Simulations of advanced combustion modes using detailed chemistry

2012-01-0145

Simulations of advanced combustion modes using detailed chemistry combined with tabulation and mechanism reduction techniques Francesco Contino Universite catholique de Louvain Tommaso Lucchini and Gianluca D'Errico Politecnico di Milano Catherine Duynslaegher , Veronique Dias and Herve Jeanmart Universite catholique de Louvain Copyright © 2012 SAE International

ABSTRACT Multi-dimensional models represent today consolidated tools to simulate the combustion process in HCCI and Diesel engines. Various approaches are available for this purpose, it is however widely accepted that detailed chemistry represents a fundamental prerequisite to obtain satisfactory results when the engine runs with complex injection strategies or advanced combustion modes. Yet, integrating such mechanisms generally results in prohibitive computational cost. This paper presents a comprehensive methodology for fast and efficient simulations of combustion in internal combustion engines using detailed chemistry. For this purpose, techniques to tabulate the species reaction rates and to reduce the chemical mechanisms on the fly have been coupled. In this way, the computational overheads related to the use of these mechanisms are significantly reduced since tabulated reaction rates are re-used for cells with similar compositions and, when it becomes necessary to perform direct integration, only the relevant set of species and reactions is taken into account. The proposed approach named tabulation of dynamic adaptive chemistry (TDAC) has been implemented in the Lib-ICE code, which is a set of libraries and applications for IC engine modeling developed using the OpenFOAM® technology. In particular, a modified version of the in-situ adaptive tabulation (ISAT) algorithm has been developed for systems with variable temperature and pressure, and the directed relation graph (DRG) method has been used to reduce the mechanism at run-time. The validation has been carried out with HCCI and Diesel cases both using a simplified case to compare the results obtained with and without TDAC, and a detailed case that is validated with experimental data. For each tested condition, a detailed comparison between computed and experimental data is provided along with the achieved speed-up factors compared to the use of direct-integration.

INTRODUCTION In the current environmental context, many research projects in internal combustion engines focus on two primary topics: meeting the stringent emission standards and searching for alternative fuels. To reduce the emissions in conventional engines, various exhaust gas after-treatment strategies have been investigated but lately innovative combustion modes are also studied to further reduce the emissions. These combustion modes include homogeneous charge compression ignition (HCCI) [1, 2], and other derived modes such as premixed compression ignition (PCI) [3], partially premixed combustion (PPC) [4] or radical controlled compression ignition (RCCI) [5]. Environmentally sustainable transportation is also a very important challenge. Biofuels are often mentioned as a possible solution to replace part of the fossil fuels but the combustion of new generation biofuels requires careful analysis and characterization. Page 1 of 19

