The detailed mathematical model we developed is a multi-physics model that takes the following phenomena into account: the reactive gas flow and temperature, chemical species and radiative heat transfer distribution in the kiln, the turbulent non-premixed combustion of hydro-carbon gasses in the burner, the insulating properties of the lining, the rotary motion of the kiln and the forced convection on the outside surface.

The material bed only occupies a small fraction of the volume of the kiln and has a negligible limited impact of temperature distribution. We therefore do not take the material bed into account in our model and simulate an empty kiln.

The most important physical phenomenon that takes place in this burner region is the turbulent non-premixed combustion of the fuel injected from the burner with the secondary air. Combustion, even without turbulence, is an intrinsically complex process involving a large range of chemical time and length scales. Some of the chemical phenomena controlling flames take place in short times over thin layers and are associated with very large mass fractions, temperature and density gradients. The full description of chemical mechanisms in laminar flames may require hundreds of species and thousands of reactions leading to considerable numerical difficulties. Turbulence itself is probably the most complex phenomenon in non-reacting fluid mechanics. Various time and length scales are involved and the description of turbulence remains to date an open questions. The modeling of the kiln therefore requires resorting to a set of assumptions that are described in the remainder of this section.

### 3.2 Grid generation

The mesh model is shown in Figure 4. Figure 4(a), Figure 4(b) and Figure 4(c) give an exterior view of the mesh in the air inlet region, an interior view of the mesh and a detailed view of the mesh on the burner, respectively. We employed a *polyhedral mesh* of 2.8 million cells with local refinement in the critical inlets and burner regions. The main difficulties in meshing this geometry were found in balancing the accuracy required capture the flow around small features in the burner with the overall computational cost. Polyhedral meshes provide a balanced solution in complex mesh generation problems of this kind.

Tetrahedra are the simplest type of volume elements. As their faces are plane segments, both face and volume centroid locations are well defined. A disadvantage is that tetrahedra cannot be stretched too much. To achieve a reasonable accuracy a much larger number of control volumes is needed than if structured meshes are used. Furthermore, as tetrahedral control volumes have only four neighbors, and computing gradients at cell centers using standard approximations can be problematic.

Polyhedra offer the same automatic meshing benefits as tetrahedra while overcoming their disadvantages. A major advantage of polyhedral cells is that they have many neighbors (typically of order 10) allowing gradients to be much better approximated. Obviously more neighbors implies more storage and computational operations per cell, but this is more than compensated by a higher accuracy. Polyhedral cells are also less sensitive to stretching than tetrahedra. A polyhedron with 12 faces for instance has six optimal directions which, together with the larger number of neighbors, leads to a more accurate solution with a lower cell count. Comparisons in many practical tests have verified that with polyhedral meshes, one needs about four times fewer cells, half the amount of memory and a tenth to a fifth of computing time compared to tetrahedral meshes to reach solutions of the same accuracy. In addition, solvers on polyhedral meshes were found to converge more robustly with respect to change in their parameters. A more detailed analysis of polyhedral meshes can be found in [6].

### 3.3 Governing reacting flow equations

In this section we present the conservation equations for reacting flows we used. The equations are derived from the Navier-Stokes (NS) equations by adding terms that account for reacting flows. The reacting gas is a non-isothermal mixture of multiple species which must be tracked individually. As heat capacities change significantly with temperature and composition, the transport coefficients require specific attention. In this subsection we will describe the Navier-Stokes and Reynolds-Averaged Navier-Stokes flow model, the non-realizable K-Epsilon turbulence model and the Standard Eddy Break Up combustion model. A more detailed derivation of these equations can be found in *e.g.* [7, 8].

