in a lower flame temperature, that is, . The first case is usually assumed to be true for autoignition in closed systems (e.g., shock tubes), while the latter is assumed to be the case for combustion in open systems at low Mach numbers (e.g., gas turbine combustors, Bunsen burners).

#### 1.2.3 Chemical Equilibrium

There are two different approaches to determine the chemical *equilibrium composition* of a reacting mixture. One is based on the application of the *Law of Mass Action* [1], and the other on the method of Lagrange multipliers [4]. If is the number of species and is the number of atomic elements in the system, then the first method requires the compilation of independent chemical reactions, followed by the simultaneous solution of the same number of equations, one for each reaction. Obviously, as the number of chemical species increases, the problem becomes more tedious. An excellent coverage of this approach can be found in Ref. [1].

The second method, which does not have the shortcomings of the first approach, consists of considering all conceivable reaction mechanisms between the chemical species, and using the maximum entropy principle without specifying explicitly any of the reaction mechanisms [4]. As pointed out by Keck [5], an equilibrium state is meaningful only when the constraints subject to which such a state is attained are carefully determined, and all equilibrium states are in fact constrained equilibrium states. At temperatures of interest to combustion, nuclear and ionization reactions can be assumed frozen, and the fundamental constraints imposed on the system are the conservation of neutral atoms. If , and represent, respectively, the number of moles of species , the number of *j*th atomic element in species , and the entropy of a closed adiabatic system, then

- (1.6)

is the total number of moles of atomic element in the system, which is conserved during chemical conversion. Therefore, the problem reduces to determining a chemical composition which maximizes subject to the relationship in Equation 1.6. Using the method of undetermined Lagrange multipliers, it can be easily shown that such a composition will be given by [5]:

- (1.7)

where is the constraint potential (Lagrange multiplier) conjugate to elemental constraint , and is the partition function of species , which is defined as follows:

- (1.8)

In Equation 1.8, represents the standard Gibbs free energy of species at temperature *T*, and *R*_{u} is the universal gas constant. As this function takes on finite values for every species, Equation 1.7 shows that, in principle, *all* species made of the declared atomic elements are present at chemical equilibrium state, no matter how small their concentration. This further explains why full conversion to carbon dioxide and water in a stoichiometric mixture, as given by Equation 1.2, is a hypothetical situation. Substituting Equation 1.7 back into Equation 1.6 forms a set of transcendental equations for . The level of reduction in the number of equations to be solved is dazzling; from which could easily go up to several thousands for heavy hydrocarbons to a maximum of five for atomic elements of carbon, oxygen, hydrogen, nitrogen and sulfur for almost any hydrocarbon fuels. This method forms the basis of the widely used equilibrium codes of STANJAN [6] and NASA [7].

### 1.3 Chemical Kinetics

- Top of page
- Introduction
- Combustion Thermodynamics
- Chemical Kinetics
- Laminar Premixed Flames
- Diffusion Flames
- Conclusions
- References

#### 1.3.1 Combustion Chemical Reactions

The development of models for describing the dynamic evolution of chemically reacting systems is a fundamental objective of chemical kinetics. This task involves identifying the chemical reactions in the most elementary level, and also the rate at which such reactions proceed. The conventional approach to this problem involves first specifying the state and species variables to be included in the model, compiling a “full set” of rate-equations for these variables based on a “full set” of *elementary* chemical reactions, and then integrating this set of equations to obtain the time-dependent behavior of the system [8]. Such models are frequently referred to as detailed kinetic models (DKMs). The most widely known and used such model is GRI-MECH 3.0 for the combustion of methane at high temperatures and low pressures [9]. A DKM can easily include several hundred chemical species and several thousand chemical reactions for heavy hydrocarbons [10]. (An extensive study of DKMs is provided in Chapter 2.) Combustion chemical reactions are in general *chain reactions*, which means that the products of one reaction serve as the reactants of other reactions. However, independent of the fuel molecule, these reactions can be classified into four groups:

