Ecient Modelling of the Regener- ative Heat Transfer in a Rotary Kiln
Project in Computational Science: Report January 2013
A rotary kiln is a common part in a kraft pulp mill where it is used for lime reburning to recycle some of the chemicals used in the kraft process. A rotary kiln consists of an approximately 90 meter long, inclined rotating cylinder where lime mud is heated by a flame. The lime mud is heated by the hot flue gases produced by the flame as well as from the rotating refractory which absorbs heat from the gases and transports the energy to the bottom of the lime mud as the refractory rotates. The latter heat transfer process is known as regenerative heat transfer.
Several Computational Fluid Dynamic (CFD) models have been made to model the kiln but due to computational limitations the regenerative heat transfer has been neglected in these models. In this project, the regenerative heat transfer has been modelled by imposing energy convection within the refractory using the commercial CFD software STAR-CCM+. Transient models with a rotating refractory region have been used to validate the stationary model since no accurate measurements of the kiln were available. It is shown that for a simplified model of the kiln the stationary approach with a moving heat conduction produces the same results as for the corresponding transient model.
1 Introduction 1
2 Aim 3
3 Problem Description 5
3.1 Simplifications . . . 5
3.2 Geometry . . . 6
4 Mathematical model 7 4.1 Energy and Transport Equations . . . 7
4.2 Turbulence . . . 9
4.3 Radiation . . . 10
4.4 Materials . . . 11
4.4.1 Refractory . . . 11
4.4.2 Gas . . . 11
4.4.3 Lime mud . . . 11
5 Discretization 12 5.1 Discretization of the Governing Equations . . . 12
5.2 Mesh . . . 13
6 Solution Methods 13 6.1 Setting Up a Linear System of Equations . . . 13
6.2 Iterative Solution of the Navier Stokes Equation (SIMPLE algorithm) . . 17
6.3 Algebraic Multigrid Methods . . . 17
6.4 Transient Solvers and Rotating Motion . . . 18
7 Simulation 18 7.1 Convergence Criteria . . . 18
7.2 Investigation of the Mesh . . . 20
7.3 Boundary and Initial Conditions . . . 20
7.3.1 Transient Simulation . . . 22
7.3.2 Stationary Simulation . . . 22
8 Results 23
9 Discussion 25
10 Conclusion 26
During the production of paper pulp many of the chemicals used are recycled to reduce expenses and diminish the impact on the environment. In one process, when converting green liquor to white liquor, calcium oxide, CaO, is used as a reactant. Apart from white liquor, the reaction produces calcium carbonate, CaCO3 which can be heated to become CaO again. The recycling of CaCO3 to CaO is called lime reburning and takes place 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 refractory wall which typically is around 90 meters long and with a diameter between 2 to 4 meters . The cylinder has a horizontal inclination between 1 to 4 degrees and rotates about 1-2 rpm. The lime mud enters the upper end of the kiln and slowly moves downwards due to the inclination and rotation of the kiln and meanwhile it is gently mixed by the kiln’s rotation. A burner is situated at the other end of the kiln, producing a flame with a 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 intake which together with the burner’s fuel produces a stream of hot flue gases flowing up through the kiln. These gases heat the lime mud and cause the mud’s CaCO3 to react into CaO according to
CaCO3 −→ CaO + CO2 (1)
which is an endotermal reaction, i.e. it requires heat . This so called calcination occurs 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 temperatures for the lime mud and the flue gases in each zone. In most modern lime kilns the lime mud is dried before entering the kiln and in the first part of the kiln the lime is heated up to the reaction temperature (800oC). In older kilns the lime mud is dried in the first section, leading to a longer total kiln length.
During the calcination the temperature of the lime mud is almost constant since all absorbed heat is used to fuel the reaction rather than heating the lime. The lime mud temperature is constant also in the drying zone since the absorbed heat is used to evaporate the moist in the mud.
After the calcination the CaO is sintered which means that the small CaO particles agglomerate and form larger particles. This is a crucial step in the retrieval of CaO since if it is heated too much or too little its reaction with the green liquor will not be optimal.