Species are defined by their mass fraction defined as

${Y}_{\ell}=\frac{{m}_{\ell}}{m}\phantom{\rule{1em}{0ex}}\text{for}\ell =1:N,$

(1)

where *N* is the number of species in the reacting mixture, ${m}_{\ell}$ the mass of species *ℓ* in a volume *V* and *m* the mass of gas in the volume, respectively. The conservation of mass can then be written as

NS: Conservation of mass

$\frac{D\rho}{Dt}=\frac{\partial \rho}{\partial t}+\frac{\partial \rho {u}_{i}}{\partial {x}_{i}}=0,$

(2)

where $\rho =m/V$ is the density of the gas and ${u}_{i}$ its three dimensional velocity field, respectively. The conservation of species *ℓ* for $\ell =1:N$ can then be written as

NS: Conservation of species

$\frac{\partial \rho {Y}_{\ell}}{\partial t}+\frac{\partial}{\partial {x}_{i}}(\rho ({u}_{i}+{V}_{\ell ,i}){Y}_{\ell})={\dot{\omega}}_{\ell},$

(3)

where ${V}_{\ell ,i}$ the *i* th component of the diffusion velocity ${V}_{\ell}$ of species *ℓ* and ${\dot{\omega}}_{\ell}$ the chemical reaction rate of species *ℓ*. The conservation of momentum for the gas can for $j=1:3$ be expressed as:

NS: Conservation of momentum

$\begin{array}{rcl}\frac{\partial}{\partial t}(\rho {u}_{j})+\frac{\partial}{\partial {x}_{i}}(\rho {u}_{i}{u}_{j})& =& \frac{\partial {\sigma}_{ij}}{\partial {x}_{i}}+\rho \sum _{\ell =1}^{N}{Y}_{\ell}{f}_{\ell ,j}\\ =& -\frac{\partial p}{\partial {x}_{j}}+\frac{\partial {\tau}_{ij}}{\partial {x}_{i}}+\rho \sum _{\ell =1}^{N}{Y}_{\ell}{f}_{\ell ,j},\end{array}$

(4)

where

*p* denotes pressure and

${\tau}_{ij}$ and

${f}_{\ell ,j}$ are the components of the Reynolds stress tensor and of the volume force acting on species

*ℓ*, respectively. We will express the conservation of energy using the sensible enthalpy of the mixture

${h}_{s}$ as independent variable. To introduce this quantify, we will first denote the enthalpy of species

*ℓ* as

${h}_{\ell}$. This quantify is the sum of the of a sensible and chemical part,

*i.e.*,

${h}_{\ell}={h}_{\ell ,s}+\mathrm{\Delta}{h}_{f,\ell}^{0},$

(5)

where the last term represents the enthalpy of formation of the species at a particular reference temperature

${T}_{0}$. The enthalpy of the mixture is then defined as a weighted average that again can be decomposed into a sensible and chemical part, as follows

$\begin{array}{rcl}h& =& \sum _{\ell =1}^{N}{Y}_{\ell}{h}_{\ell}=\sum _{\ell =1}^{N}{Y}_{\ell}{h}_{\ell ,s}+\sum _{\ell =1}^{N}{Y}_{\ell}\mathrm{\Delta}{h}_{f,\ell}^{0}\\ =& {h}_{s}+\sum _{\ell =1}^{N}{Y}_{\ell}\mathrm{\Delta}{h}_{f,\ell}^{0}.\end{array}$

(6)

We will denote the heat diffusion coefficient and the temperature by

*λ* and

*T*, respectively. The energy flux

${q}_{i}$ is the sum a heat diffusion term derived from Fourier’s Law and a term associated with the diffusion of species with different enthalpies,

*i.e.*,

${q}_{i}=-\lambda \frac{\partial T}{\partial {x}_{i}}+\rho \sum _{\ell =1}^{N}{h}_{\ell}{Y}_{\ell}{V}_{\ell ,i}.$

(7)

In the energy conservation equation the following three source terms will play a role: the heat source due to radiative heat flux denoted as

*Q*, the viscous heating term denoted as Φ, where

$\mathrm{\Phi}={\tau}_{ij}\frac{\partial {u}_{i}}{\partial {x}_{j}}$ and heat release due to combustion denoted as

${\dot{\omega}}_{T}$, where

${\dot{\omega}}_{T}=-\sum _{\ell =1}^{N}\mathrm{\Delta}{h}_{f,\ell}^{0}{\dot{\omega}}_{\ell}.$