- Initiation reactions,
- Chain-branching reactions
change the number of radicals and populate the radical pool. H + O

_{2}= OH + O is one of the most important chain-branching reactions in combustion applications. - Chain-propagating reactions
change the

*type*of radical, while conserving the*number*of radicals in the reactions. H + H_{2}O = H_{2}+ OH represents these reactions. - Three-body reactions,
sometimes known as

*chain-terminating reactions*or equivalently,*dissociation–recombination reactions*, change the number of moles of the mixture; for example, H + OH + M = H_{2}O + M. As the recombination of radicals is highly exothermic and usually involves only a small rotational energy barrier, these reactions cannot be bimolecular and require interaction with a third body (M), to which the energy of molecule formation is disposed. Otherwise, this energy would dissociate the products into the original reactants. Depending on their molecular size, different molecules have different third-body efficiencies.

A more detailed presentation of this subject is provided in Chapter 2.

#### 1.3.2 Kinetic Rate Equations

Assuming that changes in the chemical composition of the system are the results of elementary reactions of the type:

- (1.9)

then the molar rate of change of each chemical species can be expressed as

- (1.10)

where

- (1.11)

The temperature dependence of and are represented by the Arrhenius form as

- (1.12)

The pre-exponential factor or collision frequency, , is weakly temperature-dependent over the temperatures reached in combustion applications. Such temperature dependence is represented by a modified form of , in which *n* is called the “temperature exponent.” The exponential part, on the contrary, is strongly temperature-dependent. This part represents the fraction of molecules possessing enough energy to surmount the activation energy barrier, and undergo chemical reactions. The typical value of this parameter for combustion of hydrocarbons is 40–45 kcal mol^{–1}. When considering Equation 1.10, at the state of dynamic equilibrium the forward and reverse reaction rates must balance, which leads to the *Principle of Detailed Balancing*;

- (1.13)

Although this relationship is obtained under chemical equilibrium, the first equality in Equation 1.13 also holds at nonequilibrium conditions. The reason for this is that chemical reactions are assumed to be too slow to disturb the Maxwell–Boltzmann distribution of energy among internal molecular degrees of freedom, hence local thermodynamic equilibrium among internal degrees of freedom. This further allows the definition of a single temperature during chemical relaxation. The principle of detailed balance provides a tool to determine the reverse rate of reaction, based on the forward rate constant and equilibrium coefficient. A less common practice is also to assign the reverse rate independently of the forward rate [11]. However, this approach is less accurate than using the equilibrium constant, for the obvious reason that the thermodynamic data are known much more accurately than the kinetic data.

#### 1.3.3 Chemical Time Scales and Nonequilibrium Effects

Each species and reaction in a kinetic mechanism evolves based on definite time scales. Species time scales can be defined as follows:

- (1.14)

in which use has been made of Equation 1.10. In this form, the chemical time scale is determined based on the collective effect of all chemical reactions which either consume or produce species *i*. If only one reaction is considered, say the *k*th reaction, then the reaction time scale can be defined as , in which is the reaction time based on species participating in reaction *k*. Combustion chemical reactions are usually characterized by a wide spectrum of chemical time scales. When a chemical system undergoes either heat or work interaction with the surrounding environment on a time scale , depending on how the chemical time scales compare with , the system could be either in the state of local thermodynamic equilibrium (LTE), , nonequilibrium, , or frozen equilibrium, . The sudden expansion of combustion products in an internal combustion engine, or through a hypersonic nozzle and sudden cooling of combustion products through a heat exchanger with constant area, are examples of such interactions.

