Institutionen fr informationsteknologi

E�cient Modelling of the Regener-ative Heat Transfer in a Rotary Kiln

Alexander Bertholds

Project in Computational Science: Report

January 2013

PROJECTREPORT

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Abstract

A rotary kiln is a common part in a kraft pulp mill where it is used for limereburning to recycle some of the chemicals used in the kraft process. A rotary kilnconsists of an approximately 90 meter long, inclined rotating cylinder where limemud is heated by a flame. The lime mud is heated by the hot flue gases producedby the flame as well as from the rotating refractory which absorbs heat from thegases and transports the energy to the bottom of the lime mud as the refractoryrotates. The latter heat transfer process is known as regenerative heat transfer.Several Computational Fluid Dynamic (CFD) models have been made to model thekiln but due to computational limitations the regenerative heat transfer has beenneglected in these models. In this project, the regenerative heat transfer has beenmodelled by imposing energy convection within the refractory using the commercialCFD software STAR-CCM+. Transient models with a rotating refractory regionhave been used to validate the stationary model since no accurate measurementsof the kiln were available. It is shown that for a simplified model of the kiln thestationary approach with a moving heat conduction produces the same results asfor the corresponding transient model.

Contents

1 Introduction 1

2 Aim 3

3 Problem Description 53.1 Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Mathematical model 74.1 Energy and Transport Equations . . . . . . . . . . . . . . . . . . . . . . . 74.2 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.4.1 Refractory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4.2 Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4.3 Lime mud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Discretization 125.1 Discretization of the Governing Equations . . . . . . . . . . . . . . . . . . 125.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Solution Methods 136.1 Setting Up a Linear System of Equations . . . . . . . . . . . . . . . . . . 136.2 Iterative Solution of the Navier Stokes Equation (SIMPLE algorithm) . . 176.3 Algebraic Multigrid Methods . . . . . . . . . . . . . . . . . . . . . . . . . 176.4 Transient Solvers and Rotating Motion . . . . . . . . . . . . . . . . . . . . 18

7 Simulation 187.1 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.2 Investigation of the Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.3 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . 20

7.3.1 Transient Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 227.3.2 Stationary Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 22

8 Results 23

9 Discussion 25

10 Conclusion 26

References 27

1 Introduction

During the production of paper pulp many of the chemicals used are recycled to reduceexpenses and diminish the impact on the environment. In one process, when convertinggreen liquor to white liquor, calcium oxide, CaO, is used as a reactant. Apart fromwhite liquor, the reaction produces calcium carbonate, CaCO3 which can be heated tobecome CaO again. The recycling of CaCO3 to CaO is called lime reburning and takesplace in a rotary kiln, such as the one shown in Figure 1.

A rotary kiln used for lime reburning consists of a rotating cylinder with a refractorywall which typically is around 90 meters long and with a diameter between 2 to 4 meters[1]. The cylinder has a horizontal inclination between 1 to 4 degrees and rotates about1-2 rpm. The lime mud enters the upper end of the kiln and slowly moves downwardsdue to the inclination and rotation of the kiln and meanwhile it is gently mixed by thekiln’s rotation. A burner is situated at the other end of the kiln, producing a flame witha length approximately 10-15 m. The flame length depends a lot on the burner’s fuel,where oil is usually the main component. At the lower end there is also an air intakewhich together with the burner’s fuel produces a stream of hot flue gases flowing upthrough the kiln. These gases heat the lime mud and cause the mud’s CaCO3 to reactinto CaO according to

CaCO3 −→ CaO + CO2 (1)

which is an endotermal reaction, i.e. it requires heat [1]. This so called calcinationoccurs at temperatures starting from approximately 800oC.

A traditional lime kiln can be divided into four different zones: drying, heating, calci-nation and sintering. Figure 2 shows the different zones and the typical temperaturesfor the lime mud and the flue gases in each zone. In most modern lime kilns the limemud is dried before entering the kiln and in the first part of the kiln the lime is heatedup to the reaction temperature (800oC). In older kilns the lime mud is dried in the firstsection, leading to a longer total kiln length.

During the calcination the temperature of the lime mud is almost constant since allabsorbed heat is used to fuel the reaction rather than heating the lime. The lime mudtemperature is constant also in the drying zone since the absorbed heat is used toevaporate the moist in the mud.

After the calcination the CaO is sintered which means that the small CaO particlesagglomerate and form larger particles. This is a crucial step in the retrieval of CaOsince if it is heated too much or too little its reaction with the green liquor will not beoptimal.

In order to optimize the reburned lime quality, reducing stresses on the refractory andoptimizing fuel consumption it is of utmost importance to get a better understandingof the energy-consuming lime reburning process. Computational fluid dynamics (CFD)models can provide valuable information on the temperature profiles of the refractory,

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Figure 1: A lime kiln at the Iggesund pulp mill (2008).

flue gases and the lime bed. These temperatures are often difficult to measure dueto the kilns’ length and the lack of possibilities to access the interior of commercialkilns. Thermographic cameras become inaccurate due to the high amount of particlesin the flue gases and it is difficult to reach the middle of the kiln with measuring-probesinserted at the kilns’ inlet or outlet. Furthermore, a lime kiln operates around the clockand therefore maintenance and experiments are costly due to the required productionbreaks. CFD models can be used not only to optimize the kiln operation and to minimizethe stress on the refractory but also to investigate the impact of using alternative fuelsin the burner. Performing experiments with new fuels will be expensive and risky butan accurate CFD model can predict the outcome of various fuel mixtures.