(8)

The work done by the gas on the species can be expressed as $\rho {\sum}_{\ell =1}^{N}{Y}_{\ell}{f}_{\ell ,i}{V}_{\ell ,i}$. With all these quantities introduced, the conservation of energy in terms of ${h}_{s}$ can be expressed as:

NS: Conservation of energy

$\begin{array}{rcl}\rho \frac{D{h}_{s}}{Dt}& =& {\dot{\omega}}_{T}+\frac{Dp}{Dt}-\frac{\partial}{\partial {x}_{i}}\left(\lambda \frac{\partial T}{\partial {x}_{i}}\right)+\mathrm{\Phi}+\dot{Q}\\ -\frac{\partial}{\partial {x}_{i}}\left(\rho \sum _{\ell =1}^{N}{h}_{s,\ell}{Y}_{\ell}{V}_{\ell ,i}\right)+\rho \sum _{\ell =1}^{N}{Y}_{\ell}{f}_{\ell ,i}{V}_{\ell ,i}.\end{array}$

(9)

Turbulent combustion results from the two-way interaction between chemistry and turbulence. When a flame interacts with a turbulent flow, the combustion modifies the turbulence in two ways. The heat released induces high flow accelerations through the flame front and the temperature changes generate large changes in kinematic viscosity. These phenomena may either generate or damp turbulence and are referred to as flame-generated turbulence and relaminarization due to combustion, respectively. The turbulence conversely modifies the flame structure. This may either enhance the chemical reactions or completely inhibit it, leading to flame quenching. Compared to premixed flames, turbulent non-premixed flames exhibit some specific features that have to be taken into account. Non-premixed flames do not propagate as they localized on the fuel-oxidizer interface. This property is useful for safety purposes but it also has consequences on the chemistry-turbulence interaction. Without propagation speed, a non-premixed flame is unable to impose its own dynamics on the flow field and is therefore more sensitive to turbulence.

The description of the turbulent non-premixed combustion processes in a computational fluid dynamics model may be achieved using three levels of accuracy in the computations. Either a Reynolds Averaged Navier Stokes (RANS), a Large Eddy Simulations (LES) or a Direct Numerical Simulations (DNS) model can be adopted. In current engineering practice, the RANS model is extensively used because it is less demanding in terms of resources. Its validity however is limited by the closure models describing turbulence and combustion and the need for some form of callibration. Considering the complexities and the dimensions of our kiln, using the RANS model is the only feasible choice.

#### 3.3.1 RANS model

In constant density flows, Reynolds averaging consist in splitting any quantity

*ξ* in mean and fluctuating component (

$\xi =\overline{\xi}+{\xi}^{\prime}$). In variable density flow Favre [

9] mass-weighted averages are usually preferred,

*i.e.*,

$\tilde{f}=\frac{\overline{\rho f}}{\overline{\rho}}$. Any quantity

*f* can therefore be split into:

$f=\tilde{f}+{f}^{\u2033},\phantom{\rule{1em}{0ex}}\text{where}\tilde{{f}^{\u2033}}=0.$

The RANS equations derived from the reacting Navier-Stokes equation given above are then given by the equation for conservation of mass

RANS: Conservation of mass

$\frac{\partial \overline{\rho}}{\partial t}+\frac{\partial \overline{\rho}\tilde{{u}_{i}}}{\partial {x}_{i}}=0,$

(10)

the equation for conservation of species *ℓ* for $\ell =1:N$

RANS: Conservation of species

$\frac{\partial}{\partial t}(\overline{\rho}\tilde{{Y}_{\ell}})+\frac{\partial}{\partial {x}_{i}}(\overline{\rho}\tilde{{u}_{i}}\tilde{{Y}_{\ell}})={\overline{\dot{\omega}}}_{\ell}-\frac{\partial}{\partial {x}_{i}}(\overline{{V}_{\ell ,i}{Y}_{\ell}}+\overline{\rho}\tilde{{u}_{i}^{\u2033}{Y}_{\ell}^{\u2033}}),$