According to the *principle of Le Chatelier*, the internal dynamics shifts towards minimizing the effect of external change and re-establishing a new chemical equilibrium, consistent with the new values of the state variables. If the interaction lowers the gas temperature and density of a highly dissociated mixture, then the internal dynamics will shift in the exothermic direction so as to minimize the cooling effect of interaction. As a result, three-body recombination reactions – for example, H + H + M = H_{2} + M and H + O_{2} + M = HO_{2} + M – become an important part of the energy restoration process. Bimolecular reactions also shift towards the exothermic direction. From a kinetics standpoint, three-body reactions have small or zero activation energies, which makes them almost temperature-insensitive and rather highly pressure- (density) sensitive, whereas the rate of bimolecular reactions which involve activation energies are temperature-sensitive [12]. Therefore, *sudden* cooling to low temperatures and lowering of the density will depress the rate of recombination and exothermic bimolecular reactions markedly, and the exothermic processes will lag in their attempt to restore the equilibrium. A failure to release the latent energy of molecule formation enhances the cooling and puts the system farther out of equilibrium. If the expansion is fast enough, then the exothermic lag grows indefinitely and the composition becomes frozen [13]. An important situation where predictions based on equilibrium fail is the predictions of CO at the exhaust of an internal combustion engine. Here, the main reaction step in oxidation of CO to CO_{2} is CO + OH = CO_{2} + H, which involves an activation energy of about 18 kcal mol^{−1}. This energy barrier makes the reaction temperature-sensitive such that, when the temperature falls, the reaction becomes slower, and so does the energy-restoration process. Such an effect shows itself in departures from LTE predictions. Janbozorgi *et al.* have examined the expansion stroke of an internal combustion engine with an intermediate piston speed, and compared the kinetic predictions with frozen and LTE predictions, as shown in Figure 1.1 [12]. Clearly, during the early stages of expansion, where the piston speed is slow, the state of the gas follows the LTE predictions and departures emerge as the piston speed increases.

#### 1.3.4 Kinetics Simplification and Reduction

Considering the fact that a DKM involves a wide spectrum of chemical time scales associated with chemical species, the system of Equation 1.10 comprises a set of *stiff* ordinary differential equations (ODEs), which could be computationally expensive for reacting flows. As a result, a great deal of effort has been devoted to developing methods for reducing the size of DMKs.

A quasi-steady-state approximation (QSSA) [14, 15], which is usually employed for short-lived radicals, assumes that after a so-called “induction period” the reactions consuming radicals become much faster than those producing them; hence a low, stationary level of these intermediates emerges. Mathematically, this is equivalent with zero net rate of change in Equation 1.10, which is a deliberate transition from differential to algebraic equations for these intermediates. However, deciding which radicals this assumption can be applied to requires a good deal of knowledge and physical intuition on the part of the kineticist.

A partial equilibrium approximation (PEA) [16] is invoked for reactions which reach a state of dynamic equilibrium. The entropy generation due to a chemical reaction *k* can be expressed as:

- (1.15)

where λ_{k} is the progress variable of reaction *k*. The necessary and sufficient condition for reaction *k* to be in equilibrium is that the entropy must be a maximum with respect to all possible changes , and so

- (1.16)

is the constraint to be satisfied by a reaction to be in partial equilibrium. As mentioned in Ref. [17], however, the check for whether a reaction *k* satisfies partial equilibrium assumption or not, should be based on the order of magnitude of the net rate of change due to reaction *k* compared to net rate of production and net rate of consumption due to that reaction individually. The more advanced methods are based on ideas from dynamical systems, computational singular perturbation (CSP) [18] and inertial low-dimensional manifold (ILDM) [19], to automatically identify the species and reactions for which QSSA and PEA hold. Other elegant methods, such as Adaptive Chemistry [20], Directed Relation Graph (DRG) [21], the ICE-PIC method [22] and rate-controlled constrained-equilibrium (RCCE) [5, 23] have been proposed and developed. Whilst a detailed presentation of combustion kinetics modeling has been undertaken in Chapter 8, details of the RCCE method are provided in the next section.