However, the CFD simulations are complex and very time-consuming, partly due to therotation of the kiln-refractory. In previous CFD models of the kiln such as in [2] and [3],the rotation of the refractory has been neglected to facilitate the computations and thelime mud has been treated as an one-dimensional plug flow with a uniform temperatureprofile. Unfortunately this simplification reduces the accuracy of the model: In the realkiln the refractory is heated by the flue gases and as it rotates the heat is transported to

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Figure 2: The temperatures of the gas and the lime mud in the different zones of a limekiln.

the bottom of the lime mud. This process is drawn schematically in Figure 3. This meansthat the lime bed is heated from the top by the flue gases as well as from the bottomby the warm refractory, causing temperature gradients in the lime mud. According toGorog et al. [4] this so called regenerative heat transfer constitutes 7% - 13% of thetotal heat transfer to the lime mud in the pre-heat and calcination zones. Even thoughthe heat transfer from the refractory has been included in previous models ([2], [3])their treatment of the lime mud as a uniform plug flow makes it impossible to predictthe temperature profiles of the lime mud, which plays a significant role in the sinteringprocess.

2 Aim

The goal of this project is to find a computationally feasible way to model the rotation,and thereby the regenerative heat transfer, of a rotary kiln in steady-state operation.The rotation can be modelled using transient solvers which makes it possible to add

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Figure 3: A cross section of the kiln showing the heat transfers, Q, between the differentregions of the kiln.

motion to regions in STAR-CCM+, the commercial CFD software used in this project.Unfortunately, due to the small time scales, this may be very computationally heavy forthe rotary kiln. The time scales range from 104 s (the time it takes for the lime mud topass through the kiln) down to 10−3 s (the rate of the chemical reactions in the lime) andeven further down if the turbulence of the flame should be modelled accurately. Sinceit takes several hours for the kiln to reach its steady-state operation mode [1], it wouldbe computationally infeasible to make a transient model of a whole rotary kiln with atime step sufficiently small to acquire accurate results. However, using the so calledconvective heat velocity option in STAR-CCM+ it is possible to model regenerative heattransfer using only stationary solvers [5]. Thus, to find a computationally feasible way ofmodelling the regenerative heat transfer of a rotary kiln, the convective velocity optionwill be investigated in this project. Since no measurements of the temperature profilesin a rotary kiln are available, a simplified transient model will be used the verify theresults.

The simplified model consists of only a small part of the kiln, and many aspects of the

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lime reburning process which does not have a significant effect on the regenerative heattransfer, such as chemical reactions and the presence of a flame, will be neglected.

3 Problem Description

3.1 Simplifications

Since the goal of the project is to consider a stationary model of the rotation rather thanaccurately model chemical reactions etc., a lot of the processes in the kiln are neglectedto speed up the computations. For example, details about reaction rates and speciescomposition, which are significant in a full model of the kiln, are not of interest heresince their behaviour does not change much with or without rotation. However, in afinal model of the kiln where the purpose is to understand the sintering process and theeffect of changing fuel, the presence of a flame and the chemical reactions need to betaken into account.

Only a very small part of the kiln, 1m long, will be modelled. It is assumed that if acorrelation between a transient and a corresponding stationary solution is found for thisshort model the same correlation will hold for a full model of the kiln. This might seemlike a bold simplification but both Boateng [1] and Gorog et al. [4] states that the heattransfer in the refractory in the longitudinal direction of the kiln is negligible. Thereforeif a correlation between the transient and the stationary models is found for a smallsection of the kiln, it is likely that the same correlation holds for the entire kiln. Initialinspection of the kiln behaviour showed that a length of 1m was enough for avoidingboundary effects on the temperature profiles.

In the first models, the presence of flue gases and lime mud are represented by fixedtemperatures on the inner wall boundary of the refractory to further speed up the com-putations. This model is used to get an initial comparison between the results from thetransient and the stationary cases and to see how fine the mesh on the refractory hasto be. When a satisfactory correlation between the transient and the stationary modelshas been obtained, and a sufficiently fine mesh is found, the fixed temperatures on therefractory wall will be replaced by two fluid domains, one for the flue gas and one forthe lime mud. Simulations will be made for the calcination zone only. Reference valuessuch as temperatures and flow-velocities will be taken from [3] and [4].

In the full model the flue gases are modelled as air with an enhanced absorption coef-ficient (namely 100 m−1) so it gets properties similar to the flue gases which absorbsmore radiation than normal dry air. The radiation properties of the real flue gases arecomplicated and a thorough study of how to model them is beyond the scope of thisproject. The approach with modified air is deemed to not affect the regenerative heattransfer in great extent but when performing a full final model of the rotary kiln the fluegases have to be modelled more carefully.

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The simplification that the lime mud is a non-reacting homogeneous fluid, while theactual lime mud consists of a reacting inhomogeneous particle-flow, complicates theheat transfer process between the lime mud and the flue gases.