In order to optimize the reburned lime quality, reducing stresses on the refractory and optimizing fuel consumption it is of utmost importance to get a better understanding of the energy-consuming lime reburning process. Computational fluid dynamics (CFD) models can provide valuable information on the temperature profiles of the refractory,
Figure 1: A lime kiln at the Iggesund pulp mill (2008).
flue gases and the lime bed. These temperatures are often difficult to measure due to the kilns’ length and the lack of possibilities to access the interior of commercial kilns. Thermographic cameras become inaccurate due to the high amount of particles in the flue gases and it is difficult to reach the middle of the kiln with measuring-probes inserted at the kilns’ inlet or outlet. Furthermore, a lime kiln operates around the clock and therefore maintenance and experiments are costly due to the required production breaks. CFD models can be used not only to optimize the kiln operation and to minimize the stress on the refractory but also to investigate the impact of using alternative fuels in the burner. Performing experiments with new fuels will be expensive and risky but an accurate CFD model can predict the outcome of various fuel mixtures.
However, the CFD simulations are complex and very time-consuming, partly due to the rotation of the kiln-refractory. In previous CFD models of the kiln such as in  and , the rotation of the refractory has been neglected to facilitate the computations and the lime mud has been treated as an one-dimensional plug flow with a uniform temperature profile. Unfortunately this simplification reduces the accuracy of the model: In the real kiln the refractory is heated by the flue gases and as it rotates the heat is transported to
Figure 2: The temperatures of the gas and the lime mud in the different zones of a lime kiln.
the bottom of the lime mud. This process is drawn schematically in Figure 3. This means that the lime bed is heated from the top by the flue gases as well as from the bottom by the warm refractory, causing temperature gradients in the lime mud. According to Gorog et al.  this so called regenerative heat transfer constitutes 7% - 13% of the total heat transfer to the lime mud in the pre-heat and calcination zones. Even though the heat transfer from the refractory has been included in previous models (, ) their treatment of the lime mud as a uniform plug flow makes it impossible to predict the temperature profiles of the lime mud, which plays a significant role in the sintering process.
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
Figure 3: A cross section of the kiln showing the heat transfers, Q, between the different regions 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 for the rotary kiln. The time scales range from 104 s (the time it takes for the lime mud to pass through the kiln) down to 10−3 s (the rate of the chemical reactions in the lime) and even further down if the turbulence of the flame should be modelled accurately. Since it takes several hours for the kiln to reach its steady-state operation mode , it would be computationally infeasible to make a transient model of a whole rotary kiln with a time step sufficiently small to acquire accurate results. However, using the so called convective heat velocity option in STAR-CCM+ it is possible to model regenerative heat transfer using only stationary solvers . Thus, to find a computationally feasible way of modelling the regenerative heat transfer of a rotary kiln, the convective velocity option will be investigated in this project. Since no measurements of the temperature profiles in a rotary kiln are available, a simplified transient model will be used the verify the results.
The simplified model consists of only a small part of the kiln, and many aspects of the
lime reburning process which does not have a significant effect on the regenerative heat transfer, such as chemical reactions and the presence of a flame, will be neglected.
3 Problem Description
Since the goal of the project is to consider a stationary model of the rotation rather than accurately model chemical reactions etc., a lot of the processes in the kiln are neglected to speed up the computations. For example, details about reaction rates and species composition, which are significant in a full model of the kiln, are not of interest here since their behaviour does not change much with or without rotation. However, in a final model of the kiln where the purpose is to understand the sintering process and the effect of changing fuel, the presence of a flame and the chemical reactions need to be taken into account.
Only a very small part of the kiln, 1 m long, will be modelled. It is assumed that if a correlation between a transient and a corresponding stationary solution is found for this short model the same correlation will hold for a full model of the kiln. This might seem like a bold simplification but both Boateng  and Gorog et al.  states that the heat transfer in the refractory in the longitudinal direction of the kiln is negligible. Therefore if a correlation between the transient and the stationary models is found for a small section of the kiln, it is likely that the same correlation holds for the entire kiln. Initial inspection of the kiln behaviour showed that a length of 1 m was enough for avoiding boundary effects on the temperature profiles.
In the first models, the presence of flue gases and lime mud are represented by fixed temperatures 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 the transient and the stationary cases and to see how fine the mesh on the refractory has to be. When a satisfactory correlation between the transient and the stationary models has been obtained, and a sufficiently fine mesh is found, the fixed temperatures on the refractory wall will be replaced by two fluid domains, one for the flue gas and one for the lime mud. Simulations will be made for the calcination zone only. Reference values such as temperatures and flow-velocities will be taken from  and .