To investigate both these topics, multi-dimensional models are widely applied, since they can predict the engine performance and pollutant emissions. Furthermore, they allow a very detailed investigation of the main thermal and chemical phenomena taking place in the engine cylinder during fuel-air mixing and combustion processes. Within this context, the use of kinetic mechanisms has become a fundamental prerequisite to simulate complex injection strategies or advanced combustion modes [6,7], but most of these studies use skeletal or reduced mechanisms to maintain a reasonable computational time. However, complex fuel kinetics in a large range of thermo-chemical conditions can be only taken into account through the comprehensiveness provided by detailed mechanisms. When complex chemical schemes are included in combustion models, the evolution of the chemical species in each computational cell is modeled by a system of nonlinear stiff ordinary differential equations (ODE). An efficient way to solve this system is to use the Jacobian of the chemical system to linearize it and then a semi-implicit solver. However, this is still computationally demanding when hundreds of species and thousands of reactions are involved, hence limiting the application of detailed kinetics for practical ICE simulations. For this reason, different approaches were proposed to reduce the computational effort and they can be divided in two groups: methods that reduce the impact of the number of cells, hence reducing the number of times the ODE system is solved, and methods that reduce the impact of the number of species in the kinetic mechanism, hence limiting the size of the ODE system. In the first group, since many cells may be in the same state, multi-grid methods group similar cells and solve them together [8]. Similarly, previously computed results may be adequate for the current cell, therefore the in-situ adaptive tabulation (ISAT) method [9] and the piecewise reusable implementation of solution mapping (PRISM) [10] store previously computed results dynamically and retrieve them according to specific regions of accuracy. In the second group, the reduction of the chemical schemes can be performed by skeletal reduction through elimination of species and reactions that have a small contribution to the overall activity of the system. These methods include sensitivity analysis [11], computational singular perturbation (CSP) [12], directed relation graph (DGR) [13], DRG with error propagation (DRGEP) [14], dynamic adaptive chemistry (DAC) [15], path flux analysis (PFA) [16] and element flux analysis (EFA) [17]. Other techniques could further reduce the resulting skeletal mechanism such as lumping [18], chemistry-guided reduction [19] or quasi steady-state assumptions (QSSA) [20]. Combining both dimension reduction and tabulation (or multi-grid) has been proposed as a preprocessing step with the intrinsic lowdimensional manifold (ILDM) [21], further developed in flame prolongation of ILDM (FPI) [22]. It has also been proposed as dynamic methods, for example: ISAT and rate-controlled constrained equilibrium [23] or adaptive multi-grid chemistry and DAC (AMC-EDAC) [24]. A generalized technique, called TDAC (tabulation of dynamic adaptive chemistry), is presented in this work and applied to the simulation of combustion modes requiring the use of very detailed chemistry. To achieve a very high speed-up compared to direct-integration and significantly limit the operation of the ODE solver, TDAC combines on-the-fly mechanism reduction and tabulation techniques. In this way, only the relevant set of species and reactions is accounted for in each computational cell, and computed reaction rates can be re-used for cells having similar composition to the tabulated ones. A first version of TDAC, combining DAC and ISAT, was applied by the authors to simulate PCCI and HCCI combustion modes and satisfactory results were achieved [25,26]. The TDAC technique has been implemented by the authors into the Lib-ICE code, which is a set of libraries and solvers for ICE simulations based on the OpenFOAM® technology [27, 28, 29]. The object-oriented structure of the code allows to easily implement and test different tabulation and mechanism reduction methods, identifying the best combination for the different combustion modes that needs to be modeled. Two different cases were simulated to test the TDAC method: HCCI engine and diesel combustion at constantvolume conditions. For each of them the performance of the proposed technique was evaluated in terms of speed-up factor compared with direct-integration and agreement with experimental data of ignition delay, pressure trace and heat release rate profile.

COMPUTATIONAL METHODS Including combustion in CFD simulations increases the total number of operations. The operations required for solving the chemical kinetics depend on the number of cells in the mesh and the size of the oxidation mechanism. The system of stiff nonlinear ODE is integrated in every cells and the size of the mechanism defines the level of complexity to integrate this system. Reduction of the computational effort for these separated aspects is achieved by the TDAC method, which is composed of two layers: a tabulation method and a mechanism reduction method. The tabulation method intends to reuse previously computed results, hence decreasing the effect of the number of cells, whereas the mechanism reduction method finds a skeletal mechanism at runtime, hence reducing the effect of the mechanism size (see Figure 1). In the following three sections, we present the ISAT algorithm used at the tabulation layer, then we describe the mechanism reduction layer, and finally the modification needed for the coupling between these layers are introduced. Page 2 of 19

cpu cost

Ncells

Mech. Red. TDAC

Tabulation

Nspecies Figure 1 - The TDAC method takes advantage of the synergy between the tabulation method and the mechanism reduction method. It reduces the computational cost associated with the number of cells in the mesh and the number of species in the kinetic mechanism.

IN SITU ADAPTIVE TABULATION The ISAT algorithm is used when computationally demanding results, e.g. the integration of large and stiff ODE systems, can be stored for subsequent uses. A thermochemical state is defined by the composition

ψ = {Y1 , Y2 , ..., YNs , T, p},

(1)

where Yi are the species mass fractions, Ns is the number of species, T is the temperature and p is the pressure. The integration of the reaction equations for a fixed time step Δt maps the initial composition ψ0 = ψ(t0) to the reacted value ψ(t0+Δt) which is a unique function of ψ0 called the reaction mapping, R(ψ0). During computation, given a query point, ψq, ISAT computes a linear approximation of the mapping from a previously stored point:

R(ψ q ) ≈ Rl (ψ q ) = R(ψ 0 ) + δRl ,

(2)

where δRl is the difference of reaction mapping due to the difference of composition between ψ0 and ψq. δRl is computed by

δRl = A(ψ 0 )(ψ q − ψ 0 ),

(3)

where A is the mapping gradient matrix defined by

Aij (ψ 0 ) � Page 3 of 19

∂Ri (ψ 0 ) . ∂ψj

(4)

The matrix A is related to the first order sensitivity coefficients with respect to initial conditions defined by

Cij (ψ 0 , t) �

∂ψi (t) , ∂ψj0

(5)

with

A(ψ 0 ) = C(ψ 0 , t0 + ∆t) .