Letting the lime mud be represented by a non-reacting fluid causes the heat transferto be underestimated: Since the top layer will be heated up quickly by the hot gases,the decrease in temperature difference between the lime mud and the gases will de-crease the heat transfer between them. In the real flow, in the regions where reactionor phasechange occurs, a major part of the absorbed heat will be consumed by the en-dothermic chemical reactions (either calcination or evaporation of water depending onthe zone of the kiln), leaving the temperature of the lime mud, and hence also the heattransfer, relatively unaffected. To solve this issue without involving the computationallyexpensive reactions, the lime mud surface is given a constant temperature in the gasdomain, equal to the lime mud’s temperature. The amount of heat that the lime mudabsorbs on the surface equals the amount of heat removed from the gases by the coldlime bed surface. With this configuration, a change in the lime mud temperature doesnot affect the heat transfer from the lime mud to the flue gases or vice versa, but achange in flue gas temperature would affect the heat transfer between the regions.

Modelling the lime mud as a fluid also affects the heat transfer between the lime mud andthe refractory at the bottom of the lime bed for the same reasons as described above, i.e.that the energy consuming chemical reaction is not taken into account. Here, instead ofusing a boundary with a fix temperature as for the gas-to-lime mud boundary, the specificheat capacity of the lime mud is altered to get a realistic heat transfer. When increasingthe specific heat capacity of the lime mud it can absorb more heat without increasingin temperature and thus the heat transfer, induced by the temperature difference, isunaffected.

Note that the heat transfer is modelled in two different ways at the two boundaries gas-to-lime mud and refractory-to-lime mud, even though the same behaviour of the heattransfer is desired. That is, the lime mud is supposed to absorb heat without increasingin temperature due to the endothermic chemical reactions which have been neglected inthe model. The reason why two different approaches have been used is that it simplifiedthe investigation of how different gas and lime bed temperatures affect the heat transferto the lime mud.

Furthermore, the turbulence in the lime mud flow is neglected since it is assumed not toaffect the regenerative heat transfer significantly.

3.2 Geometry

Due to the kiln’s rotation the lime bed obtains an inclined surface [1]. In this projectthe geometry is based on data from the Monsteras kiln where the inclination has beenmeasured to 32 degrees compared to a vertical line through the kiln [3]. It is assumedthat this is the inclination of the lime bed throughout the kiln. Due to the high viscosity

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of the lime mud and the slow rotation of the refractory wall, it is reasonable to assumethat the lime mud and flue gases never mix [1]. Therefore, the lime mud and flue gasesare modelled as two separate regions with a fix boundary between them with the sameinclination as measured in [3]. The lime bed has a filling degree of 21% of the kiln’sdiameter as measured in [3].

Figure 4: The part of the lime kiln that is modelled. The inclined line indicates theboundary between the lime mud and the flue gases.

Figure 4 shows the part of the kiln being modelled where the gas inlet and lime mudoutlet is to the left, facing the reader. The inclined line indicates the boundary betweenthe lime mud and the flue gases. This part of the kiln is 1 meter long, has an innerdiameter of 2.8 meters and a 30 cm thick refractory wall and the inclination of the kilnin the longitudinal direction is 1.43 degrees.

4 Mathematical model

4.1 Energy and Transport Equations

The so called transport equation (2) can be used to describe how a scalar, φ, changes ina closed physical system as a consequence of diffusion and convection [6], namely

d

dt

∫VρχφdV +

∮Aρφ(v− vg) · da =

∮A

Γ∇φ · da +

∫VSφdV, (2)

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where

• φ is the transported scalar, e.g. temperature, energy or mass,

• V is the cell volume,

• A is the cell surface-area,

• ρ is the density,

• χ is the porosity,

• A is the cell area,

• Γ is the diffusion coefficient,

• a is the face area vector,

• v is the velocity,

• vg is the grid-velocity,

• Sφ is a source term for the scalar φ.

For example, using equation (2) to describe the transport of energy E in a solid weobtain the energy equation:

d

dt

∫VρCpTdV +

∮AρCpTvs · da = −

∮Aq′′ · da +

∫VsudV (3)

where

• Cp is the material’s specific heat capacity,

• q′′ is the heat flux vector [W ],

• su is a energy source term whose definition depends on the physical models selectedby the user,

• vs is the so called solid convective velocity which can be used to model rotation ofa pure body, [5]. vs can be used to do a stationary model of rotation which is thegoal of this project.

The other parameters have the same definition as before. The energy and enthalpy are

related as E = H − pρ where H = h+ |v|2

2 and h = CpT .

The Navier Stokes (N-S) equations which describe the motion of a fluid based on theconservation of mass and momentum [7] can also be derived from equation (2). Theirintegral form reads as

∂

∂t

∫VρχdV +

∮Aρ(v− vg) · da =

∫VSudV (4)

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which describes the conservation of mass, and

∂

∂t

∫VρχvdV +

∮Aρv⊗ (v− vg) · da = −

∮ApI · da +

∮AT · da +

∫VfdV (5)

which represents the conservation of momentum.

Here, I is the identity matrix and ⊗ is the tensor product operator. The evaluation ofthe viscous stress tensor depends on the chosen turbulence models since it is the sum ofthe laminar and the turbulent viscous stress tensors.

The energy equation for a fluid is slightly more complex compared to the solid case dueto the viscous stresses and body forces acting on the fluid volume:

d

dt

∫VρEdV+

∮A

[ρH(v−vg)+vgp]·da = −∮Aq′′·da+

∮T·vda+

∫Vf·vdV+

∫VsudV (6)

where

• E is the total energy [J ]

• H is the total enthalpy [J ]

• T is the viscous stress tensor which consists of a laminar and a turbulent part

• f is the body force vector

The first term in equations (2)-(6) containing the time derivative, ddt(·), is only necessary

in transient analysis of the system, i.e. when we are interested in the system’s behaviourover time.

