Combustion systems involve a complex interaction of many different physical processes. This includes fluid flow, heat transfer, chemical reactions, and po- tentially fluid-solid interactions. Some of the funda- mental equations that describe these processes are introduced in Chapters 3 and 4. Each of these pro- cesses is briefly described below in the context of nu- merical modeling.
Representation of turbulence
Large-scale combustion systems are typically char- acterized by turbulent, reacting flow conditions. The effect of turbulent flow (turbulence) on combustion processes is significant and must be considered to ac- count for this effect. As yet, it is not practical to model the full detail of the temporal and spatial fluctuations that are associated with turbulence. As computing resources become more powerful and our ability to handle the enormous amount of information that will be generated increases, it may one day be possible to model the details of turbulent flow on industrial com- bustion systems. Until that day, a simplified model representation of turbulence must be used.
Often, dealing with turbulence involves time-aver- aging the fundamental equations to eliminate the tur- bulent fluctuations and utilizing a separate turbulence model to account for the influence of turbulent fluc- tuations on the flow. The fundamental equations can then be solved for the mean quantities. Alternatively, large scale turbulent fluctuations can be directly solved while utilizing a turbulence model for the small scale fluctuations. This technique, called large eddy simulation (LES), is an important advancement in tur- bulence modeling but requires large computational re- sources compared to time-averaging.
Time-averaging is typically done either with Reynolds averaging, the conventional time-averaging, or with Favre averaging, a density-weighted averag- ing. The latter is better suited to handle the large den- sity variations experienced in combustion applications. Averaging of the conservation equations is accom- plished by first assuming that instantaneous quanti- ties are represented by mean and fluctuating portions as shown in Equation 1. By allowing φ to represent the dependant variable, this can be expressed as:
φ = φ φ+ ′ (1)
where φ is the instantaneous value, φ is the mean portion and φ′ is the fluctuating portion. Density- weighted averaging offers advantages over conven- tional time-averaging for combustion-related flows since it simplifies the treatment of large density changes. The density-weighted mean value, φ, is de- fined as:
φ ρφ ρ
= (2)
where ρφ is the time-averaged product of the instan-
taneous density (ρ) and instantaneous value (φ) and
ρ is the time-averaged density. The instantaneous value may then be written as the sum of the density- weighted average and the fluctuating value φ′′:
φ
=
φ φ
+ ′′
(3)Equation 3 can be substituted into the transport equation and then time-averaged to derive equations in terms of the mean quantities. While it is not impor- tant to detail the process here, it is important to note that the results produce additional terms in the result- ing equations. These extra terms are known as Reynolds stresses in the equations of motion and tur- bulent fluxes in the other conservation equations. Tur- bulence models are generally required to model these extra terms, closing the system of equations.
Fluid flow and heat transfer
Gas-phase transport in combustion systems is gov- erned by PDEs that describe the conservation of mass, momentum, component mass and energy. The conser- vation of mass or continuity equation is discussed in Chapter 3. The conservation of momentum is repre- sented by the Navier-Stokes equations that are also briefly discussed in Chapter 3. The Cartesian form of Navier-Stokes equations, as well as the continuity equation, can be found in the first four equations in Table 2. In these four equations, ρ is the density, u, v, and w are the velocity components, and x, y, and z are the coordinate directions, µ is the dynamic viscosity, P is the pressure, and g is the body force due to gravity. The remaining conservation equations used to de- scribe the gas-phase transport are the energy and com- ponent mass equations, expressed in Table 2 in terms of specific enthalpy and component mass fraction. The energy source terms are on a volumetric basis and rep- resent the contribution from radiative heat trans- fer,−∇iqr, energy exchange with the discrete phase par-
ticles, SHpart, and viscous dissipation, S
H. The component mass source terms include the mean production rate due to gas-phase reactions, Ri, and the net species pro- duction rate from heterogeneous reactions, Sipart.