##### 1.3.4.1 Rate-Controlled Constrained-Equilibrium (RCCE) Method

The idea of RCCE is a logical extension of chemical equilibrium constrained to the conservation of neutral atoms (as discussed in Section 1.2.3). Owing to their very high activation energies, slow ionization and nuclear reactions can be assumed frozen over the energies and time scales encountered in combustion applications, leading to conservation of atomic elements. By the same token, the cascade of constraints in a chemically reacting system can be easily extended based on the existence of classes of slow chemical or energy-exchange reactions which, if completely inhibited, would prevent the relaxation of the system to the complete chemical equilibrium. For instance, a heavy hydrocarbon does not break down into smaller fragments unless the C–C bonds are broken; the total number of moles in a reacting system does not change unless a three-body reaction occurs; and radicals are not generated in the absence of chain-branching reactions, the definition of a single temperature in a chemically reacting system is based on the observation that thermal equilibration among translation, rotation and vibration is in general faster than the chemical reactions [12]. Consistent with the perfect gas assumption and definition (Equation 1.6), the constraints imposed on the system by the reactions are assumed to be a linear combination of the mole numbers of the species present in the system:

- (1.17)

where includes kinetic constraints in addition to the elemental constraints defined earlier, and has the same meaning as before; the value of constraints *j* in species *i*. The mathematical work is exactly the same as that presented under chemical equilibrium, and the constrained-equilibrium composition of the system is, therefore, expressed by Equations 1.7 and 1.8. By taking the time derivative of Equation 1.17 and using Equation 1.10, it is possible easily to obtain:

- (1.18)

where is the number of reactions in the mechanism. Clearly, any reaction *k* that does not change all constraints *j* is in constrained-equilibrium, and not required. The working equations of RCCE in terms of constraint potentials have been derived for a constant volume, constant energy system in Ref. [8].

Since the oxidation of any heavy hydrocarbon fuel is characterized by essentially the complete fragmentation of large molecules to a mixture of small hydrocarbons, it is a mixture of the smaller fuel fragments, mostly C_{1} and C_{2}, that eventually react to form the final product of combustion and release heat [24]. On the basis of this fact, Janbozorgi *et al.* [8] considered the tail of this process – namely the oxidation of C_{1} fuels (CH_{4}, CH_{3}OH, CH_{2}O) – and determined a set of constraints that were able to accurately model the C_{1} chemistry in a unified manner over a wide range of initial temperatures, pressures, and equivalence ratios. Figures 1.2 and 1.3 show the predictions of their model against a detailed kinetic mechanism which included 29 species and 133 reactions. It was concluded that the dominant oxidation path of CH_{4} at low temperature is through the formation of methyl peroxides:

whereas at higher temperature formation of the methoxy radical, CH_{3}O, dominated the chemistry:

The competition between the following two reactions differentiates the low- and high-temperature paths:

The formation of alkyl peroxides at low temperatures is responsible for the observed *cool flame* phenomenon in higher hydrocarbons, and is the most important aspect of the low-temperature combustion chemistry of hydrocarbon fuels. Cool flames are described in detail in Chapter 13, while general low-temperature combustion chemistry is described in Chapter 2. It should be noted that, as GRI-MECH 3.0 does not involve alkyl peroxides, it is incapable of modeling low-temperature homogeneous oxidation of methane.

### 1.4 Laminar Premixed Flames

- Top of page
- Introduction
- Combustion Thermodynamics
- Chemical Kinetics
- Laminar Premixed Flames
- Diffusion Flames
- Conclusions
- References

The competition between chemical energy release and energy loss through the boundaries determines a global time scale, known as the “ignition delay time,” . Knowledge of this characteristic time is crucial in the design and operation of a number of practical and research devices, such as diesel engines, spark-ignition (SI) engines, HCCI engines and RCMs. In SI engines, the chemical activities in the *end gas* are accelerated by the isentropic compression due to piston motion and the propagating flame which, under the correct thermodynamic conditions, can lead to the well-known *knocking* phenomenon. It has been well established [25-27] that, depending on the initial temperature and gradients in temperature and mixture composition within the reactor, the reaction zone can have different speeds, ranging from the laminar premixed flame speed to infinity. Such gradients could be due to imperfect mixing. Assuming that the temperature gradient is the only nonuniformity in a motionless mixture, successive points along the gradient have different delay times, leading to the propagation of an *autoignition wave* [25]. In general, a characteristic *autoignition velocity* (*u*_{ig}) relative to the unburned gas can be defined as:

- (1.19)

It has been recognized [28, 29] that, if this velocity is comparable with the acoustic velocity, then the pressure wave generated by the combustion energy release can couple with the autoignition front, with mutual reinforcement of both fronts and a very rapid reaction. When the autoignition wave moves much faster or slower than the acoustic velocity, however, such coupling does not occur and the combustion is less intense [25]. In the extreme slow motion of this wave which, according to Equation 1.19, corresponds to a large gradient between burned and unburned gases, the overall chemical time scale becomes comparable with the diffusion time scales and *laminar flames* emerge. However, for faster velocities molecular transports do not play an important role in determining the wave structure. For this reason, laminar flames are usually called “diffusion waves,” and “detonation waves” are a shock wave coupled with an autoignition front.

A flame is a *self-sustaining* propagation of a *localized* combustion zone at *subsonic* velocities [2]. The balance between the molecular mass transport of fresh reactants into the reaction zone, chemical energy release within this zone, and the energy carried from the reaction zone back into the reactants by heat conduction over the length , determines the wave structure. In this case, the velocity defined in Equation 1.19 is a unique chemico-physical property of the mixture, known as “laminar flame speed.” A typical laminar flame structure is shown in Figure 1.4. Due to a high concentration of radicals in the reaction zone, and the steep gradient of concentrations towards the preheat zone, light species (e.g., H, H_{2}, O, and OH) diffuse back into the preheat zone and partially react with fuel and oxygen. These reactions are responsible for the partial combustion energy release in the preheat zone. For this reason, the addition of hydrogen to a fuel mixture can enhance the flame speed, as it promotes the chemistry in the preheat zone. In contrast, absorption of the highly reactive hydrogen atom in stable molecules or less-reactive radicals may result in a lowering of the flame speed. The important reactions that hydrogen atom can undergo are:

The first reaction – “chain branching” – has a high activation energy, which makes it temperature-sensitive. It is, therefore, faster within the reaction zone. The second reaction, however, is a three-body reaction and is almost temperature-insensitive and rather, as mentioned earlier, is highly pressure-sensitive. These reactions are favored in the recombination direction as the pressure is elevated. As the pressure is raised, the rate of conversion of active hydrogen radicals to less-active hydroperoxy radicals within the preheat zone is increased, and both the flame speed and flame thickness are reduced. Flame speed can also be reduced by adding *flame inhibitor* chemicals, such as I_{2} and Br_{2} which, once dissociated to I and Br radicals, will act as a sink for hydrogen by forming stable I–H and Br–H molecules.

#### 1.4.1 Governing Equations

In the absence of body forces and external energy sources, and also in the limit of low-Mach number flows, the momentum equation can be neglected and the equations governing the propagation of a laminar flame can be written as [30]:

- Mass balance
- (1.20)

- Species mass balance
- (1.21)

- Energy balance
- (1.22)

where, , , , , , and , are mixture density, convective velocity, diffusion velocity, pressure, molecular mass, mass fraction, and heat capacity at constant pressure of species *i*, respectively. is the thermal conductivity of the mixture and is the net molar rate of consumption of species *i*. These equations are closed when using the Fick's law for the diffusion velocities:

- (1.23)

where is the diffusion coefficient of species *i* into the rest of the mixture. A concise discussion of the difference between binary and multicomponent diffusion coefficients can be found in Ref. [2]. Equations 1.20-1.22 describe the dynamics of the flame propagation until it reaches the steady-state flame speed . Several standard software packages are available that can be used to solve these equations, among which PREMIX [31] is the most widely used. While it is possible to solve numerically for the flame structure using detailed kinetic mechanisms, several approximate mathematical techniques have also been developed to solve these equations. An excellent detailed description of different analytical methods for solving laminar premixed flame structure can be found in Ref. [17].

