PCC-LES M ODEL F ORMULATION

5.1. GAS-PHASE MODELLING 48

5.1 Gas-phase Modelling

5.1.1 LES Transport Equations for the Gas-phase

The basic transport equation for the gas-phase, introduced in section 3.1 and fil-tered by the procedure presented in section 4.3, including basic modelling and clos-ing strategies, are discussed next.

The Favre filtered continuity equation Eq. (4.6) including source terms to reflect mass transfer between the particle- and the gas-phase can be written as

∂ρ

∂t+ ∂

∂x_i

¡ρue_i¢

=X

n Ω_n¹, (5.1)

where P

nΩ_n ² is the mass transfer term between the dispersed and gas-phase ac-cording to a sub-model n, including contributions from pyrolysis and char conver-sion. Additionally, the transferred mass from the particle- to the gas-phase can include evaporation of moisture from coal. In this work, however, evaporation has been neglected because of the low moisture content of the considered coals. All considered source terms are summarised in Tab. 5.1.

Starting from the Favre filtered momentum equation Eq. (4.10) and applying the Boussinesq approximation, cf. Appendix A.3 for details, gives

∂ρue_i

∂t + ∂

∂x_i

¡ρue_iue_j¢

= ∂

∂x_i



µ_Eff Ã∂u_j

∂x_i +∂u_i

∂x_j −2 3

∂u_k

∂x_kδ_{i j}

!

−∂p

∂x_i +X

n Λ_in^{3, (5.2)} where P

nΛ_in is a momentum source considering gravitation and momentum transferred from or to the particle-phase by relative velocities between parti-cles and gas-phase. The filtered effective stress tensor τ_Eff is approximated as µ_Eff

³_∂

ue_i

∂x_i +^∂_∂x^ue_j^j−²₃^∂_∂x^uek_kδ_{i j}

´ with the effective viscosity µ_Eff =µ+µ_SGS. The filtered molecular viscosityµ is approximated using the filtered cell temperature based on the following equivalent of Sutherland’s law as

µ= A_sp T

³1.0+T_s/T´, (5.3)

1For the OpenFOAM implementation of this Eqn see listing Lst. C.4

2The mass transfer term corresponds toP

nΩ_n=Ω_Part from Eq. (4.4) and Eq. (4.6), here written as summation term to indicate that different sub-model contributions are considered individually

3cf. listing Lst. B.1

where A_s and T_s are mixture dependent coefficients. The SGS viscosity µ_SGS is computed by the LES turbulence model, which is discussed in detail in section 5.1.2.

The Favre filtered transport equation for the mass fraction of specieskis derived from Eq. (3.7), using the gradient assumption to model the unresolved scalar flux

ρu

:

_iSGSY_{kSGS= −}^µSGS

Sc_SGS

∂Yf_k

∂x_i, (5.4)

and reads

∂ρYe_k

∂t + ∂

∂x_i

¡ρue_iYe_k¢

= ∂

∂x_i õ

µ

Sc+ µ_SGS Sc_SGS

¶∂Ye_k

∂x_i

! +X

n Ψ_kn ⁴, (5.5) whereP

nΨ_kn is the species source term⁵, Sc, and Sc_SGS are the molecular and tur-bulent Schmidt numbers, respectively. The composition of the gas-phase is affected by homogeneous reactions and by species transfer between the coal particles and the gas-phase. The latter originates from pyrolysis reactions and char conversion.