Turbulence model
As previously mentioned, the process of time-aver- aging the conservation equations introduces extra terms into the equations. Numerous turbulence mod- els have been developed over the years to determine the values of these extra terms. One of the most com- mon and widely accepted approaches, known as the Boussinesq hypothesis, is to assume that the Reynolds stresses are analogous to viscous dissipation stresses. This approach introduces the turbulent viscosity µ and a turbulent transport coefficient σ into each equation. Most of the turbulence models currently used for fluid flow and combustion are focused on determin- ing µ. In the k-epsilon model (one of the most widely used and accepted), the turbulent viscosity is given as:
µ ρ ε µ t C k = 2 (4)
Table 2
Summary of Fundamental Differential Equations General form of the transport equation:
Physical Transport Source
Equation Parameter Coefficient Term
Γ S Continuity 1 0 Sm part X-Momentum ~u Y-Momentum ~v Z-Momentum w~ Enthalpy H~ H Turbulent k Energy k Dissipation ε Rate Species Y~i i
Other terms appearing in general form:
Nomenclature Subscripts/Superscripts
S part = source term accounting for exchange between e = effective discrete phase particles and gas phase t = turbulent
u, v, w = velocity components x, y, z = directional component
H = enthalpy i = ith
chemical specie
k = turbulent kinetic energy ~ = Favre (density weighted) average
ε = turbulent kinetic energy dissipation − = time-average
= effective viscosity part = discrete phase particle component = turbulent viscosity
g = gravitational vector (x, y, z)
= density
c1, c2 = model constants Ri = reaction rate Yi = species mass fraction
where Cµ is a model parameter, k is the turbulent ki- netic energy, and ε is the turbulent kinetic energy
dissipation. The turbulent kinetic energy and the dis- sipation are determined by solving an additional par- tial differential equation for each quantity as given in Table 2.
Discrete phase transport
Many combustion applications, including pulverized coal, oil, black liquor and even wood involve small solid or liquid particles moving through the combustion gases. The combustion gases are described by assuming that they represent a continuum, whereas a description of the solid and liquid fuel involves discrete particles. Describing the motion of this discrete phase presents unique modeling challenges. There are two basic refer- ence frames that can be used to model the transport of the discrete phase particles, Eulerian and Lagrangian.
The Eulerian reference frame describes a control volume centered at a fixed point in space. Conserva- tion equations similar to the ones used for gas trans- port are used to describe the transport mass and en- ergy of particles passing through this control volume. The interaction of the particle phase and the gas phase is accomplished through source terms in the respec- tive transport equations.
The Lagrangian reference frame considers a con- trol volume centered on a single particle. This ap- proach tracks the particle on its trajectory as it trav- els through space and interacts with the surrounding gases. The motion of a particle can be described by:
m du
dt F F
part part D g
= + (5)
where mpart represents the mass of the particle, upart is the particle vector velocity, t is time, and F D and
Fg represent
drag and gravitational forces. Aerodynamic drag is a function of the relative differences between particle and gas velocities, Reynolds number and turbulent fluctua- tions in the gas. Consideration is also given for mass loss from the particle due to combustion.1,2,3
Turbulence has the effect of dispersing or diffus- ing the particles. This dispersion effect has been iden- tified with the ratio of the particle diameter to turbu- lence integral scale. For large particle sizes, particle migration will be negligible, while at small sizes par- ticles will follow the motion of the gas phase. This ef- fect can be modeled using the Lagrangian stochastic deterministic (LSD) model.4 The LSD model computes
an instantaneous gas velocity which is the sum of the mean gas velocity and a fluctuating component. The instantaneous gas velocity is used in computing the right-hand side of Equation 5.
From the particle velocity the particle position, xpart, is expressed as: dx dt u part part = (6)
This equation, along with appropriate initial condi- tions, describes the particle trajectory within the com- putational domain.
Combustion
Homogeneous chemical reactions Homogeneous or
gas-phase combustion involves the transport and chemical reaction of various gas species. During this process, heat is released and combustion product spe- cies are formed. As mentioned, a transport equation for each of the chemical species involved is solved. The main objective of a gas-phase combustion model is to determine the mean production rate, Ri for turbulent combustion.
Various methods can be used to determine the pro- duction rate. One common method known as the Eddy Dissipation Combustion Model (EDM) was developed by Magnussen and Hjertager5 and is based on the
eddy break-up model.6 This model assumes that the
rate of combustion is controlled by the rate of mixing of the reactants on a molecular scale. The reaction rate is given by: W v v C k Y W v k RCT i ij ij j=1 N A k k kj j rc ( ′′ − ′) min : ′ ∈
∑
ρεTerm 1 Term 2 Term 3
Ri = Wi iω =
ρ
(7)
where Wi is the component molecular weight, v′ij and
′′
vijare the reactant and product stoichiometric coeffi-
cients for the ith species and the jth reaction, ε is the
turbulent dissipation, k is the turbulent kinetic energy,
CA is the model dependent mixing constant and RCTj denotes the set of species that are reactants for the jth
reaction. Term 1 represents the stoichiometric coeffi- cients in the particular reaction, Term 2 represents the molecular mixing rate, and Term 3 limits the reaction to the availability of individual reactants.