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 absorbs more radiation than normal dry air. The radiation properties of the real flue gases are complicated and a thorough study of how to model them is beyond the scope of this project. The approach with modified air is deemed to not affect the regenerative heat transfer in great extent but when performing a full final model of the rotary kiln the flue gases have to be modelled more carefully.
The simplification that the lime mud is a non-reacting homogeneous fluid, while the actual lime mud consists of a reacting inhomogeneous particle-flow, complicates the heat transfer process between the lime mud and the flue gases.
Letting the lime mud be represented by a non-reacting fluid causes the heat transfer to 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 reaction or 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 on the zone of the kiln), leaving the temperature of the lime mud, and hence also the heat transfer, relatively unaffected. To solve this issue without involving the computationally expensive reactions, the lime mud surface is given a constant temperature in the gas domain, equal to the lime mud’s temperature. The amount of heat that the lime mud absorbs on the surface equals the amount of heat removed from the gases by the cold lime bed surface. With this configuration, a change in the lime mud temperature does not affect the heat transfer from the lime mud to the flue gases or vice versa, but a change 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 and the 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 of using a boundary with a fix temperature as for the gas-to-lime mud boundary, the specific heat capacity of the lime mud is altered to get a realistic heat transfer. When increasing the specific heat capacity of the lime mud it can absorb more heat without increasing in temperature and thus the heat transfer, induced by the temperature difference, is unaffected.
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 heat transfer is desired. That is, the lime mud is supposed to absorb heat without increasing in temperature due to the endothermic chemical reactions which have been neglected in the model. The reason why two different approaches have been used is that it simplified the investigation of how different gas and lime bed temperatures affect the heat transfer to the lime mud.
Furthermore, the turbulence in the lime mud flow is neglected since it is assumed not to affect the regenerative heat transfer significantly.
Due to the kiln’s rotation the lime bed obtains an inclined surface . In this project the geometry is based on data from the M¨onster˚as kiln where the inclination has been measured to 32 degrees compared to a vertical line through the kiln . It is assumed that this is the inclination of the lime bed throughout the kiln. Due to the high viscosity
of the lime mud and the slow rotation of the refractory wall, it is reasonable to assume that the lime mud and flue gases never mix . Therefore, the lime mud and flue gases are modelled as two separate regions with a fix boundary between them with the same inclination as measured in . The lime bed has a filling degree of 21% of the kiln’s diameter as measured in .
Figure 4: The part of the lime kiln that is modelled. The inclined line indicates the boundary between the lime mud and the flue gases.
Figure 4 shows the part of the kiln being modelled where the gas inlet and lime mud outlet is to the left, facing the reader. The inclined line indicates the boundary between the lime mud and the flue gases. This part of the kiln is 1 meter long, has an inner diameter of 2.8 meters and a 30 cm thick refractory wall and the inclination of the kiln in 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 in a closed physical system as a consequence of diffusion and convection , namely
ρχφdV + I
ρφ(v − vg) · da = I
Γ∇φ · da + Z
• 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 we obtain the energy equation:
ρCpT dV + I
ρCpT vs· da = − I
q00· da + Z
• Cp is the material’s specific heat capacity,
• q00 is the heat flux vector [W ],
• suis a energy source term whose definition depends on the physical models selected by the user,
• vs is the so called solid convective velocity which can be used to model rotation of a pure body, . vs can be used to do a stationary model of rotation which is the goal 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 the conservation of mass and momentum  can also be derived from equation (2). Their integral form reads as
ρχdV + I
ρ(v − vg) · da = Z
which describes the conservation of mass, and
ρχvdV + I
ρv ⊗ (v − vg) · da = − I
pI · da + I
T · da + Z
which represents the conservation of momentum.
Here, I is the identity matrix and ⊗ is the tensor product operator. The evaluation of the viscous stress tensor depends on the chosen turbulence models since it is the sum of the laminar and the turbulent viscous stress tensors.
The energy equation for a fluid is slightly more complex compared to the solid case due to the viscous stresses and body forces acting on the fluid volume:
ρEdV + I
[ρH(v−vg)+vgp]·da = − I
f·vdV + Z
sudV (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, dtd(·), is only necessary in transient analysis of the system, i.e. when we are interested in the system’s behaviour over 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, it is impossible to solve equations (2)-(6) analytically and one has to resort to the use of numerical methods such as finite volume methods (FVM), finite element methods or finite differencing methods. The commercial software STAR-CCM+ used in this project employs FVM to solve equations (2)-(6), their discretization and the solution methods used in STAR-CCM+ are described in sections 5 and 6 respectively.