To ensure that the sum of species mass fractions equals unity, Eq. (5.5) is solved for k−1 species only. The mass fraction of the inert N₂species is then calculated from the remainder of the sum of reacting species mass fractions

Y_N2=1−X

k−1Y_k. (5.6)

Starting from the simplified transport equation for sensible enthalpy Eq. (3.6), using the gradient assumption to model the unresolved enthalpy flux terms in anal-ogy to the unresolved filtered species flux term, yields

∂ρhe_S

∂t + ∂

∂x_i

¡ρue_ihe_S¢

= ∂

∂x_i µ

c_pα_Eff∂Te

∂x_i

¶ +X

n Θ_n, (5.7)

where the cumulative enthalpy source term is expressed as P

nΘ_n. The filtered effective heat diffusion coefficientα_Effis calculated from the molecular and the SGS viscosity using the following relation

c_pα_Eff=c_p¡

α+α_SGS¢

=c_p µ µ

Pr+µ_SGS Pr_t

¶

≈λ_Eff⁶, (5.8)

4cf. listing Lst. C.7

5Here, again considering the contributions of then-th sub-model, thusP

nΨ_kn=Ψ_k,Hom+Ψ_k,Part

in Eq. (3.7).

6Here, the OpenFOAM convention that the density is absorbed into the heat diffusion coefficient is used. Thus, Eq. (5.8) is written in the formc_pα=λinstead ofc_pα=λ/ρ.

5.1. GAS-PHASE MODELLING 50 where Pr and Pr_SGS denote the laminar and turbulent Prandtl numbers. For better stability and to reduce computational costs the diffusion term in Eq. (5.7) is approx-imated by α_Eff^∂_∂x^heS

i

7. Additionally, the enthalpy diffusion induced by species gradi-ents _∂x^∂

i

³ ρP

kh_k,SY_kV_ik´

has been neglected in Eq. (5.7). Thus the filtered sensible enthalpy transport equation can be written as

∂ρhe_S

∂t + ∂

∂x_i

¡ρue_ihe_S¢

= ∂

∂x_i Ã

α_Eff∂he_S

∂x_i

! +X

n Θ_kn^{. (5.9)}

For better readability source terms in the transport equations have been pre-sented by a summation term. A detailed list of individual source terms is listed in Tab. 5.1.

Table 5.1: Individual transport equation source terms.

Term Comment Section

Mass Source TermsP

nΩ_n,Eq. (5.1)

Ω_CC Particle char conversion 5.3.5

Ω_Dev Particle devolatilisation 6.2

Momentum Eqn. Source TermsP

nΛ_in,Eq. (5.2)

g_i Gravitational force

-Λ_i,Dr Particle drag forces 5.3.2

Λ_i,Disp Particle dispersion forces 5.3.2

Species Source TermsP

nΨ_sn,Eq. (5.5)

Ψ_k,Hom Homogeneous gas-phase reactions 5.1.3

Ψ_k,CC Particle char conversion (e.g. CO, CO₂) 5.3.5 Ψ_k,Dev Particle devolatilisation (e.g. CH₄, H₂, . . . ) 6.2

Enthalpy Source TermsP

nΘ_kn,Eq. (5.9)

Θ_Hom Homogeneous gas-phase reactions 5.1.3 Θ_Cond,G Conductive heat transfer from coal particles 5.3.3 Θ_Rad Combined particle and gas-phase radiation 5.3.4 Θ_CC Enthalpy of particle char conversion reactions 5.3.5 Θ_Py Enthalpy of particle pyrolysis reactions 6.2

7Using the following relation

∂

∂x_i µ

c_pα∂T

∂x_i

¶

= ∂

∂x_i µ

α∂c_pT

∂x_i −αT∂c_p

∂x_i

¶ ,

and replacingc_pT=h_S, it becomes clear that the termT∂c_p/∂x_ihas been omitted in Eq. (5.7).

5.1.2 Turbulence Model

In the process of creating the results presented in the chapters 7 and 8, several tur-bulence models have been employed. For turbulent flows which are not bounded by walls, cf. Chap. 7 or non-resolved wall modelled flows, cf. Chap. 8, the OpenFOAM Smagorinsky-Lilly type model was applied. Theσ-model, Nicoud et al. (2011), was used for simulations of pipe flows with the purpose of creating LES inflow data, having resolved near wall flows.