(6)

To compute A, the following linear system of ODE is integrated implicitly from t0 to t0+Δt :

d C(ψ 0 , t) = J(ψ(t))C(ψ 0 , t) , dt

(7)

where J is the Jacobian and the initial condition C(ψ0, t0) = I.

The linear approximation defined by equation (2) is only valid in the region

|R(ψ q ) − Rl (ψ q )| = |δR − δRl | ≤ εISAT ,(8) where εISAT is a user-specified tolerance and δR= R(ψq) - R(ψ0). The computation of this region is however prohibitive. It is therefore approximated by a conservative hyper-ellipsoid in the composition space called ellipsoid of accuracy (EOA):

˜ T BT BAδψ ˜ EOA � δψ T A ≤ ε2ISAT ,

(9)

where B is an optional scaling matrix, Ã is the modified A matrix and δψ = ψq - ψ0. B is used to manually modify the shape of the hyper-ellipsoid which can modify the retrieval tolerance for specific species (e.g. when the precision on a species of interest should be increased). Ã is obtained by limiting the smallest singular values to prevent them from generating very large principal axes (see [7, 30]) . During the simulation, the computed results are stored in a binary tree consisting of leafs and nodes : the leafs store ψ, R(ψ), A(ψ) and the EOA description; the nodes store the hyperplanes in the composition space that are used to scan the binary tree and to retrieve the closest stored composition. For each query, one of the following three operations is performed: retrieve if ψq is in the EOA, and growth or addition if ψq is not in the EOA. ISAT retrieves the reaction mapping of ψq using equation (2). For growths and additions, the direct integration of the reaction equations should be first performed. Then, ISAT checks the local error using equation (8). If the error is less than εISAT, the current EOA is too conservative and it is grown to include ψq. Otherwise a new leaf is added to the binary tree to store R(ψq). Further details about ISAT can be found in [9]. In a previous work, some modifications have been included to improve the efficiency of ISAT in engine cases [25]. For unsteady cases, the binary tree should be kept updated to properly predict the onset of the combustion. Therefore, we introduced two parameters to control the maximum number of retrievals and growths for any stored point. These parameters keep the binary tree updated without relying on εISAT only, which significantly increases the performance during the main combustion. In this paper, we have also introduced a cleaning and balancing procedure following the principles given in the DOLFA method [31]. This procedure first scans the binary tree after a given interval to remove points that are deemed too old. Then, it reshapes the binary tree so that the hyperplane of the root node goes through the mean composition and is perpendicular to the direction with maximum variance.

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MECHANISM REDUCTION TECHNIQUES The first version of TDAC only included the DAC method as the mechanism reduction layer [25]. The performances of other reduction techniques (DRG, DRGEP, PFA, and EFA) were subsequently assessed and compared to DAC within the TDAC framework [32]. The first three methods were initially developed for preprocessing the mechanism. We have therefore modified them to reduce the mechanism at runtime. In HCCI simulations, the DRG method was the most efficient and it is used in the following simulations. However, different reduction methods might have better performances in different cases because the interactions between the tabulation algorithm and the reduction methods are not trivial. This is beyond the scope of this paper and will not be detailed in the following sections. The mechanism reduction methods reduce kinetic mechanisms based on a quantitative variable for the coupling between species. This coupling is characterized by the impact of removing one species on the production rate of another species. These methods abstract the couplings by constructing a graph with each node representing a species and each edge representing a strong coupling. If an edge goes from a species A to a species B, it indicates that the removal of B induces a significant error on the production rate of A. If A is kept in the mechanism, then B should also be kept. Figure 2 shows a portion of an illustration graph constructed by the DRG method. As an example, a graph generated by the DRG method for a real oxidation mechanism is shown in Figure 3.

A F

E B D C

Figure 2 - Given a starting species A, the species B, C and D are kept in the mechanism while E and F are removed.