Even though the Navier Stokes equations (4) and (5) theoretically should fully describe a fluid flow, it is hard to use numerical methods to determine the behaviour of a flow which is turbulent . This has to do with the chaotic nature of turbulence where there are large differences in for example velocity over very short distances. The latter makes the problem stiff and thus it is hard to find a stable solution without resorting to
various simplifications when modelling the turbulence . Many turbulence models have been developed over the years and many of them are available in STAR-CCM+. The available 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 . In this project the k-ε model has been employed. It is described in  as a good compromise between robustness and accuracy when dealing with turbulent flows where heat transfer is present. The particular model used is called Realizable k-ε Two-Layer model, its definition can be studied in  under → Modeling Turbulence and Transition → Using K-Epsilon Turbulence → K-Epsilon Turbulence Formulation → Two-Layer Formulation and in the paper by Rodi  where the two layer k-ε model was first presented.
All these models are so called Reynolds Averaged Navier Stokes (RANS) equations which separates the flow-field parameters into two parts: one for the mean of the parameter and one for its fluctuations which thus are considered as deviations from the mean. This results in a turbulent part of the viscous stress tensor, T, in the momentum equation (5), known as the Reynolds stress tensor . The realizable two-layer k-ε model solves the transport equation (2) for the kinetic energy, k, of the turbulence and uses algebraic expressions based on the wall distance to evaluate the dissipation rate, ε, which is the rate of the decrease in kinetic energy. This differs from the standard k-ε model where also the dissipation rate is evaluated using the transport equation . In equation (2), the scalar is the kinetic energy when the equation is solved for the k-ε model and among the source terms we find for example turbulence production terms based on various empirical observations and the dissipation rates . The two layer approach which is used here, solves the transport equation in two layers: one near the walls and one beyond the wall boundary layer region .
Heat transfer in the kiln is dependent on both convection and thermal radiation. The former takes place when heat diffuses between adjacent molecules while the latter is when thermal radiation is absorbed and scattered by the molecules. The amount of energy absorbed and scattered depends a lot on the participating media   . For example dry air, consisting mostly of N2and O2 absorbs very little thermal radiation while three- atomic molecules such as CO2and H2O absorbs more  . Since the flue gases contain a 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 Ordinates Method (DOM) is available . With DOM, each cell in the participating media can absorb, emit and scatter radiation in a number of directions specified by solid angles or ordinates. The accuracy of the radiation modelling depends on the number of ordinates, of which there are four different accuracy levels available in STAR-CCM+. Here, the second-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 are invariant to the wavelength. In reality, particles absorb different amounts of energy of
the radiation depending on the wavelength .
The results of the DOM radiation model, based on the involved materials radiative properties, contributes to the heat flux vector q00 in the energy equation (8).
The governing equations behind the radiation models in STAR-CCM+ are beyond the scope of this report. Details about the equations and their discretization are found in  under Participating Media Radiation Formulation.
The refractory wall of a rotary lime kiln is usually made out of brick which available in the STAR-CCM+ material database. It has a low thermal conductivity, 1.0 [mKW ], and a specific heat of 960 [kgKJ ].
The flue gases consist of several different gas-molecules as well as soot particles. As mentioned before the flue gases are greatly simplified; the soot is neglected and the flue gases are modelled as air with an enhanced absorption coefficient of 100 [m−1] to mimic the presence of the three-atomic molecules. This heuristic approach will introduce some errors in the model but should not affect the relation between the stationary and the transient modelling of the kiln very much, as long as the same coefficient is used in both models. 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 to be defined prior to the simulations. It is modelled as a fluid with the material properties found in  namely a density of 1180[mkg3] and a thermal conductivity of 1.5 [mKW ]. As described in Section 3.1, the specific heat of the lime mud is set to a very high value, 100000 [kgKJ ], to represent the case where it absorbs a lot of energy without being heated due to the energy-consuming lime reburning reaction (1) that occurs.
5.1 Discretization of the Governing Equations
Finite volume methods are based on discretizations of the volume and surface integrals of the governing equations (2)-(6), which results in a set of linear equations (non-linear equations are linearized using other numerical methods) .
The discretization is made by partitioning the domain of the problem into cells. The set of cells, which has to cover the entire domain, is called a mesh. By replacing volume integrals by cell volume multiplications and surface integrals with the summation over all cell-faces in equations (2)-(6), they are discretized as follows.