#### 1.4.2 Experimental Approach

Laminar burning speed is a fundamental thermo-physico-chemical property of each fuel–air mixture, which depends only on the temperature, pressure, and mixture composition. It characterizes the rate at which the unburned reactants are consumed or, equivalently, the production rate of the burned gas. As laminar burning speed is a mixture property, its measurement is of fundamental importance in several respects. Of particular interest in practical applications is to know how fast the flame will propagate within an internal combustion engine. It may also serve as a benchmark data for testing the predictive capabilities of chemical kinetic mechanisms developed for several fuels over a wide range of thermodynamic conditions (pressure and temperature). The reliability of such data depends critically on the accuracy of the measurements and the experimental methods. The accuracy requirement of the laminar burning speeds becomes far more demanding at higher temperatures and pressures, under which conditions gas turbine combustors and internal combustion engines operate. Another application of laminar burning speed measurement is in turbulent flame speed correlations. Abdel-Gayed and Bradley, [32], have shown that the turbulent flame speed of a mixture can be estimated reasonably accurately by knowing the laminar burning speed of the mixture at the corresponding temperature and pressure [32]. Consequently, these authors proposed a correlation with the following form:

- (1.24)

where and are the turbulent and laminar burning speeds, respectively, and are model constants, and is the turbulent fluctuations velocity. A thorough explanation of theoretical basis of turbulent combustion can be found in Chapter 9.

##### 1.4.2.1 Laminar Burning Speed Measurement Techniques

Laminar burning speed measurement is a sensitive process, which requires careful design of the experimental apparatus and accurate data acquisition system. Some of the more advanced methods to measure this parameter are described in the following section.

###### 1.4.2.1.1 Counter Flow Flame Methods

In this method, two opposing nozzles are used to obtain a plug flow, in which configuration two streams of fresh reactants are impinged against each other so as to produce two stationary flames and a stagnation point; this configuration is termed “counter flow twin flames.” A second version involves one nozzle blowing hot inert gases, while another nozzle drives out any unburned premixed gases. The advantage of the counter flow twin flame is that it lowers the aerodynamic stretch effects; the configuration is shown in Figure 1.5. In this technique, the laminar burning speed is obtained by precisely mapping the flow velocity field using an optical method such as digital particle image velocimetry (DPIV). The velocity field is composed of axial and radial components; the gradients in the axial and radial components introduce stretch and strain effects, which could considerably affect the burning speed.

After mapping the velocity field by using the DPIV method, the minimum axial velocity is defined as the reference “stretched flame speed.” The stretch rates at the reference point are then modified by changing the upstream velocity. After having achieved the burning speeds at different stretch rates, the laminar burning speeds can be defined by extrapolating the stretched burning speeds to zero-stretched burning speed [32, 33].

###### 1.4.2.1.2 Outwardly Propagating Constant Pressure Spherical Flame Method

In this method, an outwardly propagating spherical flame is used to measure the laminar burning speed [34-37], with the pressure and temperature of the unburned gas remaining constant while the flame expands. In order to achieve this, a spherical vessel with optical windows is located in a Schlieren set-up to record the location of the flame front, and consequently to measure the flame front speed, . A typical snapshot of a propagating spherical flame is shown in Figure 1.6, where the stretched laminar burning speed can be defined as:

- (1.25)

Here, is the stretched laminar burning speed, is the density of burned gas, and is the density of the unburned gas zone. The stretch rate in spherical flames is

- (1.26)

which is a function of flame radius, and flame speed, *S _{f}*.

After measuring , it is plotted against stretch rate , and the zero-stretch laminar burning speeds are then obtained using an extrapolation method. The predictions of this method are most reliable when flame radius is too large for stretch effects to be unimportant.

###### 1.4.2.1.2.1 Extrapolation of Stretched Laminar Burning Speeds

In the methods explained above for burning speed measurement, stretch is important; hence, an extrapolation process is required to obtain a zero-stretch laminar burning speed. Especially in counter flow twin flames, the stretch rate is high, usually greater than 300 s^{−1}. Consequently, different extrapolation functions are normally used, including linear, polynomial, and logarithmic functions [38-40], each of which provides a different prediction for the zero-stretch laminar burning speed. As yet, however, there is no consensus as to which method gives the most accurate prediction.