Figure 3 - Network generated with Gephi [33]. It represents the active part computed with the directed relation graph method (adapted for runtime reduction) Page 5 of 19

[13] of the n-heptane mechanism from Curran et al. [34] near top dead center.

The difference between many reduction methods relies on the way to compute the coupling. DRG only characterize direct coupling, the strength of the path between two species is therefore characterized by its weakest link [14]. The reduction algorithm is executed before every call to the ODE solver. It first computes the strength of the direct links between species. Then, the algorithm computes the strength of the paths connecting all the species to a user-defined search initiating set of species. In the following simulations, fuel, CO, and HO2 are used as the search initiating set. In the final step, the reduction scheme removes from the mechanism the species with a value of path strength below a userdefined threshold, εMR, and all the reactions containing at least one disabled species. As mentioned above, we have modified the DRG method such that the reduction can be performed at run-time. It implies that when a species is deemed not important at a given time, it is not totally discarded but rather disabled. If a species is disabled since the beginning of the computation and has zero initial mass fraction, it is kept in the definition of the mechanism for subsequent reduction where it might be active, but neither its chemical source term nor its transport term are solved. If a species is disabled but has a non-zero mass fraction (this also includes inert species), only its transport term is solved. In addition, these disabled species still contribute to the reaction rates of active species through "thirdbody" reactions. The principal advantage of reducing the mechanism at runtime for the local thermochemical conditions is that this reduced mechanism is smaller on average than through the preprocessing of the mechanism which generally largely compensates the overheads of the method.

TABULATION OF DYNAMIC ADAPTIVE CHEMISTRY

retrieve

add grow

Rl (ψ q ) R(ψ q )

R(ψaq )

ψaq

ODE solver

ψq

Mech. Reduct.

ψq ISAT

CFD solver

The TDAC method is the coupling between a tabulation algorithm and a reduction method. It can be visualized as successive layers (see Figure 4): when ISAT receives a query ψq that needs to integrate the ODE set (growth and addition operations), it provides ψq to the reduction algorithm, which then finds the reduced mechanism for the local thermochemical conditions and provides the reduced set of active species ψaq to the stiff ODE solver. This solver computes the reaction mapping for the reduced set R(ψaq) that is used by ISAT to build the reaction mapping R(ψq) in the full composition space. Using simplification methods at distinct levels combines their effects and provides a significant reduction of the computational cost.

Figure 4 - The ISAT algorithm first attempts to retrieve the mapping Rl(ψq) of the query composition ψq through a linear approximation. If it fails, the mechanism reduction method simplifies the mechanism at runtime. It then provides the composition with active species ψaq to the stiff ODE solver, which computes the mapping R(ψaq). ISAT grows or adds a point and extend this mapping to the full composition space R(ψq) . The coupling is more than the serial use of two methods. In particular, ISAT has been adapted to manipulate and to store chemical points with a varying number of species. It may not be necessary to add new points to the binary tree when the Page 6 of 19

number of active species changes. Therefore, the EOA needs to be checked and grown in the full composition space and ISAT should keep track of what species is active for each stored point and activate some species if necessary. Because A is computed through the Jacobian, it only takes active species into account and is more compact. The inactive species are included in the definition of the EOA by only adding the difference of species concentration weighted by the corresponding scale factor to the error definition (8). Equation (9) is applied to all active species while for inactive species, only the diagonal term is considered:

˜ Ta BTa Ba A ˜ a δψa + δψa A δψi BTi Bi δψi ≤ ε2ISAT ,

(10)

where the subscript a indicates active species and subscript i inactive species. During the growth procedure, the algorithm first loops through all inactive species to check if some new dimension should be added to the composition space. Then the procedure proceeds as presented in [30]. This method does not rely on specific tabulation and mechanism reduction methods. Alternative reduction methods have been analyzed (see above) while keeping ISAT but other tabulation algorithm could be used. Moreover, the concept could further simplify the system through other level of reduction such as QSSA.

RESULTS This section presents the results obtained with the TDAC method applied to HCCI and Diesel cases both using a simplified case to compare the results obtained with and without TDAC, and a detailed case that is validated with experimental data.