Magnussen7 later proposed the eddy dissipation con-
cept (EDC) to overcome some limitations of other mod- els. Specifically, the EDC model is applicable to non- premixed and premixed combustion and can be used with simplified or detailed chemistry to describe the reaction process. A detailed description of the EDC model can be found in Magnussen,7 Lilleheie et al.,8
Magnussen9 and Lilleheie et al.10
Magnussen’s premise is that chemical reactions oc- cur in the fine structures of turbulence where the tur- bulent energy is being dissipated. Within these struc- tures, molecular mixing occurs and the reactions can be treated at the molecular level. The EDC model is based on the concept of a reactor defined by a reaction zone in these fine turbulence structures. The length and time scales from the turbulence model are used to characterize these fine turbulence structures. The re- action rates within these fine structures can be defined with the specification of an appropriate chemical kinet- ics mechanism. These reaction rates are then related to the average reaction rates in the bulk fluid and then applied to the time-averaged transport equations.
While some of the simpler models mentioned above have been utilized extensively, the EDC provides a means of more accurately treating the complexities of coal combustion and modern combustion systems. This is particularly important as the sophistication of the heterogeneous combustion models improves.
Heterogeneous chemical reactions Simulation of coal combustion must account for a complex set of physical processes including drying, devolatilization, and char oxidation. When a coal particle enters the combustion zone, the rapid heatup causes moisture to evaporate.
Coal→Dry Coal + Water Vapor (8)
Evaporation is followed by devolatilization to produce volatiles and char.
Dry Coal→Gaseous Fuel + Char (9)
The volatiles consist of light gases (primarily hydro- gen, carbon monoxide, carbon dioxide, and methane), tars and other residues. The devolatilization rate can not be adequately represented with a single first-or- der kinetic expression. Ubhayaker et al.11 suggested
a two first-order kinetic rate expression:
Dry Coal Gaseous Fuel
Char K 1 i i d i i i i → +
(
−)
= α α 1 2, (10) where Kid is the kinetic rate of reaction and a
i is the vola- tiles’ mass fraction. The kinetic rates are first-order in the mass of coal remaining and are expressed in an Arrhenius form. The total devolatilization rate becomes:
d i
i id
K = K
∑
α (11)A more advanced model known as the Chemical Percolation Devolatilization (CPD)12,13,14 has been de-
veloped and is described elsewhere. Unlike the empiri- cal formulation of Ubhayaker et al.,11 the CPD model
is based on characteristics of the chemical structure of the parent coal.
Following devolatilization the remaining particle consists of char residue and inert ash. Char is assumed to react heterogeneously with the oxidizer:
Char + Oxidant→Gaseous Products + Ash (12)
A basic approach to char oxidation was described by Field.15 The effective char oxidation rate is a func-
tion of the kinetic rate of the chemical reaction and the diffusion rate of the oxidizer to the particle.15,16
Char + Oxidant Gaseous Products + Ash i i i ch K i → =1,22 (13)
where
K
ichis the effective char oxidation rate. The to-tal char oxidation rate is expressed as:
ch i
i ch
K K=
∑
(14)The Carbon Burnout Kinetic (CBK) model has been developed by Hurt et al.17 specifically to model the
details of carbon burnout. The model has a quantita- tive description of thermal annealing, statistical kinet- ics, statistical densities, and ash inhibition in the late stages of combustion.
Radiative heat transfer
Radiative heat transfer in combustion systems is an important mode of heat transfer and is described by the radiative transfer equation (RTE):
i Ω Ω Ω Φ Ω Ω ∇
(
) ( )
= −(
+)
( )
+( )
+ ′ → I r I r Ib r I λ λ λ λ λ λ λ κ σ κ σ π , , ( ) 4 r, ′ d( )
′∫
Ω Ω Ω (15)where κλ is the spectral absorption coefficient, σλ is the scattering coefficient, and Ibλ is the black body ra-
diant intensity.
This equation describes the change in radiant in- tensity, I rλ
( )
,Ω, at locationr
in direction Ω. The three terms on the right-hand side represent the decrease in intensity due to absorption and out-scattering, the increase in intensity due to emission, and the increase in intensity due to in-scattering.Radiative heat transfer information is obtained by solving the RTE (Equation 15) which is coupled with the thermal energy equation by the divergence of the radiant flux vector −∇ ℑi . The divergence can be ob- tained from: ∇ ℑ = −
∫
∫
∫
∞ ∞ i 4 0 4 0 0 κλ λ λ κλ λ λ π E T db ( ) I ( )Ω Ωd d (16)The two terms on the right-hand side account for emission and absorption, respectively.