###### 1.4.2.1.3 Flat Flame Burner Method

In this method, a flat flame burner is used to produce a perfect unstretched laminar flame [41, 42]. In fact, the burning speed can correlate with stream velocity. However, as a large heat transfer occurs from the flame to the burner and cooling water, the results obtained must be corrected. Unfortunately, when using this method the laminar burning speeds can be determined over only a limited range of temperatures and pressures.

###### 1.4.2.1.4 Constant Volume Spherical Vessel Method

In this technique, the propagating flame compresses the unburned gas isentropically. The main advantages of this method compared to others are:

Laminar burning speeds can be measured at high temperatures and pressures.

The stretch effects are very small due to large flame radii and, therefore, no zero-stretch extrapolation is required.

Spherical chambers have been used by Metghalchi and Keck [41] to measure burning speeds for a wide range of fuels, equivalence ratios, diluents concentrations, pressures, and temperatures. A more comprehensive explanation of this approach is presented in the following section.

###### 1.4.2.1.5 Thermodynamic Model

The theoretical model used to calculate the burning speed from the pressure rise is based on that previously developed by Metghalchi and Keck [41], but has been modified to include corrections for energy losses due to electrodes and radiation from the burned gas to the wall, as well as the temperature gradient in the preheat zone. In this case, it is assumed that gases in the combustion chamber can be divided into burned and unburned gas regions, separated by a reaction layer of negligible thickness. The burned gas in the center of chamber is divided into *n* shells, where *n* is proportional to the combustion duration. Although the burned gas temperature of each shell is different, all of the burned gases are in chemical equilibrium with each other. The burned gases are surrounded by a preheat zone having a variable temperature, and this is itself surrounded by unburned gases. A thermal boundary layer separates the unburned gas from the wall. The effect of energy transfer from the burned gas to the spark electrodes is considered by a thermal boundary layer. A schematic of the model used for this method is shown in Figure 1.7, and further details are available in Refs [43, 44]. The system of equations comprises the constancy of volume and energy balance, namely

- (1.27)

- (1.28)

where and are the initial specific volume and energy of the unburned gas in the chamber, is the specific volume of isentropically compressed burned gas, and is the specific volume of isentropically compressed unburned gas. , and are the displacement volume of the wall boundary layer, the displacement volume of the preheat zone ahead of the reaction layer, and the displacement volume of the electrode boundary layer, respectively. , , and are the specific energy of isentropically compressed burned gas, specific energy of isentropically compressed unburned gas, specific heat ratio of unburned gas and radiation energy loss from the burned gas zone, respectively. The above equations have been solved for two unknowns; burned mass fraction and the burned gas temperature of the last layer. Given pressure, , as a function of time, the equations can be solved numerically using the method of shells to obtain the burned mass fraction, , as a function of time and temperature distribution . Ultimately, the burning speed may be defined as:

- (1.29)

where is the area of a sphere having a volume equal to that of the burned gas. In this method, the measured laminar burning speed data from spherical vessel (pressure method) can be fitted to the following power law relationship:

- (1.30)

where is the burning speed, at an arbitrary thermodynamic reference point , in cm s^{–1}. is the mixture equivalence ratio, is the unburned gas temperature (in K), and is the mixture pressure in atmosphere. , , and are model constants that are different for the different mixtures. The values of the model constants and parameters for three hydrocarbon fuels, JP8, JP10, and ethanol, are listed in Table 1.1. Also shown (in Figure 1.8) are the model predictions for laminar burning speeds of JP8–air mixture at three different equivalence ratios, an initial temperature of 500 K, and an initial pressure of 1 atm.