HCCI COMBUSTION In the following HCCI combustion simulations, we used iso-octane as a fuel and its combustion was modeled by the third version of the mechanism from Lawrence Livermore National Laboratory, which includes 874 chemical species and 3796 elementary reactions [35, 36]. The TDAC method is first applied to a simplified geometry (1350 cells). Inhomogeneous initial conditions were set for both the equivalence ratio and the temperature (see Figure 5) to increase the level of chemical complexity while keeping the number of cells sufficiently low to obtain results using direct integration (i.e. without TDAC).

Figure 5 - The first HCCI case uses a simplified geometry but the initial fields for temperature Page 7 of 19

and equivalence ratio are inhomogeneous so that it includes some level of chemical complexity while keeping the number of cells low. The boundary condition for the temperature was modeled with the wall function from Han and Reitz [37], the engine cycle was simulated from -180 to 180 after top dead center (ATDC) at 1200 rpm with a time step of 0.25 crank angle degree (CAD) during compression and expansion (from -180 to -20 and from 20 to 180) and a time step of 0.05 CAD during the combustion phase. Based on previous works [25, 32], εISAT was set to 10-4 and εMR was set to 2 10-2. For the turbulence, we used the RNG k-ε model modified by Han and Reitz for engine cycles [38]. Both cylinder pressure and heat release rate profiles predicted by TDAC agrees rather well with data computed by direct integration (see Figure 6). Also the predicted evolution of the main chemical species during combustion is satisfactory (see Figure 7). This good agreement is particularly important when using TDAC in more detailed cases since many other parameters or models of the simulations might affect the final results. On a cluster with dual quad core Intel Xeon 2.5 GHz interconnected by gigabit ethernet, the computational time with TDAC on this case was around 5 hours on a single central processing unit (cpu), compared to 270 hours on 16 cpu without TDAC. Based on the scale-up factor (around 12.5 on this architecture), this makes an overall speed-up factor of around 700, which is significant and makes TDAC very suitable to simulate premixed combustion. Because of this speedup on a non-uniform equivalence ratio, this method might also be efficient for partially premixed combustion.

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Pressure [bar]

50

40

60

30

45

20

30

10

15

0 ï40

ï20

0

0 40

20

CAD ATDC

Heat release rate [J/CAD]

60

Figure 6 - The pressure and the heat release rate computed with direct integration (lines) and TDAC (symbols). ï1

10

Species mass fraction

ï2

10

ï3

10

ï4

10

HO2 OH CO2 iC8H18 H2O CO

ï5

10

ï6

10 ï40

ï30

ï20

ï10

0

10

CAD ATDC

20

30

40

Figure 7 - The mass fraction for major and minor species computed with direct integration (lines) and TDAC (symbols).

The TDAC method is further validated in a more detailed case using an axisymmetric mesh with around 12000 cells at TDC and including simplified regions for the crevices and the gasket (see Figure 8). The engine used for the experiments is a medium-duty diesel engine converted to HCCI with a displacement of 0.98 liters/cylinder and a compression ratio of 13.3. An automatic layer addition/removal technique is used to keep the number of cells minimal. The same boundary condition for temperature is used as in the previous case. The homogeneous initial conditions are specified in Table 1: temperature 474°C, pressure 1.38 bar and equivalence ratio 0.2. The swirl intensity is set to 0.9 and the swirl profile factor to 3.11 with an engine speed of 1200 rpm. The engine cycle was simulated from -160 to 120 ATDC with a time step of 0.05 CAD during compression and expansion and a time step of 0.025 CAD during combustion. More details can be found in Hessel et al. [39]. The same settings for the TDAC method and for the turbulence as in the simplified case were used. Table 1 – Initial conditions for the detailed HCCI case based on Hessel et al. [39]. Equivalence ratio Temperature [K] Page 9 of 19

0.12 475.6

0.20 474.1

0.28 468.3

Pressure [bar]

1.361

1.378

1.357

Figure 8 - The mesh of the detailed HCCI simulation at -20 ATDC. It consists of around 12000 cells at TDC and includes simplified crevice and gasket regions. In this case, the numerical simulation without the use of TDAC is prohibitive and therefore, we validate the method with experimental data from Hessel et al. [39]. The results obtained with TDAC are in good agreement with the experimental data for the ignition timings (see Figure 9), the maximum pressure is however slightly underestimated. This can be explained by the heat exchange at the wall that is overestimated. The speed-up factor has been estimated by computing several time steps without TDAC at four different CAD. The speed-up factor for this HCCI case is around 500, which is one order of magnitude better compared to what has been reported in the literature.