The behaviour of the physical system can be determined by solving equations (2)-(6),together with suitable physical models (e.g. for turbulence and radiation) which con-tribute to the source terms. However, apart from the case of very easy geometries, itis impossible to solve equations (2)-(6) analytically and one has to resort to the use ofnumerical methods such as finite volume methods (FVM), finite element methods orfinite differencing methods. The commercial software STAR-CCM+ used in this projectemploys FVM to solve equations (2)-(6), their discretization and the solution methodsused in STAR-CCM+ are described in sections 5 and 6 respectively.

4.2 Turbulence

Even though the Navier Stokes equations (4) and (5) theoretically should fully describea fluid flow, it is hard to use numerical methods to determine the behaviour of a flowwhich is turbulent [6]. This has to do with the chaotic nature of turbulence wherethere are large differences in for example velocity over very short distances. The lattermakes the problem stiff and thus it is hard to find a stable solution without resorting to

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various simplifications when modelling the turbulence [6]. Many turbulence models havebeen developed over the years and many of them are available in STAR-CCM+. Theavailable turbulence models in STAR-CCM+ can be divided into four main groups: k-ε,Spalart - Allmars models, k - ω models and Reynolds stress transport models [5]. In thisproject the k-ε model has been employed. It is described in [5] as a good compromisebetween robustness and accuracy when dealing with turbulent flows where heat transferis present. The particular model used is called Realizable k-ε Two-Layer model, itsdefinition can be studied in [5] under → Modeling Turbulence and Transition → UsingK-Epsilon Turbulence → K-Epsilon Turbulence Formulation → Two-Layer Formulationand in the paper by Rodi [8] where the two layer k-ε model was first presented.

All these models are so called Reynolds Averaged Navier Stokes (RANS) equations whichseparates the flow-field parameters into two parts: one for the mean of the parameterand one for its fluctuations which thus are considered as deviations from the mean. Thisresults in a turbulent part of the viscous stress tensor, T, in the momentum equation(5), known as the Reynolds stress tensor [5]. The realizable two-layer k-ε model solvesthe transport equation (2) for the kinetic energy, k, of the turbulence and uses algebraicexpressions based on the wall distance to evaluate the dissipation rate, ε, which is therate of the decrease in kinetic energy. This differs from the standard k-ε model wherealso the dissipation rate is evaluated using the transport equation [5]. In equation (2),the scalar is the kinetic energy when the equation is solved for the k-ε model and amongthe source terms we find for example turbulence production terms based on variousempirical observations and the dissipation rates [5]. The two layer approach which isused here, solves the transport equation in two layers: one near the walls and one beyondthe wall boundary layer region [8].

4.3 Radiation

Heat transfer in the kiln is dependent on both convection and thermal radiation. Theformer takes place when heat diffuses between adjacent molecules while the latter is whenthermal radiation is absorbed and scattered by the molecules. The amount of energyabsorbed and scattered depends a lot on the participating media [9] [10] . For exampledry air, consisting mostly of N2 and O2 absorbs very little thermal radiation while three-atomic molecules such as CO2 andH2O absorbs more [9] [10]. Since the flue gases containa lot of CO2 and H2O, these have to be accounted for in the radiation model. In STAR-CCM+ the so called Participating Media Radiation based on the Discrete OrdinatesMethod (DOM) is available [9]. With DOM, each cell in the participating media canabsorb, emit and scatter radiation in a number of directions specified by solid angles orordinates. The accuracy of the radiation modelling depends on the number of ordinates,of which there are four different accuracy levels available in STAR-CCM+[9]. Here, thesecond-coarsest one with 24 ordinates has been used (called S4 in STAR-CCM+).

The radiation is assumed to be gray which means that the radiation properties areinvariant to the wavelength. In reality, particles absorb different amounts of energy of

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the radiation depending on the wavelength [10].

The results of the DOM radiation model, based on the involved materials radiativeproperties, contributes to the heat flux vector q′′ in the energy equation (8).

The governing equations behind the radiation models in STAR-CCM+ are beyond thescope of this report. Details about the equations and their discretization are found in[9] under Participating Media Radiation Formulation.

4.4 Materials

4.4.1 Refractory

The refractory wall of a rotary lime kiln is usually made out of brick which available inthe STAR-CCM+ material database. It has a low thermal conductivity, 1.0 [ WmK ], anda specific heat of 960 [ J

kgK ].

4.4.2 Gas

The flue gases consist of several different gas-molecules as well as soot particles. Asmentioned before the flue gases are greatly simplified; the soot is neglected and the fluegases are modelled as air with an enhanced absorption coefficient of 100 [m−1] to mimicthe presence of the three-atomic molecules. This heuristic approach will introduce someerrors in the model but should not affect the relation between the stationary and thetransient modelling of the kiln very much, as long as the same coefficient is used in bothmodels. Nor should it affect the regenerative heat transfer process in great extent.

4.4.3 Lime mud

The lime mud was not available in the standard material database and therefore it had tobe defined prior to the simulations. It is modelled as a fluid with the material propertiesfound in [3] namely a density of 1180[ kg

m3 ] and a thermal conductivity of 1.5 [ WmK ]. Asdescribed in Section 3.1, the specific heat of the lime mud is set to a very high value,100000 [ J

kgK ], to represent the case where it absorbs a lot of energy without being heateddue to the energy-consuming lime reburning reaction (1) that occurs.