Smagorinsky-Lilly Type Turbulence Models

This section discusses the procedure which is used to compute the turbulent viscosi-ties and its relation to the model proposed by Lilly (1962) and Smagorinsky (1963), which is commonly known as Smagorinsky-Lilly or Smagorinsky model. Turbulent viscositiesµ_Turb, e.g.µ_SGS, needed for the closure of Eq. (5.2), Eq. (5.5), and Eq. (5.9) are often defined based on dimensional grounds. For example, Tennekes and Lum-ley (1972) give the following relation based on a mixing lengthl_Turb

µ_Turb∝ρu_Turbl_Turb, (5.10) whereu_Turb is a turbulent velocity scale.

The relation Eq. (5.11), formulated by Prandtl and Wieghardt (1946) µ_Turb∝ρl_Turbq

k_Turb, (5.11)

wherek_Turb⁸is the turbulent kinetic energy, was a precursor for modern one equa-tion models of turbulence, Speziale (1991). This approach, of using the turbulent kinetic energy, is more flexible since a transport equation for the turbulent kinetic energy can be used to solve fork_Turb. This can increase the accuracy while increas-ing the computational costs. In the LES context this can be formulated as

µ_SGS=c_kρ∆^qk_Turb, (5.12) where l_Turb is replaced by the product of a model constant c_k and the grid spacing

∆. Another way of representing µ_Turb uses the turbulent kinetic energy k_Turb and its dissipation rate² as discussed by Menon et al. (1996) or Yoshizawa (1987).

8The turbulent kinetic energyk_Turbis a mass specific quantity and has the units m²s⁻².

5.1. GAS-PHASE MODELLING 52 To formulate an equation for k_Turb a local equilibrium of the influx of turbu-lent kinetic energy from the resolved scales and its dissipation ² is assumed. The equilibrium can be written as

−τ_{i j,SGS}:Se_{i j}=²(∆), (5.13)

where τ_{i j,SGS} models the subgrid scale viscous stress tensor, cf. Eq. (4.9), and : denotes the Frobenius inner product, cf. Eq. (A.1). The grid spacing dependent cut off dissipation rate²(k) can be written as

²(k)=c²ρk^3/2_Turb∆^{−1, (5.14)} with the model constant c², Fureby (1996). The subgrid scale viscous stress tensor τ_{i j,SGS} is modelled by the Boussinesq approximation. After inserting Eq. (A.9) and Eq. (5.14) into Eq. (5.13) and replacingµ_Turb with Eq. (5.12) the algebraic equation for k_Turb can be written as

−c_²ρk^3/2_Turb∆⁻¹=ρ2

3k_Turbδ_{i j}:Se_{i j}+2ρc_kk^1/2_Turb∆S^e_{i j,Dev}:Se_{i j}, (5.15) or reformulated

c_²

∆^ρkTurb+ 2

3ρk^1/2_Turbδ_{i j}:Se_{i j}−2ρc_k∆S_{i j,Dev}:Se_{i j}=0⁹, (5.16) where Eq. (5.16) is a quadratic equation forp

k_Turb andS_{i j,Dev}the deviatoric strain rate tensor, cf. Eq. (A.9).

In the constant density case the transport equations can be divided by ρ and µ_Turb/ρsubstituted byν_Turb, Eq. (5.15) is then be simplified to

k_Turb=2c_k c²

∆²S²_{i j}, (5.17)

sinceδ_{i j}:Se_{i j}=0, as a consequence of the continuity equation¹⁰.

By inserting Eq. (5.17) into Eq. (5.12) the commonly used Smagorinsky constant C_Smagcan be expressed as

C_Smag= Ãc³_k

c_²

!1/4

. (5.18)

9cf. listing Lst. B.9

10Additonally, the relation Se_{i j,Dev}:Se_{i j}=Se²_{i j} holds, since Se_{i j,Dev}=Se_{i j} for incompressible flows.