[ρφ(v · a − G)]f =X
(Γ∇φ · a)f + (SφV )0, (7)
• 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:
[ρH(v − vg) + q00· 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 transport equation where the convected scalar now is the enthaply H. The heat flux vector is given by −kef f∇T , where T is the temperature (should not be confused with the viscous stress tensor, T!) and kef f is the effective thermal conductivity. The transient term is computed as for the transport equation and the viscous viscous stress tensor is calculated in the segregated flow equations (Eq 4 and 5).
Solid Energy Equation: It is discretized analogously to the fluid energy equation.
Navier Stokes (momentum equation):
[vρ(v − vg) · a]f = −X
(pI · a)f +X
T · a. (9)
Navier Stokes (conservation of mass): Instead of discretizing equation (4) as for the other 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, to fulfil continuity, a mass flow correction is calculated and added to the mass flow of the cell. This process is described in more detail in  under Modeling Flow Using a Seg- regated Approach → Segregated Flow Formulation → Continuity Equation in Discrete Form.
The construction of a mesh plays an essential part in finite volume approximations since a too coarse mesh will give inaccurate results while a very fine mesh makes the computations very time consuming. Thus, there is a trade-off between the accuracy and the speed of the computations. It is important to verify that the results do not change significantly when the mesh is refined and Section 7 begins with an investigation of different mesh shapes and sizes. The meshes that have been used to obtain the results of this report are shown in Figure 5. It was necessary to use a finer mesh in the refractory for the stationary model than for the transient one. See Section 7.2 for more details about this.
Two thin boundary layer cells, such as the one seen in the gas domain of Figure 6b, were added to all surfaces in both models. The same layer is present in Figure 6a but is not visible 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 performing transient analysis, the time derivative in equations (7)-(9) has to be approximated using a first- or second order scheme . In this project the following second-order scheme is used in transient analysis;
dt(ρχφV )0 = 3(ρ0φ0)n+1− 4(ρ0φ0)n+ (ρ0φ0)n−1
2∆t V0, (10)
(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.
(a) The mesh at the boundary between the gas and refractory domain for the stationary model with 60 layers in the refractory.
(b) The mesh at the boundary between the gas and refractory domain for the transient model with 22 layers in the refractory.
Figure 6: A close-up of the meshes in Figure 5. As can be seen the major difference lies in the number of layers in the refractory. The two layers of the boundary layer mesh is visible in 6b.
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), Pf[ρφ(v · a − G)]f, can be in- terpreted as the multiplication of the cell-face value of φ with the mass flow rate at the same face, f , i.e. Pf[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 . In this project a hybrid between the second-order upwind scheme and the central differencing scheme have been used. In STAR-CCM+ it is called Hybrid Second-Order Upwind/Central , its formulation is beyond the scope of this report.
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 cell on the opposite side of the face of cell 0, the distance vector between the centres of cell 1 and cell 0 is ds. ~α = a
a·ds and ∇φ = ∇φ0−∇φ2 1. In this project, the gradients ∇φ are evaluated using the Hybrid Gauss-Least Squares Method with a Venkatakrishan limiter method . The details of these methods are beyond the scope of this project but can be studied further in .
Using equation (10) and the gradient evaluation described above, equation (7) gives rise to a linear system of equations of the following form:
anφk+1n = b, (12)
assuming that we have iterated k steps and then have the result vector b. ap and anare the coefficients obtained from equation (7) for the cell p and all its neighbours n. The same 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 and under-relaxation factor ω . Using this concept equation (12) becomes
anφk+1n = b +ap
ω(1 − ω)φkp. (13)
The so called residuals, r, which is one of the tools that can be used to determine whether the solution have converged or not, can be computed from equation (12):
r = b − apφkp−X
and can be read as the difference between the expected result and the result from the current iteration.
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 steps of the SIMPLE algorithm are :
1. Using the pressure field values of the previous iteration, solve the momentum equation to obtain an estimate for the velocity field.
2. Solve the pressure correction equation (A discrete version of the Poisson equation which depends on the velocity. It is described in ).
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 pressure fields).
For more details see  and .
6.3 Algebraic Multigrid Methods
Algebraic multigrid (AMG) methods are used together with the Gauss-Seidel method  to solve the linear equation systems, c.f. equation (12), that arise from the above equations . These have the advantage of being less costly than Gauss Elimination or LU factorization for sparse matrices, which is what we often get when setting up this type of system of equations . Only a brief explanation of the theory of multigrid methods is presented here, the interested reader can see for example  for more details.