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5 Discretization

5.1 Discretization of the Governing Equations

Finite volume methods are based on discretizations of the volume and surface integralsof the governing equations (2)-(6), which results in a set of linear equations (non-linearequations are linearized using other numerical methods) [5].

The discretization is made by partitioning the domain of the problem into cells. Theset of cells, which has to cover the entire domain, is called a mesh. By replacing volumeintegrals by cell volume multiplications and surface integrals with the summation overall cell-faces in equations (2)-(6), they are discretized as follows.

Transport Equation:

d

dt(ρχφV )0 +

∑f

[ρφ(v · a−G)]f =∑f

(Γ∇φ · a)f + (SφV )0, (7)

where

• subscript f is the face quantity,

• G is the grid flux, Gf = (a · vg)f ,

• subscript 0 refers to the current cell,

• other notations are analogous to those in Section 4.

Fluid Energy Equation:

d

dt(ρEV0) +

∑f

([ρH(v− vg) + q′′ · a− (T · v)] · a

)f

= (f · v + s)V0. (8)

The convective term ΣfρH(v − vg) is discretized in the same way as for the transportequation where the convected scalar now is the enthaply H. The heat flux vector is givenby −keff∇T , where T is the temperature (should not be confused with the viscous stresstensor, T!) and keff is the effective thermal conductivity. The transient term is computedas for the transport equation and the viscous viscous stress tensor is calculated in thesegregated flow equations (Eq 4 and 5).

Solid Energy Equation: It is discretized analogously to the fluid energy equation.

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Navier Stokes (momentum equation):

∂

∂t(ρχvV )0 +

∑f

[vρ(v− vg) · a]f = −∑f

(pI · a)f +∑f

T · a. (9)

Navier Stokes (conservation of mass): Instead of discretizing equation (4) as for theother equations (2, 3, 5 and 6) the conservation of mass criteria is fulfilled by calcu-lating an uncorrected mass flow rate after the momentum equation is solved. Then, tofulfil continuity, a mass flow correction is calculated and added to the mass flow of thecell. This process is described in more detail in [5] under Modeling Flow Using a Seg-regated Approach → Segregated Flow Formulation → Continuity Equation in DiscreteForm.

5.2 Mesh

The construction of a mesh plays an essential part in finite volume approximationssince a too coarse mesh will give inaccurate results while a very fine mesh makes thecomputations very time consuming. Thus, there is a trade-off between the accuracy andthe speed of the computations. It is important to verify that the results do not changesignificantly when the mesh is refined and Section 7 begins with an investigation ofdifferent mesh shapes and sizes. The meshes that have been used to obtain the results ofthis report are shown in Figure 5. It was necessary to use a finer mesh in the refractoryfor the stationary model than for the transient one. See Section 7.2 for more detailsabout this.

Two thin boundary layer cells, such as the one seen in the gas domain of Figure 6b, wereadded to all surfaces in both models. The same layer is present in Figure 6a but is notvisible on the gas surface.

6 Solution Methods

6.1 Setting Up a Linear System of Equations

The discrete equations described in Section 5 are solved iteratively. When performingtransient analysis, the time derivative in equations (7)-(9) has to be approximated usinga first- or second order scheme [5]. In this project the following second-order scheme isused in transient analysis;

d

dt(ρχφV )0 =

3(ρ0φ0)n+1 − 4(ρ0φ0)

n + (ρ0φ0)n−1

2∆tV0, (10)

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(a) The mesh used for the stationary model, containing 162000 cells.

(b) The mesh used for the transient model, containing 82000 cells.

Figure 5: The mesh of the stationary and transient model.

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(a) The mesh at the boundary between the gas and refractory domain for the stationary model with 60layers in the refractory.

(b) The mesh at the boundary between the gas and refractory domain for the transient model with 22layers in the refractory.

Figure 6: A close-up of the meshes in Figure 5. As can be seen the major difference liesin the number of layers in the refractory. The two layers of the boundary layer mesh isvisible in 6b.

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where ∆t is the size of the time step and n is the current iteration in time. In non-transient analysis this term is equal to zero.

The discretization of the second term in equation (7),∑f [ρφ(v · a − G)]f , can be in-

terpreted as the multiplication of the cell-face value of φ with the mass flow rate atthe same face, f , i.e.

∑f [ρφ(v · a − G)]f = mfφf . Several techniques are available to

calculate the face value φf from the cell-centroid values φ, the easiest being the first-order upwind scheme [5]. In this project a hybrid between the second-order upwindscheme and the central differencing scheme have been used. In STAR-CCM+ it is calledHybrid Second-Order Upwind/Central [5], its formulation is beyond the scope of thisreport.

The face-gradient term, ∇φf , in equation (7) is calculated as

∇φf = (φ1 − φ0)~α+∇φ− (∇φ · ds)~α, (11)

where subscript 0 corresponds to the cell where φ is evaluated and subscript 1 to a cellon the opposite side of the face of cell 0, the distance vector between the centres of cell1 and cell 0 is ds. ~α = a

a·ds and ∇φ = ∇φ0−∇φ12 . In this project, the gradients ∇φ are

evaluated using the Hybrid Gauss-Least Squares Method with a Venkatakrishan limitermethod [5]. The details of these methods are beyond the scope of this project but canbe studied further in [5].