This follows from the definition given by Eq. (A.7)

This demonstrates that the standard Smagorinsky-Lilly model, i.e.

µ_SGS=ρ¡

C_Smag∆^{¢2¯¯ e}S_{i j}¯

¯, (5.19)

can be recovered in the constant density case from the generalised variable den-sity formulation based on the local equilibrium assumption for the kinetic energy.

Hence, the model based on Eq. (5.12) in combination with the quadratic equation for pk_Turb, Eq. (5.16), is referred to as generalised compressible Smagorinsky model.

σ-Model

An alternative LES turbulence model, used within this work, is the σ turbulence model, proposed by Nicoud et al. (2011). Theσcalculates the turbulent SGS contri-butions to the viscosity as

ν_SGS=¡

Cσ∆^¢2σ3

¡σ₁−σ₂¢ ¡

σ₂−σ₃¢

σ²₁ , (5.20)

whereσ_i are the square roots of the Eigenvalues of the matrixG_{i j}=^∂u_∂x^k_i ^∂u_∂x^k_j.

5.1.3 Turbulence Chemistry Interaction Models

Since the instantaneous concentrations and the temperature needed for Eq. (3.14) are generally not known in a turbulent flow and applying mean or filtered values instead would lead to inaccurate mean reaction rates, discussed in appendix A.5, an approach to solve this fundamental problem of turbulence chemistry interaction (TCI) is needed.

For the closure of the LES species transport equation, the filtered reaction rate Ψ_k,Hom is needed. In the present work the eddy break-up (EBU) model of Hu et al.

(2006) has been implemented. The basic assumption of the EBU model are infinitely fast chemical reactions in comparison to the turbulent mixing, i.e. τ_Chem<<τ_Mix. Hence, turbulent mixing is the factor that controls the reaction rate. The filtered fuel consumption rateΨ_Fuis defined as

Ψ_Fu=c₁ρ¯

¯ eS_{i j}¯

¯min Ã

Ye_Fu,Ye_Ox

s ,c₂ Ye_Pr 1+s

!

, (5.21)

whereYe_Ox,Ye_FuandYe_Prare the filtered mass fractions of the oxidiser, fuel, and prod-ucts. Furthermore, s=¡

ν_OxMW_Ox/ν_FuMW_Fu¢

St is the mass stoichiometric factor.

5.2. SOLID PHASE MODELLING 54 The default values of the model coefficients c₁andc₂are 4.0 and 0.5. After evaluat-ing the fuel consumption rate, the individual species source terms can be calculated by Ψ_{k,Hom= η}^ηk

Fuω_Fu, where η_k =ν_StMW_k is the mass stoichiometric coefficient of the species k and η_Fu the mass stoichiometric coefficient of the corresponding fuel species.

OnceΨ_k,Hom is known, Θ_n,Hom, needed for the closure of the sensible enthalpy transport equation, can be calculated.

Θ_n,Hom=HV_FuΨ_Fu,Hom, (5.22)

where HV_Fu is the heating value of the fuel for the corresponding reactionn.

Besides its low computational costs and comparably simple implementation, the EBU model has the advantage that no Arrhenius rate coefficients are needed since the reaction rates are only dependent on the turbulent time scales. This makes the EBU model suitable whenever the exact nature of the reacting species is unknown, e.g. whenever tars or species mixtures are approximated by postulate substances.

5.2 Solid Phase Modelling

The particle position x_i,Partchanges according to d

dtx_i,Part=u_i,Part. (5.23)

Hence, the position of each particle is determined by integrating Eq. (5.23) for each particle individually. The unknown particle velocityu_i,Partcan be found using New-ton’s second law

m_Part d

dtu_i,Part=F_i,Tot, (5.24)

where F_i,Tot is the sum of various forces acting on the particle and neglecting any momentum changes due to changes in the particle mass. Particles are assumed to be spherical and non-rotating. Thus, in the present study only drag, gravitational, and dispersive forces arising from the modelled turbulent fluctuations are consid-ered,

F_i,Tot=F_i,Dr+F_i,G+F_i,Disp. (5.25)

The dispersive force F_Disp in Eq. (5.25) is a force introduced by the filtering proce-dure, reflecting the dispersive nature of the subgrid scale gas-phase velocity fluctu-ations on the particle motion.