(11)

the equation for conservation of momentum for $j=1:3$

RANS: Conservation of momentum

$\frac{\partial}{\partial t}(\overline{\rho}\tilde{{u}_{j}})+\frac{\partial}{\partial {x}_{i}}(\overline{\rho}\tilde{{u}_{i}}\tilde{{u}_{j}})+\frac{\partial \overline{p}}{\partial {x}_{j}}=\frac{\partial}{\partial {x}_{i}}(\overline{{\tau}_{ij}}-\overline{\rho}\tilde{{u}_{i}^{\u2033}{u}_{j}^{\u2033}}),$

(12)

and finally the equation for conservation of momentum

RANS: Conservation of energy

$\begin{array}{rcl}\frac{\partial}{\partial t}(\overline{\rho}\tilde{{h}_{s}})+\frac{\partial}{\partial {x}_{i}}(\overline{\rho}\tilde{{u}_{i}}\tilde{{h}_{s}})& =& {\overline{\dot{\omega}}}_{T}+\frac{\overline{Dp}}{Dt}\\ +\frac{\partial}{\partial {x}_{i}}(\overline{\lambda \frac{\partial T}{\partial {x}_{i}}}-\overline{\rho}\tilde{{u}_{i}^{\u2033}{h}_{s}^{\u2033}})\\ +\overline{\mathrm{\Phi}}-\frac{\partial}{\partial {x}_{i}}\left(\overline{\rho \sum _{k=1}^{N}{h}_{s,\ell}{Y}_{\ell}{V}_{\ell ,i}}\right).\end{array}$

(13)

The averaging procedure introduces unclosed quantities that have to be modeled. Without entering in the details we list here the two main unclosed terms that will be described in the next sections:

#### 3.3.2 Turbulence model

Using the turbulence viscosity assumption by Boussinesq [

10], the Reynolds stresses can be represented as

$\begin{array}{rcl}\overline{\rho {u}_{i}^{\u2033}{u}_{j}^{\u2033}}& =& \overline{\rho}\tilde{{u}_{i}^{\u2033}{u}_{j}^{\u2033}}\\ =& -{\mu}_{t}(\frac{\partial {\tilde{u}}_{i}}{\partial {x}_{j}}+\frac{\partial {\tilde{u}}_{j}}{\partial {x}_{i}}-\frac{2}{3}{\delta}_{ij}\frac{\partial {\tilde{u}}_{K}}{\partial {x}_{K}})+\frac{2}{3}\overline{\rho}k,\end{array}$

(14)

where

${\mu}_{t}=\overline{\rho}{\nu}_{t}$ is the turbulent dynamic viscosity and

${\delta}_{ij}$ the Kronecker delta. The turbulent kinetic energy

*k* in turn can be expressed as

$k=\frac{1}{2}\sum _{j=1}^{3}\tilde{{u}_{j}^{\u2033}{u}_{j}^{\u2033}}.$

(15)

Modeling the turbulent viscosity

${\mu}_{t}$ is the central problem in turbulence computations. Many approaches exist. In this work we use a classical turbulence model developed for non-reacting flows, namely the

*Realizable K-Epsilon model* [

11]. Heat release effects on the Reynolds stresses are not explicitly taken into account in this approach and the turbulent viscosity is modeled as

${\mu}_{t}=\overline{\rho}{C}_{\mu}\frac{{k}^{2}}{\epsilon},$

(16)

where *ε* is the rate of energy dissipation. In this model the critical coefficient ${C}_{\mu}$ is a function of mean flow and turbulence properties, rather than assumed to be constant as in the standard model. This allows to satisfy certain mathematical constraints on the normal stresses consistent with the physics of turbulence and is referred to as *realizability*.