Using equation (10) and the gradient evaluation described above, equation (7) gives riseto a linear system of equations of the following form:

apφk+1p +

∑n

anφk+1n = b, (12)

assuming that we have iterated k steps and then have the result vector b. ap and an arethe coefficients obtained from equation (7) for the cell p and all its neighbours n. Thesame type of equation systems are formed by equations (8) and (9).

To increase the stability of the solution STAR-CCM+ uses implicit under-relaxation.This means that the difference between two following iterations is reduced using andunder-relaxation factor ω [5]. Using this concept equation (12) becomes

apωφk+1p +

∑n

anφk+1n = b+

apω

(1− ω)φkp. (13)

The so called residuals, r, which is one of the tools that can be used to determine whetherthe solution have converged or not, can be computed from equation (12):

r = b− apφkp −∑n

anφkn, (14)

and can be read as the difference between the expected result and the result from thecurrent iteration.

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6.2 Iterative Solution of the Navier Stokes Equation (SIMPLE algorithm)

The discrete N-S equations are solved using the SIMPLE algorithm which uses a predictor-corrector approach to fulfil the conservation of mass- and momentum criteria. The stepsof the SIMPLE algorithm are [6]:

1. Using the pressure field values of the previous iteration, solve the momentumequation to obtain an estimate for the velocity field.

2. Solve the pressure correction equation (A discrete version of the Poisson equationwhich depends on the velocity. It is described in [5]).

3. Update the pressure field using the result from step 2.

4. Update the mass flow rates.

5. Update the velocity field.

6. Update density

7. Return to step 2 until convergence (i.e. no more change in velocity- and pressurefields).

For more details see [5] and [6].

6.3 Algebraic Multigrid Methods

Algebraic multigrid (AMG) methods are used together with the Gauss-Seidel method[6] to solve the linear equation systems, c.f. equation (12), that arise from the aboveequations [5]. These have the advantage of being less costly than Gauss Elimination orLU factorization for sparse matrices, which is what we often get when setting up this typeof system of equations [5]. Only a brief explanation of the theory of multigrid methodsis presented here, the interested reader can see for example [6] for more details.

The structure of an AMG method is to solve the linear equation system for the samedomain using meshes of different sizes. This is convenient since relaxation techniques,such as the Gauss-Seidel used here, effectively reduces high-frequency errors which iswhy the techniques are also known as smoothers. Therefore, if a few iterations are firstmade on a fine mesh, the residuals can be transferred to a coarser mesh. What waslow-frequency errors on the fine grid will be high-frequency errors on the coarser gridand thus they are effectively reduced using Gauss-Seidel. Then if necessary, the residualscan be transferred to an even coarser mesh, repeating the same procedure. The processof going from a fine to a coarse mesh is called restriction, referring to the residualsof the fine grid which are restricted onto the coarser one. When the coarsest level isreached, its solution is interpolated back onto the second-coarsest mesh and the systemof linear equations is solved again using a smoother (e.g. Gauss-Seidel). The result fromthe coarser mesh is used as the previous iteration value in the smoother. This process,

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known as prolongation, is repeated until the finest mesh grid from where the processstarted is reached again.

6.4 Transient Solvers and Rotating Motion

To model the rotation of the refractory, a motion is imposed on its mesh. This meansthat in each time step τ , the mesh of the refractory will rotate ωτ radians where ω isthe rotational speed, as shown in Figure 7. When the mesh rotates the cells’ parametervalues, such as the temperature, will move along with the cell and the interfaces betweenthe moving regions are updated so that the cells can access the neighbouring cell-valuesin the computations. Between each iteration in time, equations (7)-(9) are evaluatedin the so called inner iterations to see how the change of refractory position affects thesolution. The more inner iterations and the smaller the time step τ is, the better willthe resemblance to a real continuous rotation be.

Since the goal of the simulations in this project is to find the temperature profiles of thekiln when it is operating in steady state it is necessary to continue the iterations untilthe variations in time are negligible.

7 Simulation

7.1 Convergence Criteria

A solution is considered to have converged when the following criteria are fulfilled:

1. The total heat transfer equals zero which means that the heat that goes into thedomain equals the heat leaving the domain.

2. The outer surface temperature and the temperatures along the probe-lines shownin Figure 9 are constant.

3. The residuals for continuity, energy and momentum are constant and below 10−4

for the stationary simulations.

4. The residuals for continuity, energy and momentum are constant and below 0.01at the end of the inner iterations for the transient simulations.

The reason why the convergence criteria regarding the residuals is less strict for thetransient case is due to behaviour of the transient solver. In each time step the meshis moved according to the rotational speed and the chosen time step. That naturallyincreases the residuals significantly and it was not computationally feasible to increasethe number of inner iterations to match the stationary residuals.

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(a) The mesh at time ti. P1 and P2 are adjacent to each other.

(b) The mesh at time ti+1 = ti + τ . Point P2 has now travelled the distance ωτ while P1 has remainedstationary.

Figure 7: Illustration of how the rotation is modelled using a moving mesh. The blueregion is the refractory. P1 and P2 are fixed in the mesh meaning that they move inspace if the mesh does.

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7.2 Investigation of the Mesh

Figure 8 shows three different meshes for the refractory and Table 1 shows the tem-perature at different locations of the refractory. The temperatures are the average often measuring points along a line in the longitudinal direction at the locations shownin Figure 9. The surface temperature is the temperature average over the refractory’souter surface. To speed up these calculations, the lime mud and the flue gases werereplaced by fix temperatures on the respective surfaces. This gives slightly different re-sults compared to the full model but should not affect the requirements on the refractorymesh.

