Combustion is the burning of materials (Fuels), usually by reaction with the oxygen in air. During the reaction heat is released, and some of the common chemical products of burning are Carbon Dioxide, water vapor and *soot*.

Combustion is probably the oldest science, since soot deposited about 800,000 years ago on the roof of a cave near Beijing in China could only have been due to fires deliberately created by man for warmth and to cook his food. The primitive fuel technologists who lit these fires must have been aware of the role of draughts of air and the buoyancy of the products. They would also quickly discover the role of radiant Heat Transfer since a lone piece of charcoal will not remain alight, whereas a group of pieces of charcoal would radiate heat to each other and thus maintain a viable fire.

Combustion may be classified as:

Intentional, where the purpose is to release the chemical energy.

Unintentional, as in property destroying fires.

Naturally-occurring, as in forest fires caused by lightning.

Intentional combustion is one of the most important technologies in moderm society as illustrated by the fact that developed countries spend more on fuel than on food. The main applications are: electrical power production, warmth for buildings and industrial process heat. In these applications, the fuel and air for the reaction are fed to burners fitted in Boilers or Furnaces. After the heat is removed, the products of combustion pass into the flue and are normally dispersed high into the atmosphere from the Chimney.

*Combustion products* may include a number of pollutants such as: Carbon Monoxide, oxides of nitrogen, Sulfur Dioxide, particulates, unburned hydrocarbons and dioxins (if Chlorine is present). The permissible levels of the various *pollutants* emitted to the atmosphere are governed by legislation and for the foreseeable future, these levels will be progressively reduced. It is now strongly suspected that the increasing level of carbon dioxide in the atmosphere is leading to a significant increase in atmospheric temperature by the so-called Greenhouse Effect. For this reason, carbon dioxide is now regarded as a pollutant even though it occurs naturally in the atmosphere. It is the task of the combustion engineer to maximize the efficiency of the combustion process and simultaneously minimize the production of pollutants. (See also Air Pollution.)

The science of combustion involves complex interactions between many constituent disciplines, including: chemical kinetics; fluid dynamics—including Turbulent flow; Thermodynamics; heat and Mass Transfer; and Two-phase flow (i.e., the gas phase, and a liquid or solid fuel phase).

Each of these topics may be represented mathematically by equations, however, since all the processes are interdependent, these governing equations must be solved simultaneously to model the overall combustion process. Furthermore, since most of the equations are nonlinear partial differential equations, it is necessary to evaluate the equations on a digital computer at a number of grid nodes distributed throughout the geometrical domain of interest.

Nowadays it is generally considered to be more satisfactory to use a numerical model to study combustion systems since other techniques, such as physical models, can only represent a part of the problem. For example, there is no simple method to scale a combustion system from small- to large-scale since simple correlation parameters such as geometry, residence time, velocity or radiant heat transfer follow different *scaling* laws and hence do not result in satisfactory scaling criteria. Nevertheless, at the present time mathematical modeling must always be checked experimentally to verify that the equations used correctly represent all the relevant physical and chemical phenomena.

Many of the equations governing the behavior of combustion systems are well established and they can be used with reasonable confidence. These include the equations of conservation of mass, momentum and energy. (See Conservation equations.) There is more doubt about the equations of Turbulence and Mixing since these are models which attempt to use tractable equations to represent the fluctuating phenomena in the combustor.

In the case of two-phase flow when using liquid or solid fuels, the governing equations are well established for certain well-defined processes. Thus the evaporation rate of a drop of pure fluid can be computed with some confidence, but the drag and devolatilization rate of a complex coal particle are not so well known and some discrepancies can be expected between computed and experimental results. It is important to recognize that such discrepancies do not invalidate mathematics! Rather, they indicate areas in which further research is required to improve the particular governing equation which gives rise to the discrepancy.

The governing equations for the mean motion of a fluid are based on the following laws of conservation:

Conservation of Mass;

Conservation of Momentum;

Conservation of Energy.

The Continuity Equation. Applying the Conservation of Mass to a fluid passing through an infinitesimal, fixed-control volume, the following equation is obtained:

where ρ is fluid density, u velocity and t time.

The Momentum Equation. Conservation of momentum states that the total momentum within a system remains constant during the exchange of momentum between two or more masses in the system. For a fluid passing through an infinitesimal, fixed-control volume, the following momentum equation applies:

The stress tensor δ_{ij} represents the external stresses, consisting of normal stresses and shearing stresses. For a Newtonian fluid, the *stress tensor* is related to pressure and velocity in the following tensor form:

where η is viscosity and γ_{ij} is the Kronecker delta function (γ_{ij} = 1 if i = j and γ_{ij} = 0 if i ≠ j). The equations of conservation of mass and conservation of energy are known as the *Navier-Stokes equations*.