An updated particle velocity can be obtained by integration over Eq. (5.24). The implementation of this integration step, however, involves further transformations of the differential equation. The details are discussed in appendix C.3.1.

The particle temperature is calculated by balancing the energy absorbed from the surrounding gas-phase via convection, conduction, radiation, and additionally from processes such as evaporation, pyrolysis, or surface reactions. The resulting equation can be written as

d dt

¡m_Partc_p,PartT_Part¢

=Θ_Cond,Part+Θ_Rad+Θ_Reac^{, (5.26)} where Θ_Cond,Part is the conductive heat transfer term between particle- and gas-phase, Θ_Rad the heat transferred by radiation, and Θ_Reac the heat released due to chemical processes. The conductive heat transfer termΘ_Cond,Partis given by

Θ_Cond,Part=Nuπd_Partλ_EffG

³T_G−T_Part´

, (5.27)

where Nu is the Nusselt number andλ_Eff the effective thermal conductivity of the gas-phase.

Equivalently to the particle motion equation the particle temperature equation is subject to further transformation discussed in appendix C.3.2.

The final, fundamental particle property that needs to be known is the change in particle mass, which can be expressed as

dm_Part

dt =Ω_Ev+Ω_Dev+Ω_CC, (5.28)

whereΩ_Ev,Ω_Dev, andΩ_CC are the mass transfer terms to account for evaporation, devolatilisation and surface reactions respectively between particle- and gas-phase.

The total mass of a particle is considered as the sum of the individual masses of moisture, fixed matter, and volatile content

m_Part=m_Moi+m_Vol+m_FM. (5.29) The fixed matter mass component m_FM consists of the individual masses of ash, char, and coal in the intermediate state.

m_FM=m_Ash+m_Char+m_Int. (5.30)

5.3. INTERPHASE TRANSFER MODELLING 56 Here, intermediate state refers to the particle mass which is convertible to char or volatile matter by pyrolysis reactions. Initially, a particle consists only of m_Intand m_Ash since m_Vol(t=0)=m_Char(t=0)=0. m_Vol and m_Char are then formed by the thermolysis model. The mass of volatiles m_Vol specifies the mass of species which are potentially released by devolatilisation, but are not emitted to the gas-phase yet. The default implementation in OpenFOAM requires an a priori specification of m_Vol. Here the author introduced the thermolysis model which allows to form volatile species from the intermediate state. The change of m_Intis given by

dm_Int

dt = −dm_Char dt

¯

¯

¯thermolysis−dm_Vol dt

¯

¯

¯thermolysis. (5.31) Hence char and volatiles are formed by consuming m_Int. The actual form of Eq. (5.31) depends on the selected submodel, discussed in detail in section 5.3.5 and the implementation in appendix C.5.1.

5.3 Interphase Transfer Modelling

This section focuses on the presentation of the transfer terms between particle- and gas-phase. For each source in the gas-phase equations given in Tab. 5.1 a sink in a particle equation exists, i.e. Φ_G = −Φ_Part and vice versa. In the context of the discretised transport equations a general source term for the cell i can, thus, be written as

Φ_i,G= − 1 V_i

X

n Φ_n,Part (5.32)

where nis the number of the particles inside the cell i.