The conclusion that can be made from the investigation of the mesh is that it is importantto have thin layers at the refractory wall-boundaries to get an adequate resolution ofthe heat transfer. If a too coarse mesh is used at the boundaries, a lot of energy needsto go in or out of the cell if a temperature change is to occur. Since the heat transferis determined by temperature differences, this means that if the temperature of the celldoes not change as heat flows in or out of it, nor will the heat transfer to the cell changewhich gives inaccurate results.

Since the difference between using 22 and 60 layers was small for the transient simulationsit was deduced that it is sufficient to have 22 homogeneously distributed layers on therefractory wall. For the stationary case there is a 4% difference at the surface averagetemperature between the homogeneous 22 and 60 layer cases, giving an over predictedsurface temperature when the mesh is coarser. Therefore in the following simulations thestationary model will have 60 equally thick layers in the refractory while the transientwill have 22 layers of the same kind. The whole mesh, including the gas and lime muddomains, is shown in Figure 5. A fine boundary layer mesh is present at all domainboundaries to accurately capture the heat transfer between the regions while the meshis coarser in the middle of the gas and lime mud domains. The coarse mesh of the innerdomains is not expected to affect the solution significantly since the flow is relativelyuniform and turbulence is not expected to have much effect on the results since it playsa very small role in the process of heat transfer.

For the transient model the mesh in the refractory contains approximately 60 000 ele-ments and the entire mesh has 82 000 elements when the gas and lime mud is included.For the stationary model these numbers are 139 000 and 162 000 respectively.

7.3 Boundary and Initial Conditions

The model parameters were chosen to represent the calcination zone of the kiln (c.fSection 3 in Figure 2). 5 kg/s of lime mud enters the lime mud-inlet (facing away fromthe reader in Figure 4) at a temperature of 1200 K and 7.0 kg/s of flue gases enters thegas inlet (facing the reader in Figure 4) at a temperature of 1920 K. The walls satisfythe no-slip criteria. The ambient temperature is set to 300 K and heat is transfered from

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(a) A part of the refractory when the mesh consists of 60 equally thin layers.

(b) A part of the refractory when the mesh consists of 22 equally thin layers.

(c) A part of the refractory when the mesh consists of 22 layers where the layer thickness increases by10% for each layer in the radial direction, i.e. from the inner to the outer surface.

Figure 8: Three different types of mesh for the refractory.

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Table 1: Temperatures at five different locations for five different meshes.

Model BottomTemp.(K)

PostTemp.(K)

PreTemp.(K)

TopTemp.(K)

SurfaceTemp.(K)

Stationary, homogeneous mesh w. 60layers (cf. Figure 8a)

1572 1541 1621 1596 412

Stationary, homogeneous mesh with 22layers (cf. Figure 8b)

1570 1539 1620 1594 429

Stationary, fine to coarse mesh with 22layers (cf. Figure 8c)

1548 1520 1626 1619 535

Transient, homogeneous mesh with 22layers

1570 1509 1652 1600 409

Transient, homogeneous mesh with 60layers

1571 1515 1650 1602 408

the refractory wall to the surroundings which acts as a heat sink, i.e. it is unaffected bythe heat it absorbs. The heat transfer coefficient form the wall is set to 20 [ W

m2K].

7.3.1 Transient Simulation

The rotation was set to 1.6 rpm with the time step 0.5 s and 20 inner iterations was madein each time step. This means that after the wall rotated the distance correspondingto 0.5 s of real time in each time iteration, 20 iterations were made to update all themodel parameters such as temperature, velocity, pressure etc. The time step was chosenso that the rotation in each step would not exceed the length of a cell.

Using 6 cores on a computer with two 2.7 GHz processors, each with 4 cores, having 24GB RAM, the transient simulation took approximately 65 hours to converge.

7.3.2 Stationary Simulation

Setting the convective heat velocity option to 1.6 rpm the model converged after ap-proximately 4 hours using the same computer as in the transient simulation. To see howthe temperature profile in the refractory changed with the convective heat velocity, fouradditional simulations were made with different convective heat velocities: 1.2 rpm, 1.4rpm, 1.6 rpm and 2.0 rpm. In these four simulations the lime mud and flue gases werereplaced by fix temperatures as in the mesh investigation to save time.

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Figure 9: The refractory with the positions where the temperature is measured. Thepink surface at the bottom represents the part of the refractory which is covered by limemud. The refractory rotates counter-clockwise.

8 Results

In Figures 10 and 11 the temperature profiles of a cross section in the kiln are shownfor the transient and stationary cases with a rotation of 1.6 rpm and Figures 12 and 13shows the temperature profile in the refractory for the same cases. Note that in Figures12 and 13 the lime mud and flue gases are present in the model but have been removedfrom the plot to give a better view of the refractory temperature profile.

Table 2 compares the temperatures and time to convergence at the locations specifiedin Figure 9 between the transient and the stationary case and Table 3 shows the tem-peratures at the same locations for the different convective heat velocities.

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Figure 10: Temperature profile of the refractory at a cross section of the kiln’s calcinationzone for the transient model with a rotational speed of 1.6 rpm.

Figure 11: Temperature profile of the refractory at a cross section of the kiln’s calcinationzone for the stationary model with a convective heat velocity of 1.6 rpm.