The Energy Equation. The *First Law of Thermodynamics* states that the increase of energy in the system is equal to heat added to the system plus the work done on the system. If the flow in a *Cartesian coordinate system* is assumed to be incompressible with a constant coefficient of thermal conductivity, then the energy equation is:

where

and the last term, S_{h}, the source term, includes the heat of chemical reaction, radiation, and any interphase exchange of heat.

The Equation of State. As there are many unknowns in the conservation equations, additional equations are required for their solutions. The relationships between the thermodynamic variables pressure, density, temperature, internal energy and enthalpy (p, ρ, T, e, h), and the relationships between transport properties viscosity and thermal conductivity (γ, λ) are utilized to generate variables. The equations utilized are known as equations of state, examples of which for a PERFECT gas are as follows:

Where
is the universal gas constant and c_{v} and c_{p} the specific heat capacity at constant volume and constant pressure, respectively.

Turbulent Motion. Turbulent fluid motion is an irregular condition of flow in which various quantities show a random variation with time and space. In practical systems, turbulent flows are inevitable even in the absence of chemical reaction, causing considerable difficulties in practical flow simulation. When the unsteady Navier-Stokes equations are applied to turbulent flows, the time and space scales of the turbulent motion are so small that the number of grid points and the small size of the time steps required in the calculation are outside the realm of current computer technology. Present-day computational fluid mechanics is thus achieved through the use of time-averaged Navier-Stokes equations; the gross effects of turbulence on time-mean flow is considered, while the detailed structure of the turbulence is disregarded. (See also Turbulence.)

The time-averaged Navier-Stokes equations are referred to as the *Reynolds equations of motion*. They are of the same form as the fundamental Navier-Stokes equations, but incorporate the terms of apparent stress gradients and heat-flux quantities associated with turbulent motion. As these quantities must be related to the mean flow variables through turbulence models, the Reynolds equations are derived by decomposing the dependent variables in the conservation equations into time mean and fluctuating components. The entire equation is then time averaged. Thus,

and Θ' are the time mean and fluctuating components of dependent variables u, v, w, T, p, ρ. Fluctuations in other fluid properties such as viscosity, specific heat and thermal conductivity are usually negligible.

Taking the continuity equation, for example, the instantaneous fluid velocity ui, is first decomposed into its time mean and fluctuating components. The velocity components are substituted into the equation. By definition, the time-average of a fluctuating quantity is zero, thus the continuity equation after time-averaging is transformed to:

For steady incompressible flows, ρ' = 0 and the above equation can be reduced to:

The time-averaged momentum equation can also be written as follows:

where the effect of turbulence is incorporated into the fundamental momentum equation through the *Reynolds stresses*
.

In turbulent flows, the velocity at a point is considered as the sum of the mean and fluctuating components. In the time-averaged momentum equation, the effect of the fluctuating components is to introduce the effect of turbulence, through a term involving the Reynolds stresses, . Turbulence modeling relates the Reynolds stresses to mean flow quantities so that the turbulent flow field can be calculated without calculating the full detail of the fluctuating flow. (See also Turbulence Models.)

In the k — ε model, Reynolds stresses are related to the mean flow via the *Boussinesq hypothesis*:

The effective or "turbulent" viscosity (η_{t}) is computed from a velocity scale (k^{1/2}), and a length scale (k^{3/2}/ε) which are predicted by the solution of transport equations for k and ε:

where

These well-known equations involve empirical constants which have the following values:

In highly-swirling flows such as are used in the near-field region of a burner, μ_{t} is strongly directional and the isotropic k – ε model may be inadequate. For such cases, we may solve additional differential equations for the Reynolds stresses.

The conservation equation for a chemical species (see Chemical Reaction) can be written as:

where X_{i} is the mass fraction of chemical species i; R_{i} is the mass rate of creation or depletion by chemical reaction; and S_{i} is the rate of creation by addition from a dispersed phase. J_{i} is the diffusion flux of species i, and is given by:

where
is the thermal diffusion coefficient and D_{im}, is the diffusion coefficient of species i in the mixture. In multistep and multispecies reactions, the source of chemical species i due to reaction, Ri is computed as the sum of the reaction sources over the k reactions that the species may participate in. The overall rate of production of species i is:

Reaction may occur in the gas phase between gaseous species, or at surfaces resulting in the surface deposition of a chemical species. The reaction rate, Rik is controlled either by kinetics or mixing. The Arrhenius reaction rate (see Arrhenius Equation) is calculated as:

When the reaction is mixing-controlled, the influence of turbulence on the reaction rate can be taken into account by employing the *Magnussen and Hjertager, 1976, model*. In this model, the rate of reaction R_{ik} is computed as:

where A is an empirical constant.