From the Boussinesq relationship in Equation (

14) and the eddy viscosity definition in Equation (

16) it is possible to obtain the following expression for the normal Reynolds stress

$\overline{{u}^{2}}$ in an incompressible strained mean flow

*U* $\overline{{u}^{2}}=\frac{2}{3}k-2{\nu}_{t}\frac{\partial U}{\partial x},$

(17)

where

${\nu}_{t}=\frac{{\mu}_{t}}{\rho}$. It can be shown that

$\overline{{u}^{2}}$, which by definition is a positive quantity, become negative (

*non-realizable*), when the strain is large enough to satisfy

$\frac{k}{\epsilon}\frac{\partial U}{\partial x}>\frac{1}{3{C}_{\mu}}\approx 3.7.$

(18)

The easiest way to ensure the realizability is to make ${C}_{\mu}$ in (16) variable [12].

The critical coefficient

${C}_{\mu}$ can be expressed as a function of the mean strain and rotation rates, the angular velocity of the system rotation and the turbulence fields as follows

${C}_{\mu}=\frac{1}{{A}_{0}+{A}_{s}\frac{k{U}^{\ast}}{\epsilon}},$

(19)

where

${\overline{\mathrm{\Omega}}}_{ij}$ is the mean rate of rotation tensor viewed in a rotating reference frame with the angular velocity

${\omega}_{k}$. The parameters

${A}_{0}$ and

${A}_{s}$ in (19) can be computed as

The turbulent kinetic energy

*k* and its dissipation rate

*ε* in Equation (

16) are described by the following two balance equations

where ${P}_{k}$ is the production term of turbulent kinetic energy due to the mean velocity gradients, ${P}_{b}$ the production of turbulent kinetic energy due to buoyancy, ${Y}_{M}$ the dilatation dissipation term that accounts for the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, ${S}_{k}$ and ${S}_{\epsilon}$ user defined source terms for turbulent kinetic energy and dissipation, and ${\sigma}_{k}$ and ${\sigma}_{\epsilon}$ the turbulent Prandtl numbers for *k* and *ε*, respectively. ${C}_{1\epsilon}$, ${C}_{2\epsilon}$ and ${C}_{3\epsilon}$ are model constants.

Another weakness of traditional K-Epsilon turbulence models is their modeling of the dissipation rate *ε*. Indeed, the well-known spreading (or dispersion) rate anomaly refers to the fact that traditional models do reasonably well in predicting the spreading rate of a planar jet but perform unexpectedly poor for rounds jets. This weakness can be traced back to a deficiency in traditional *ε*-equations. The realizable model proposed by Shih [11] was developed to repair this deficiency and addresses as such an issue that is of primary importance in our study.

#### 3.3.3 Combustion model

The averaged equation for conservation of species (11) can be rewritten in compact form for

$\ell =1:N$ as

$\frac{\partial}{\partial t}(\overline{\rho}\tilde{{Y}_{\ell}})+\frac{\partial}{\partial {x}_{i}}(\overline{\rho}\tilde{{u}_{i}}\tilde{{Y}_{\ell}})=-\mathrm{\nabla}\cdot {J}_{\ell}+{\overline{\dot{\omega}}}_{\ell},$

(28)

where ${J}_{\ell}$ is the mass diffusion flux of species *ℓ*. The previous equation is solved in a CFD code for $N-1$ species where *N* is the total number of fluid phase chemical species present in the system. Since the mass fraction of the species must sum to unity, the *N* th mass fraction is determined as one minus the sum of the $N-1$ solved mass fractions. To minimize numerical error, the *N* th species should be selected as that species with the overall largest mass fraction.

In turbulent flows the mass diffusion flux is computed as

${J}_{\ell}=-(\overline{\rho}{D}_{\ell}+\frac{{\mu}_{t}}{{\mathit{Sc}}_{t}}),$

(29)

where ${\mathit{Sc}}_{t}$ is the turbulent Schmidt number and ${D}_{\ell}$ is the molecular diffusivity of species *ℓ*.