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Figure 12: Temperature profile of the refractory at a cross section of the kiln’s calcinationzone for the transient model with a rotational speed of 1.6 rpm.

Figure 13: Temperature profile of the refractory at a cross section of the kiln’s calcinationzone for the stationary model with a convective heat velocity of 1.6 rpm.

9 Discussion

It is a bit problematic to use one computational model to verify another, as is donehere where the transient model is used to verify the stationary one. However, for sucha simple model as the kiln investigated here the transient model is deemed to be anappropriate option for validation as no real measurements are available.

From the temperature profiles in Figures 10 to 13 we can make a qualitative comparisonbetween the transient and the stationary model and see that the results seem to besimilar. Table 2 further confirms this as the temperature difference is between 0% and

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Table 2: Comparison between the temperatures and time to convergence of the transientand the stationary model.

Model BottomTemp.(K)

PostTemp.(K)

PreTemp.(K)

TopTemp.(K)

SurfaceTemp.(K)

Time to conver-gence (hours)

Transient 1489 1460 1702 1765 473 65

Stationary 1477 1470 1685 1795 475 4

Difference 1% 1% 1% 2% 0% 94%

Table 3: Temperatures for different convective heat velocities for the stationary model(transient model included for comparison).

Convective heat velocity (rpm) BottomTemp.(K)

PostTemp.(K)

PreTemp.(K)

TopTemp.(K)

SurfaceTemp.(K)

1.2 1563 1522 1638 1602 410

1.4 1568 1532 1629 1599 411

1.6 1571 1541 1621 1596 412

2.0 1576 1552 1611 1594 414

Transient 1.6 rpm 1571 1515 1650 1602 408

2% at the different measuring points.

From Table 3 we can see that the temperatures changes only slightly as the convectiveheat velocity is varied. This stable behaviour is a desirable feature of the convective heatvelocity option since the temperatures in a real kiln are not expected to change muchwhen the rotational speed is varied slightly.

As expected, the highest temperatures of the refractory are at its inner boundary (c.f.Table 2) and it is also here the biggest temperature differences, approximately 325K,in the transversal direction are found. It is this temperature difference that is requiredto determine the stresses in the refractory which in turn are necessary to be able todetermine the refractory’s durability. The stresses arise when the refractory expandsand contracts as it changes temperature, which finally causes failure due to fatigue [1].Without including the contact between the lime mud and the refractory, such as inprevious models [2], [3], the temperature differences in the transversal direction of therefractory, and thus its durability, becomes very hard to predict.

10 Conclusion

The convective heat velocity option is a good approach for modelling the regenerativeheat transfer in a computationally feasible way, assuming that the results from the

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transient model are accurate and can be used to validate the stationary model. The longcomputational time for the transient simulations, even for this very simple model, makesthe transient approach computationally infeasible. In a full model were the chemicalreactions and the turbulence of the flame are included, a much smaller time step haveto be used which would make the model even slower.

To accurately model the heat transfer between the refractory and the fluid regions itis important to have a very thin mesh at the boundaries. It takes too long time forthe boundary cells to update their temperature if their volume is large due to the FVMapproach where the cell-centroid temperature represents the temperature for the wholecell.

To get a reliable model for the regenerative heat transfer, as well as other processesin the kiln, many of the simplifications made here has to be reviewed. Especially thenature of the lime mud which here have been modelled as a fluid although it ratherconsists of a particle flow. This has consequences both for the mixing of the mud, andhence its temperature profile, and for the heat conductivity in the mud, since inter-particle radiation will play an important role on the mud’s heat conductivity. Also theradiation in the flue gases needs to be more carefully modelled, with the absorptivityand emissivity of the gases based on the composition of the flue gases. Furthermorethe endothermic chemical reactions have to be included instead of giving the mud anunnaturally high specific heat capacity.

References

[1] A.A. Boateng, Rotary Kilns - Transport Phenomena and Transport Processes,Butterworth-Heinemann, 2008, USA

[2] K. Svedin, C. Ivarsson, R. Lundborg 1096. Lime kiln modeling & One-dimensionalsimulations. Varmeforsk (www.varmeforsk.se), 2009

[3] F. Ohman, F. Ishaq, K. Svedin, F& U-aktiviteter kring LignoBoost efterFRAM-programmet/Fortsatt arbete med mesaugnsmodellering. To appear at Aforsk(www.aforsk.se)

[4] J.P. Gorog, T.N. Adams, J.K. Brimacombe Regenerative Heat Transfer in RotaryKilns. Metallurgical Transactions B, Volume 13B, June 1982.

[5] STAR-CCM+ v. 7.04.11 Manual : Modeling Flow and Energy

[6] T.J. Chung, Computational Fluid Dynamics 2nd Ed., Cambridge University Press,2010, USA

[7] P.J. Pritchard, Fox and McDonald’s Introduction to Fluid Mechanics 8th Ed., JohnWiley and Sons, 2011, USA

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[8] Rodi, W. 1991. Experience with Two-Layer Models Combining the k-e Model with aOne-Equation Model Near the Wall, 29th Aerospace Sciences Meeting, January 7-10,Reno, NV, AIAA 91-0216.

[9] STAR-CCM+ v. 7.04.11 Manual : Modeling Physics→Modeling Radiation

[10] S.R. Turns, An Introduction to Combustion - Concepts and Applications 3rd Ed.,McGraw-Hill, 2012, Singapore

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