Whenever possible the transfer models are generalised so that instead of the source or sink term a model variable is calculated. For example, instead of comput-ing theΘ_Cond,Part the model for the conductive heat transfer computes the Nusselt number, Nu. In a second step, outside the scope of the transfer model, the particle and gas-phase source and sink terms are calculated. This detail shall be stated here to allow for a better understanding of the implementation of the individual sub-models. Table 5.2 gives an overview of the most relevant sub-model implemen-tations. Here, the aforementioned variable which is solved rather than the source or sink term is given in the “particle model term” column. Once the particle model has been evaluated, in some cases several, dependent, gas-phase source or sink terms can be evaluated. These are given in the “gas-phase source term” column in the Tab. 5.2.

Table 5.2: Coal particle sub-model terms Sub-model Particle Model

Term Gas-phase Source

Term Implementation

Momentum Transfer Models, section 5.3.2

Sphere drag c_Dr Λ_i,DrEq. (5.2) •Schiller and Naumann (1933) Particle

dispersion F_i,Disp Λ_i,DispEq. (5.2) •No model

•Bini and Jones (2008) Heat Transfer Models, section 5.3.3 and 5.3.4

Conductive

heat transfer Nu Θ_Cond,GEq. (5.7) •Ranz and Marshall (1952) Radiation E_Part,κ_Part,Θ_Rad,Part Θ_RadEq. (5.7) •P₁with fixed constants

Particle Reaction Models, section 5.3.5 and section 6.2 Char

conversion ^dtdm_Char

Ω_CCEq. (5.1), Ψ_k,CCEq. (5.5), Θ_CCEq. (5.7)

•Baum and Street (1971)

Pyrolysis _dt^dm_Vol

Ω_DevEq. (5.1), Ψ_k,DevEq. (5.5), Θ_PyEq. (5.7), Θ_ReacEq. (5.7)

•Badzioch and Hawksley (1970)

•Kobayashi et al. (1977)

•Richards and Fletcher (2016)

5.3.1 The Parcel Approach

To reduce the computational costs individual particles with identical diameter are combined to parcels. As a consequence all particles of a parcel are identical and thus all properties, e.g. velocity, position, temperature, and diameter are the same.

Source terms transferred to the gas-phase, e.g. mass and energy, are calculated as the product of the number of particles per parcel and the individual source term, i.e. Ω_Parc=n_PartΩ_Part. This approach is easy to implement since all submodels are formulated on a per particle basis and only the source terms need to be adapted by the total number of particles per parcel. A drawback of this method is that for a large number of particles per parcel on fine computational grids, the source terms are concentrated in a limited number of cells. This can reduce the accuracy of the simulation and potentially cause instabilities.

5.3.2 Particle Momentum Transfer Models

Particle Drag Model

The relevant particle sub-model term to model momentum transfer between parti-cle and gas-phase is the partiparti-cle drag coefficient c_Dr. In this work, the drag

coeffi-5.3. INTERPHASE TRANSFER MODELLING 58 cient of a coal particle is modelled with a modified version of the Reynolds number dependent correlation of Schiller and Naumann (1933)¹¹

c_Dr=







0.424 if Re_Part>1000

Re24_Part

µ

1+¹₆Re_Part²³

¶

else , (5.33)

where Re_Partis the particle Reynolds number defined as Re_Part=u_Reld_Part

ν_G , (5.34)

where u_Rel is the relative velocity between particle and gas-phase and d_Part the particle diameter. Equation (5.33) considers two distinct flow regimes. At high Reynolds numbers, based on the relative particle velocity, a constant particle drag coefficient value of 0.424 is considered. In the low Reynolds number regime a mod-ified version of the Stokes relation, c_Dr=24/Re_Part, with a multiplicative correction factor is considered.

After calculating the particle drag coefficientc_Dr the particle drag force for the n-th particleF_in,Drand therefore the gas-phase source term can be evaluated as

Λ_{i,Dr= −}¹ V

X

n F_in,Dr, (5.35)

with

F_in,Dr=1

2ρ_GA_n,Cr,Partu²_i,Relc_Dr, (5.36)

where A_n,Cr,Partis the cross sectional area of a sphere.