The structure of an AMG method is to solve the linear equation system for the same domain using meshes of different sizes. This is convenient since relaxation techniques, such as the Gauss-Seidel used here, effectively reduces high-frequency errors which is why the techniques are also known as smoothers. Therefore, if a few iterations are first made on a fine mesh, the residuals can be transferred to a coarser mesh. What was low-frequency errors on the fine grid will be high-frequency errors on the coarser grid and thus they are effectively reduced using Gauss-Seidel. Then if necessary, the residuals can be transferred to an even coarser mesh, repeating the same procedure. The process of going from a fine to a coarse mesh is called restriction, referring to the residuals of the fine grid which are restricted onto the coarser one. When the coarsest level is reached, its solution is interpolated back onto the second-coarsest mesh and the system of linear equations is solved again using a smoother (e.g. Gauss-Seidel). The result from the coarser mesh is used as the previous iteration value in the smoother. This process,
known as prolongation, is repeated until the finest mesh grid from where the process started 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 means that in each time step τ , the mesh of the refractory will rotate ωτ radians where ω is the rotational speed, as shown in Figure 7. When the mesh rotates the cells’ parameter values, such as the temperature, will move along with the cell and the interfaces between the moving regions are updated so that the cells can access the neighbouring cell-values in the computations. Between each iteration in time, equations (7)-(9) are evaluated in the so called inner iterations to see how the change of refractory position affects the solution. The more inner iterations and the smaller the time step τ is, the better will the resemblance to a real continuous rotation be.
Since the goal of the simulations in this project is to find the temperature profiles of the kiln when it is operating in steady state it is necessary to continue the iterations until the variations in time are negligible.
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 the domain equals the heat leaving the domain.
2. The outer surface temperature and the temperatures along the probe-lines shown in 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 0.01 at the end of the inner iterations for the transient simulations.
The reason why the convergence criteria regarding the residuals is less strict for the transient case is due to behaviour of the transient solver. In each time step the mesh is moved according to the rotational speed and the chosen time step. That naturally increases the residuals significantly and it was not computationally feasible to increase the number of inner iterations to match the stationary residuals.
(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 remained stationary.
Figure 7: Illustration of how the rotation is modelled using a moving mesh. The blue region is the refractory. P1 and P2 are fixed in the mesh meaning that they move in space if the mesh does.
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 of ten measuring points along a line in the longitudinal direction at the locations shown in Figure 9. The surface temperature is the temperature average over the refractory’s outer surface. To speed up these calculations, the lime mud and the flue gases were replaced 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 refractory mesh.
The conclusion that can be made from the investigation of the mesh is that it is important to have thin layers at the refractory wall-boundaries to get an adequate resolution of the heat transfer. If a too coarse mesh is used at the boundaries, a lot of energy needs to go in or out of the cell if a temperature change is to occur. Since the heat transfer is determined by temperature differences, this means that if the temperature of the cell does not change as heat flows in or out of it, nor will the heat transfer to the cell change which gives inaccurate results.
Since the difference between using 22 and 60 layers was small for the transient simulations it was deduced that it is sufficient to have 22 homogeneously distributed layers on the refractory wall. For the stationary case there is a 4% difference at the surface average temperature between the homogeneous 22 and 60 layer cases, giving an over predicted surface temperature when the mesh is coarser. Therefore in the following simulations the stationary model will have 60 equally thick layers in the refractory while the transient will have 22 layers of the same kind. The whole mesh, including the gas and lime mud domains, is shown in Figure 5. A fine boundary layer mesh is present at all domain boundaries to accurately capture the heat transfer between the regions while the mesh is coarser in the middle of the gas and lime mud domains. The coarse mesh of the inner domains is not expected to affect the solution significantly since the flow is relatively uniform and turbulence is not expected to have much effect on the results since it plays a 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.f Section 3 in Figure 2). 5 kg/s of lime mud enters the lime mud-inlet (facing away from the reader in Figure 4) at a temperature of 1200 K and 7.0 kg/s of flue gases enters the gas inlet (facing the reader in Figure 4) at a temperature of 1920 K. The walls satisfy the no-slip criteria. The ambient temperature is set to 300 K and heat is transfered from
(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 by 10% 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.