Discretization of equations
In the preceding sections, a mathematical descrip- tion of combustion modeling, consisting of a fundamen- tal set of algebraic relations and differential equations of various forms, has been described. This includes fluid transport, particle transport, combustion and ra- diative heat transfer. Because this system of equations is too complex to solve with analytic methods, a numeri- cal method must be employed. The methods of discretizing the fluid transport and radiative heat trans- fer are of particular interest and are presented here. Finite volume approach
It should be recognized that many of the partial dif- ferential equations are of a single general form as pro- vided in Table 2 and can be expressed as:
∂ ∂ + ∂∂ + ∂∂ + ∂∂ = ∂ ∂ ∂ ∂ + ∂∂ t x u y v z w x x y (ρφ) (ρ φ) (ρ φ) (ρ φ) φ φ Γ Γφφ φ φ φ φ ∂ ∂ y + ∂∂zΓ ∂∂z+S (17)
Since many of the equations share this form, a single method can be used to solve all of the associ- ated equations. Most of these methods involve divid- ing the physical domain into small sub-domains and obtaining a solution only at discrete locations, or grid points, throughout the domain. The well-known finite
difference method is one such method. Another very
powerful method, that is particularly suited for use in combustion modeling, is the finite volume approach. The basic idea of the finite volume approach is very straightforward and is detailed in Patankar.18 The
entire domain is divided into non-overlapping control volumes with a grid point at the center of each. The differential equation in the form of Equation 17 is in- tegrated over the entire control volume and after some rearrangement becomes:
(18)
Carrying out the integrations, the resulting equation is:
∂
∂t
( )
∆ +V∑(
Cf f −Df)
= S ∆V fρφ φ ( )φ φ (19)
where ∆V is the volume of the control volume, Cf is the mass flow rate out of the control volume, Df is the diffusive flux into the control volume, and the sum- mation is made over all the control volume faces, f. The temporal derivative in the first term of Equation 19 can be expressed using a first-order backward differ- ence scheme: ∂ ∂
( )
= − ∆ +∆ t t t t t t ρφ ρ φ φ (20)The mass flow rate Cf is determined from the solu- tion of the mass and momentum equations while the diffusive flux Df is based on the effective diffusivity and the gradient at the control volume face. Combin- ing Equations 19 and 20 with the definitions of Cf and Df and an interpolated value for φf results in an alge- braic expression in terms of the dependant variable φi at grid point i and the neighboring grid points. This is expressed as:
ai i an n b
n i
φ =
∑
φ + (21)where ai and an are coefficients for the control volume and its neighbors respectively and bi represents the remaining terms. The number of neighboring values that appear in Equation 21 is a function of the mesh, the method used to interpolate the dependant vari- able to the control volume face, and the method used to determine gradients at the control volume face.
Following this procedure for each grid point in the entire domain produces a coupled set of algebraic equations. This set of equations can be solved with an appropriate method from linear algebra. Many differ-
ent techniques are possible and can be found in a ref- erence on numerical methods.
There are two advantages to the finite volume ap- proach. First, the dependant variable in the resultant discretized equation is a quantity of fundamental in- terest such as enthalpy, velocity or species mass frac- tion, and the physical significance of the individual terms is maintained. Second, this approach expresses the conservation principle for the dependant variable over a finite control volume in the same way the con- servation equation expresses it for an infinitesimal control volume. By so doing, conservation is main- tained over any collection of control volumes and is enforced over the entire domain.
Discrete ordinates method
Several radiative heat transfer models have been developed and many are described by Brewster19 and
Modest.20 A recent review of radiative heat transfer
models21 states that the discrete ordinates method
coupled with an appropriate spectral model provide the necessary detail to accurately model radiative heat transfer in combustion systems. This is one of the most common methods currently used to model radiative heat transfer.
The discrete ordinates method (DOM)22,23 solves the
radiative transport equation for a number of ordinate directions. The integrals over direction are replaced by a quadrature and a spectral model is used to de- termine radiative properties of κand σ. This results in a set of partial differential equations given by:
µ η ξ κ σ κ σ π m m m m m m m b m I x I y I z I I S ∂ ∂ + ∂ ∂ + ∂ ∂ = −
(
+)
+ + 4 (22)where µ η ξm, m, mare the direction cosines of the cho- sen intensity Im and Sm is the angular integral. This set of equations is solved by a method outlined by Fiveland22 to find the radiative intensities throughout
the combustion space. The source term for the energy equation can be found by summing over all directions:
∇ = − ′ ′ ′
∑
iqr T w Im m m 4κ σ 4 κ (23)Mesh generation
Once discretized, the transport equations must be solved at individual points throughout the domain. This requires that the individual points be specified and the relationship between other points be identi- fied. Displaying the points along with the connections between them creates a pattern that looks something like a woven mesh. The process of creating the mesh