Particle Dispersion Model

To reflect the dispersive nature of the subgrid scale fluctuations the LES particle dispersion model of Bini and Jones (2008) has been implemented in this work. The model of Bini and Jones (2008) calculates a dispersion velocity as

du_i,Disp= s

C₀k_SGS

τ_t dW_i, (5.37)

whereτ_t is the interaction timescale between particles and the unresolved vortices, k_SGSthe kinetic energy of the subgrid fluctuations,C_oa coefficient, andWa Wiener

11This is the default drag coefficient for spherical particles of OpenFOAM.

term. For the implementation of the dispersion model within the framework of OpenFOAM, a dispersive force acting onto a Lagrangian particle is calculated by applying Newton’s second law to Eq. (5.37)

F_i,Disp=m_Partdu_i,Disp

dt . (5.38)

Substitutingdv_Dispin Eq. (5.38) with Eq. (5.37) yields

F_i,Disp=m_Part r

C₀^kSGS_τt dW_i

dt . (5.39)

For the interaction timescale τ_t multiple definitions are given by Bini and Jones (2008). In this work, however, only the following relation is used

τ_t= τ^2α_Part

¡∆^/pk_SGS¢2α−1, (5.40)

where∆is the cell width,τ_Part the particle timescale, and the coefficientαis set to 0.8.

Analogously to Eq. (5.35) the momentum transfer term for the particle disper-sion force can then be written as

Λ_i,Disp= −1 V

X

n F_in,Disp. (5.41)

5.3.3 Particle Heat Transfer Model

The Ranz and Marshall Model

Based on the work of Frössling et al. (1938), the model of Ranz and Marshall (1952) estimates the Nusselt number, which is needed for closure of Eq. (5.27), as a func-tion of the Reynolds and Prandtl number as

Nu=2+0.552Re^1/2_PartPr^1/3. (5.42) Even though developed to model the heat transfer of evaporating droplets, the model of Ranz and Marshall (1952) is also widely employed to compute the con-vective heat transfer of coal particles.

After calculating the particle Nusselt number Nu the conductive heat transfer terms for the individual particles Eq. (5.27) can be solved for. In consequence the

5.3. INTERPHASE TRANSFER MODELLING 60 heat transfer for term for the gas-phase can be calculated as

Θ_{Cond,G= −}¹ V

X

n Θ_Cond,Part (5.43)

5.3.4 Radiation Modelling

A number of methods to solve the RTE Eq. (3.12) have been developed. These in-clude: discrete ordinates methods (DOM), implicit Monte Carlo (IMC), and Flux Methods to name only a small selection. Due to the enormous amount of different methods and their complexity these solution methods cannot be presented in detail.

The interested reader is referred to the review of Viskanta and Mengüç (1987) or the book of Modest and Haworth (2016) for further details. The approach of choice to solve the RTE in the present report is the spherical harmonics (P_N) method of order n=1.

The P₁Method allows for the transformation of the RTE, Eq. (3.12), which is an integro-differential equation with five independent variables, into a set of regular ODEs depending only on spatial coordinates. The full procedure of deriving the transport equation for the incident radiation G, i.e. G =R

IdΩ, from the RTE, which is based on Legendre polynomials, is rather lengthy. Therefore, the interested reader is referred to Modest (2003) for a detailed discussion.

The resulting P₁transport equation for the incident radiation is given as

∂

∂x_i µ1

β

∂G

∂x_i

¶

= −3κ¡

4πI_BB−G¢

, (5.44)

where βis the extinction coefficient. Furthermore, from Eq. (5.44) the source term for the sensible enthalpy transport equation is given as

Θ_Rad= ∂

∂x_i µ1

β

∂G

∂x_i

¶

. (5.45)

To account for particle contributions in Eq. (5.44) total absorption and extinction coefficients are used. Further details on this and the corresponding implementation are given in appendix B.7.

Im Dokument Large eddy simulation of pulverised coal combustion (Seite 87-103)