The species chemical reaction rate unclosed term ${\dot{\omega}}_{\ell}$ must be modeled with a combustion model. A combustion model describes the two-way interaction between properties of the turbulent flow produced by the flame and the chemical reactions. It serves to compute the reaction state space and the quantities it influences, namely the fluid density, viscosity, and temperature. It accounts for the processes that occur at length and time scales that we cannot resolve on a grid either in space or time due to limitations in computational resources. The choice of combustion model is decided by knowing the Damkohler number, defined as $\mathit{Da}=\frac{{t}_{\mathrm{mix}}}{{t}_{\mathrm{rxn}}}$, where ${t}_{\mathrm{mix}}$ is the mixing time scale and ${t}_{\mathrm{rxn}}$ is the reaction time scale. When the Damkohler number is very large, as in the case of the kiln, the reaction rate is controlled by the turbulent mixing that brings reactants together at the molecular scale. In this limit, the Standard Eddy Break Up (EBU) [13] model is fairly accurate because it assumes that the reaction occurs instantaneously upon micromixing.

The EBU combustion model tracks individual mean species concentrations on the grid through transport equations. The reaction rates used in these equations are calculated as functions of the mean species concentrations, turbulence characteristics and, depending on the specific model used, temperature. A mean enthalpy equation is solved in addition to the species transport equations. The mean temperature, density and viscosity are then calculated knowing the mean enthalpy and species concentrations. In the EBU used, the individual species in the global reaction are assumed to be transported at different rates according to their own governing equations.

The reaction rate is modeled through an expression that takes the turbulent micromixing process into account. This is done through dimensional arguments. Thus, for a reaction of the form

${v}_{F}F+{v}_{O}O\u27f6{v}_{P1}{P}_{1}+{v}_{P2}{P}_{2}+\cdots +{v}_{Pj}{P}_{j},$

(30)

where

*F* stands for fuel,

*O* oxidiser and

*P* products of the reaction, the reaction rate for reaction

*ℓ* is assumed to be

${\overline{\dot{\omega}}}_{\ell}=\frac{\overline{\rho}}{M}\left(\frac{1}{{\tau}_{\mathrm{mix}}}\right){A}_{\mathrm{ebu}}min\{{\overline{Y}}_{F},\frac{{\overline{Y}}_{O}}{{s}_{O}},{B}_{\mathrm{ebu}}(\frac{{\overline{Y}}_{P1}}{{s}_{P1}}+\cdots +\frac{{\overline{Y}}_{Pj}}{{s}_{Pj}})\},$

(31)

where ${s}_{O}=\frac{{v}_{O}{M}_{O}}{{v}_{F}{M}_{F}}$, ${s}_{Pj}=\frac{|{v}_{Pj}|{M}_{Pj}}{{v}_{F}{M}_{F}}$, *v* is the molar stoichiometric coefficient for species *j* in reaction *ℓ*, *M* is molecular weight of species. Equation (30) essentially states that the integrated micromixing rate is proportional to the mean (macroscopic) concentration of the limiting reactant divided by the time scale of the large eddies ($\frac{k}{\epsilon}={\tau}_{\mathrm{mix}}$). ${\overline{Y}}_{F}$, ${\overline{Y}}_{O}$, ${\overline{Y}}_{P}$ are respectively the mean concentrations of fuel, oxidizer, and products. ${A}_{\mathrm{ebu}}$ and ${B}_{\mathrm{ebu}}$ are model constants with typical values of 0.5 and 4.0 respectively. The values of these constants are fitted according to experimental results and they are suitable for most cases of general interest.

In our simulations we used a reduced combustion mechanism with 6 species and 4 reactions to account for a fuel that is a mixture of different alkanes. This mixture consists for 95% of CH_{4} and for 5% of C_{2}H_{6}, C_{3}H_{8} and C_{4}H_{10}.

The above models are discretized by a finite volume technique using second order upwinding for the convective terms [14–16]. The flow equations are solved in a segregated approach in which the SIMPLE algorithm realizes the velocity-pressure coupling. The energy equation is solved for the chemical thermal enthalpy using again a segregated approach. The temperature is computed from the enthalpy according to the equation of state. At each outer non-linear iteration the resulting linear systems are solved using an algebraic multigrid preconditioner for a suitable Krylov subspace acceleration [17].