Table 1: Temperatures at five different locations for five different meshes.
(K) Stationary, homogeneous mesh w. 60
layers (cf. Figure 8a)
1572 1541 1621 1596 412
Stationary, homogeneous mesh with 22 layers (cf. Figure 8b)
1570 1539 1620 1594 429
Stationary, fine to coarse mesh with 22 layers (cf. Figure 8c)
1548 1520 1626 1619 535
Transient, homogeneous mesh with 22 layers
1570 1509 1652 1600 409
Transient, homogeneous mesh with 60 layers
1571 1515 1650 1602 408
the refractory wall to the surroundings which acts as a heat sink, i.e. it is unaffected by the heat it absorbs. The heat transfer coefficient form the wall is set to 20 [mW2K].
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 made in each time step. This means that after the wall rotated the distance corresponding to 0.5 s of real time in each time iteration, 20 iterations were made to update all the model parameters such as temperature, velocity, pressure etc. The time step was chosen so 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 24 GB 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 how the temperature profile in the refractory changed with the convective heat velocity, four additional simulations were made with different convective heat velocities: 1.2 rpm, 1.4 rpm, 1.6 rpm and 2.0 rpm. In these four simulations the lime mud and flue gases were replaced by fix temperatures as in the mesh investigation to save time.
Figure 9: The refractory with the positions where the temperature is measured. The pink surface at the bottom represents the part of the refractory which is covered by lime mud. The refractory rotates counter-clockwise.
In Figures 10 and 11 the temperature profiles of a cross section in the kiln are shown for the transient and stationary cases with a rotation of 1.6 rpm and Figures 12 and 13 shows the temperature profile in the refractory for the same cases. Note that in Figures 12 and 13 the lime mud and flue gases are present in the model but have been removed from the plot to give a better view of the refractory temperature profile.
Table 2 compares the temperatures and time to convergence at the locations specified in 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.
Figure 10: Temperature profile of the refractory at a cross section of the kiln’s calcination zone 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 calcination zone for the stationary model with a convective heat velocity of 1.6 rpm.
Figure 12: Temperature profile of the refractory at a cross section of the kiln’s calcination zone 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 calcination zone for the stationary model with a convective heat velocity of 1.6 rpm.
It is a bit problematic to use one computational model to verify another, as is done here where the transient model is used to verify the stationary one. However, for such a simple model as the kiln investigated here the transient model is deemed to be an appropriate option for validation as no real measurements are available.
From the temperature profiles in Figures 10 to 13 we can make a qualitative comparison between the transient and the stationary model and see that the results seem to be similar. Table 2 further confirms this as the temperature difference is between 0% and
Table 2: Comparison between the temperatures and time to convergence of the transient and the stationary model.
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) Bottom Temp.
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 convective heat velocity is varied. This stable behaviour is a desirable feature of the convective heat velocity option since the temperatures in a real kiln are not expected to change much when 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 325 K, in the transversal direction are found. It is this temperature difference that is required to determine the stresses in the refractory which in turn are necessary to be able to determine the refractory’s durability. The stresses arise when the refractory expands and contracts as it changes temperature, which finally causes failure due to fatigue .
Without including the contact between the lime mud and the refractory, such as in previous models , , the temperature differences in the transversal direction of the refractory, and thus its durability, becomes very hard to predict.
The convective heat velocity option is a good approach for modelling the regenerative heat transfer in a computationally feasible way, assuming that the results from the
transient model are accurate and can be used to validate the stationary model. The long computational time for the transient simulations, even for this very simple model, makes the transient approach computationally infeasible. In a full model were the chemical reactions and the turbulence of the flame are included, a much smaller time step have to be used which would make the model even slower.
To accurately model the heat transfer between the refractory and the fluid regions it is important to have a very thin mesh at the boundaries. It takes too long time for the boundary cells to update their temperature if their volume is large due to the FVM approach where the cell-centroid temperature represents the temperature for the whole cell.
To get a reliable model for the regenerative heat transfer, as well as other processes in the kiln, many of the simplifications made here has to be reviewed. Especially the nature of the lime mud which here have been modelled as a fluid although it rather consists of a particle flow. This has consequences both for the mixing of the mud, and hence 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 the radiation in the flue gases needs to be more carefully modelled, with the absorptivity and emissivity of the gases based on the composition of the flue gases. Furthermore the endothermic chemical reactions have to be included instead of giving the mud an unnaturally high specific heat capacity.
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