S_{b∘} (cm s^{–1}) | a_{1} | a_{2} | T_{∘} (K) | P_{∘} (atm) | |||
---|---|---|---|---|---|---|---|

JP8 | 93.6 | 0.22 | −4.4 | 2.13 | −0.18 | 500 | 1 |

JP10 | 62 | 1.51 | −0.89 | 2.02 | −0.16 | 450 | 1 |

Ethanol | 35.5 | −1.89 | 2.09 | 1.85 | −0.2 | 300 | 1 |

### 1.5 Diffusion Flames

- Top of page
- Introduction
- Combustion Thermodynamics
- Chemical Kinetics
- Laminar Premixed Flames
- Diffusion Flames
- Conclusions
- References

In many practical combustion systems, the fuel and oxidizer (air) are not mixed before combustion. In these cases, the fuel and oxidizers mix due to convection or diffusion and the reaction takes place instantaneously; these flames are called *diffusion flame* or *nonpremixed flame*. Examples of diffusion flames are flames in furnaces, candle flames, ramjet, and jet engines. A typical one-dimensional diffusion flame, where the fuel and oxidizer flow in different directions, mix, and react, is shown in Figure 1.9. The energy of combustion is then transported in both directions into cold regions; profiles of the temperature and concentrations of fuel and oxidizer are shown in Figure 1.9. Diffusion flames may either be laminar or turbulent. As diffusion flames have been discussed in detail in many classical combustion texts [45-52], only laminar diffusion flames will be reviewed at this point.

A classical example of a diffusion flame, initially presented by Burk and Schumann [53], describes the steady-state coaxial of a gaseous fuel issuing into an oxidizing environment. The flame shape and its height have been investigated, on a theoretical basis, in this situation. One practically important aspect of a diffusion flame is that of *droplet burning*, which occurs in many combustors. In most devices, the fuel is sprayed into the combustor, where shear forces between the fuel and oxidizer cause the spray to break up into droplets. A typical spherical droplet combustion describing the liquid drop, fuel vapor, and reaction (flame) zone, is shown in Figure 1.10.

The first stage is the evaporation of a droplet, which follows the law that has been verified experimentally:

- (1.31)

where is the evaporation coefficient, *d* is the diameter of liquid drop, and is the initial diameter. The evaporation coefficient can be determined as:

- (1.32)

where , , and are the density of the liquid and vapor, and the thermal conductivity at the surface of the drop, respectively. is the Spalding transfer number, which is defined as:

- (1.33)

where , and . Also is the mass fraction of fuel at the surface of the liquid, and is the mass fraction of the fuel far away. The mass burning rate, flame position, and flame temperature of a single fuel droplet have been reviewed extensively by Kuo [45]. The mass burning rate is given by:

- (1.34)

where , , and represent, respectively, the radius of the droplet, the density, and the diffusion coefficient at the droplet surface. The flame location can be determined as:

- (1.35)

where is the stoichiometric fuel–air ratio, is the concentration of the oxidizer far away from flame, and is the flow speed at the surface, which can be obtained as:

- (1.36)

The flame temperature can be calculated using the following relationship:

- (1.37)

where

- (1.38)

A more in-depth discussion of the subject, droplet and spray combustion is presented in Chapter 7.

### 1.6 Conclusions

- Top of page
- Introduction
- Combustion Thermodynamics
- Chemical Kinetics
- Laminar Premixed Flames
- Diffusion Flames
- Conclusions
- References

The fundamentals of combustion have been reviewed in this chapter. Combustion thermodynamics, chemical equilibrium, and time-dependent chemical kinetics in general, and rate-controlled constrained-equilibrium in particular, have been described. The laminar premixed flame structure, and the experimental methods used to determine burning speeds have also been reviewed in depth. Finally, the details of diffusion flames have been presented under an extremely fast chemistry approximation. For reasons of space limitation in this chapter, many other subjects are reviewed in much greater detail in the following chapters of this volume.

### References

- Top of page
- Introduction
- Combustion Thermodynamics
- Chemical Kinetics
- Laminar Premixed Flames
- Diffusion Flames
- Conclusions
- References

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