Laminar flame structures are reasonably well-understood for both premixed and diffusion flames. (See Flames.) On the other hand, turbulent flame structures are much more complicated. The basic problem in reaction rate models is due to the nonlinearity of the chemical equations, thus:

Solutions to this problem can be based on stirred reactor models, Reynolds decomposition, eddy break-up models and statistical descriptions. Over the last few years, attention has turned to the relationship between the turbulent flame and details of the turbulence structure, such as length scales and the frequency spectrum. These aspects have been surveyed by Bradley (1992)

The effect of turbulence is to increase the area of laminar flamelets due to wrinkling of the flame. In an un-stretched flame this gives an increase in the flame speed.

At sufficiently high rates of stretch, the flame is quenched; however, the effects of stretching in turbulent flames is difficult to generalize.

The laminar flamelet model, with assumed probability density functions (PDFs) which are functions of the reaction progress variable and the stretch rate, appears to be valid over a wide range. It has been used successfully to predict lift-off heights of turbulent diffusion flames, the combustion field in premixed swirling flow, and flame blow-off.

The energy equation is solved in the form of a transport equation for static enthalpy, h:

where k is the molecular conductivity, k_{t} is the effective conductivity due to turbulent transport (k_{t} = η_{t}/Pr_{t}) and S_{h} consists of source terms including heat of chemical reaction, any interphase exchange of heat, and other user-defined heat sources.

Enthalpy is defined as:

where m_{j}, is the mass fraction of species j',

Enthalpy sources due to reaction are defined as:

where
is the enthalpy of formation of species j' and R_{j'} is the volumetric rate of creation of species j'.

The laminar flow heat transfer to the fluid from the wall is calculated by a first order approximation of the heat flux:

In turbulent flows, the above equation is replaced by a log-law formulation based on the Analogy Between Heat and Momentum transfer:

where Pr is Prandtl Number and Pr_{t} turbulent Prandtl number. Within solid conducting regions, a simple conduction equation is used that includes the heat flux due to conduction and volumetric heat sources within the solid:

where is the rate of heat production per unit volume. This equation is solved simultaneously with the static enthalpy equation to yield a fully coupled conduction/convection heat transfer prediction.

Surface-to-surface radiation with participating media may be calculated by use of a Discrete Transfer Radiation Model (DTRM). The DTRM usually assumes no wavelength dependency (grey radiation), with the radiant intensity representing an integrated intensity across all wavelengths. (See also Radiative Heat Transfer.) The change in radiant intensity, dI, along a path ds, is given by:

where α_{abs} and α_{s} are coefficients which account for absorption and scattering and σ is the Stefan-Boltzmann constant.

This equation is integrated along a series of rays emanating from a single point in each discrete control volume on a surface. This series of rays defines a hemispherical solid angle about that point. The integrated intensity along each ray is:

where I_{0} is the radiant intensity at the start of the incremental path ds.

Enthalpy produced by radiation in the fluid is computed by summing the change in intensity along the path of each ray through each fluid control volume.

The intensity of radiation approaching a point on a wall is integrated to yield the incident radiant heat flux,

where Ω is the hemispherical solid angle and I is the intensity of the ray approaching the wall. The net radiation heat flux leaving the wall surface, , is computed as the sum of the reflected portion of and the emissive power of the surface:

where T_{p} is the temperature of the surface at point P, and E is the wall emissivity. The radiation heat flux calculated in this equation may be incorporated in the prediction of the wall surface temperature, T_{p}. From this equation, the radiation intensity I of a ray emanating from point P is:

The radiative absorptivity, α, of a gas is related to the absorption coefficient by:

where L is the radiation path length.

This set of equations together with any additional equations required to describe the behavior of the second phase (such as the evaporation of liquid drops) are generally evaluated using a computer package. The particular geometry and flow boundary conditions are input to define the particular problem to be modeled. The output from the calculation is available in both graphical and numerical form and can be used for experimental investigations, or to guide the design of burners installed in combustion systems.

#### REFERENCES

Magnussen, B. F. and Hjertager, B. H. (1976) *"On Mathematical Models of Turbulent Combustion with Special Emphasis on Soot Formation and Combustion"* 16th International Symposium on Combustion. Cambridge, MA. DOI: 10.1016/S0082-0784(77)80366-4

Bradley, D. (1992) *"How Fast Can We Burn"* 24th International Symposium on Combustion. Sydney. DOI: 10.1016/S0082-0784(06)80034-2

#### References

- Magnussen, B. F. and Hjertager, B. H. (1976)
*"On Mathematical Models of Turbulent Combustion with Special Emphasis on Soot Formation and Combustion"*16th International Symposium on Combustion. Cambridge, MA. DOI: 10.1016/S0082-0784(77)80366-4 - Bradley, D. (1992)
*"How Fast Can We Burn"*24th International Symposium on Combustion. Sydney. DOI: 10.1016/S0082-0784(06)80034-2