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70

Pressure [bar]

60

50

40

30

20

10 ï30

TDAC Exp eriments ï15

TDC

CAD ATDC

15

30

Figure 9 - Pressure traces of HCCI combustion obtained with TDAC (dashed line) or from the experiments reported in Hessel et al. [39] (solid line) at 1200 rpm using iso-octane at equivalence ratio 0.12, 0.2 and 0.28.

DIESEL COMBUSTION Direct-integration of detailed chemistry was successfully applied in the past to model the combustion process in Diesel engines [40, 41]. While the interaction between turbulence and chemistry is neglected, such approach still provides reasonable results in terms of pressure trace, heat release rate, flame structure and soot distribution. For a detailed study of the combustion process in modern diesel engines, a very detailed mechanism is necessary for several reasons: -

The use of multiple injections (three or more) and high EGR rates require prediction of fuel auto-ignition under a wide range thermochemical conditions. Usually, reduced mechanisms perform rather well in a defined range of ambient temperatures (900-1300 K) and EGR (0 - 15%). A suitable surrogate, instead of a single component fuel, should be used to describe the diesel fuel. In this way, significant advantages can be achieved both on the spray and combustion modeling side. Furthermore, the effect of fuel formulation on performance and pollutant emissions can be predicted. Not only the mass of soot, but also the particle distribution should be predicted, since future emission standards will impose a limitation on both these quantities.

A methodology similar to the one previously illustrated for the HCCI engine was also adopted to test the performance of TDAC for Diesel combustion. Simulations were carried out at constant-volume conditions using n-heptane as a fuel. This choice was mainly motivated by the availability of both validated chemical mechanism and experimental data available for such fuel. In this way, a wide range of operating conditions was used to test the proposed approach, including effects of ambient temperature and EGR. The possibility to test multi-component fuels mixtures will be investigated in a future work. Two different mechanisms were used in the following simulations: a simplified mechanism from Lu and Law (68 species and 283 reactions) [42] and a more detailed mechanism from Ranzi et al. (285 species and 8126 reactions) [43]. To identify the best TDAC parameters, preliminary tests were performed using a two-dimensional, axisymmetric geometry with fuel injected in the gas phase close to the symmetry axis. N-heptane is injected through a hole of 0.1 mm radius at 373 K and at a maximum speed of 620 m/s. The vessel contains air at 1250 K and 42.1 bar. The time-step used for the simulation is 10-7 s and we used the RNG k-ε model for the turbulence. Figure 10 shows that the results obtained with and without using TDAC are in very good agreement for both the simplified and the detailed mechanisms. Page 11 of 19

Furthermore, the detailed mechanism predicts a shorter ignition delay, which is defined as the duration between start of injection and maximum pressure rise rate.

42.4

dp/dt [bar/s]

Pressure [bar]

42.3 42.2 42.1 42

2000 1500 1000 500 0

0.3 0.37

0 1

Time [ms]

Figure 10 - The pressure and the pressure derivative computed with (line) and without (symbols) TDAC for the simplified (circle symbols) and the detailed mechanism (square symbol). From 0 to 1 ms, the speed-up factor is around 9 with the detailed mechanism and around 5 for the simplified mechanism. The increase of the speed-up factor with the number of species in the mechanism is lower than what can be expected from previous studies [25,32]. It is mainly due to the particular technique applied by Ranzi et al. to obtain the detailed mechanism, resulting in a moderate number of species involved in a comprehensive set of reactions. Whereas many mechanisms have a ratio between the number of reactions and the number of species around 5, for the detailed mechanism used in these simulations, this ratio is around 30. It increases the overheads of the reduction method, hence reducing the speed-up factor. This speed-up factor is smaller in Diesel cases due to the very large range of thermochemical conditions. After the ignition, the retrieve rate of the ISAT method decreases significantly around 70% of the queries while being above 95% on average in HCCI cases. This proportion remains at this level during the whole mixing controlled combustion (see Figure 11, for the simplified mechanism as illustration). On this figure, it can also be seen that the growth and the addition proportions reach 15% during the mixing controlled combustion. 1 0.9

Proportion [-]

0.8

with scale factors

0.7 without scale factors

0.6 0.5

retrieve retrieve + growth

0.4 0.3 0.2 0.1 0 0

0.37

1

Time [ms]

2

Figure 11 - The retrieve proportion decreases significantly after the ignition. The use of scale factors slightly increases this proportion and reduces the overall number of addition. Page 12 of 19

3

For Diesel cases, the best settings for ISAT are somewhat different to what can be used in HCCI. While in HCCI the use of the scale factors, i.e. the B matrix (see ISAT section above), is not essential for the performance of the method, in Diesel cases, it becomes a fundamental parameter. It can be explained by the dramatic change in the range of equivalence ratios and the need for the ISAT algorithm to focus on particular species. In these cases, scale factors of 0.01 were specified for the fuel, CO, CO2, and H2O. As can be seen on Figure 11, the retrieve rate is 5% higher when using scale factors, with the same growth rate, this reduces the addition proportion by 5%. This higher retrieve rate reduces the computational cost by around 10%. To further illustrate the combustion part of a Diesel case, we have applied the TDAC method to the simulation of nheptane combustion experiments in the SANDIA combustion chamber. It consists of a constant volume vessel with optical access and a common rail injector located at the center of one of its sides. Experimental data are provided through the Engine Combustion Network database (ECN), consisting of a well documented set of diesel combustion experiments performed for different fuels and ambient conditions (pressure, temperature, EGR). These experiments provide a valuable set of validation data for spray and combustion models [44,45]. The complete vessel geometry was modeled and the injected fuel spray was modeled with the Lagrangian approach. The proposed set-up in terms of spray sub-models, mesh size and turbulence models used is illustrated Table 2. Table 2 – Spray sub-models used for the SANDIA combustion chamber simulation [46]. Atomization Secondary breakup Injection Evaporation Heat transfer Collision Minimum mesh size Turbulence model

Bianchi [47] Kelvin-Helmholtz [48] Huh-Gosman [49] Frossling Ranz-Marschall Neglected 1 mm (ALMR) Realizable k-ε [50]

Simulations were carried out using the Adaptive Local Mesh Refinement technique (ALMR), where the grid is dynamically refined only where the spray evolves and fuel-air mixing takes place [51]. ALMR enables a high mesh resolution where the fuel air mixing process takes place, while keeping the overall grid size minimal. A geometric field is chosen as an error estimator and when its values lie in a user-specified interval the parent cell is split into eight child cells by introducing new nodes at the cell centroid and at the mesh face centers. An arbitrary level of refinements can be chosen by the user as well as a maximum number of cells to control the mesh size. The geometric field used as a refinement criterion is represented by the total fuel mass fraction (liquid and gas) in each computational cell. Figure 12 displays the significant reduction in terms of grid size when ALMR is applied, compared to the case where a uniform mesh with the same minimum size is used. In this way, the CPU time required for spray simulations is significantly reduced, without any loss in terms of grid quality. In the simulated case, from 0 to 3ms, the number of cells varies from 2500 cells to around 40000. A cutting through the injector in the middle of the mesh better illustrates the mesh structure (see Figure 13). .

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4

4.5

x 10

4

Number of cells

3.5 3

2.5 2

1.5 1 0.5 0 0

0.5

1

1.5

2

2.5

3

Time [ms]

Figure 12 - With the automatic refinement technique only the required number of cells for a specific time is used (solid line) instead of taking the number of cells required in the most limiting case for the whole simulation (dashed line).

Figure 13 - The cutting of the mesh after 3ms in the middle of the injector illustrates the automatic refinement technique. In past works, spray calculations at non-reacting conditions were extensively carried out by the authors to validate the proposed set-up in terms of spray sub-models and mesh size. In [46] a detailed comparison was performed between computed and experimental data in terms of: -

spray liquid and vapor penetration shape of the spray vapor region radial distribution of mixture fraction and its variance at different distances from the injector.

Such validation was considered a fundamental pre-requisite to perform combustion calculations on the same geometry. Page 14 of 19

The initial conditions follow the specification of [45]: the pressure in the vessel is 42.1 bar and the temperature is 1000 K (950K at the walls). The fuel is injected at 373K with a mass flow rate that reaches 2.8 g/s after 0.03 ms. In addition to air in the vessel, residual CO2 (9%) and H2O (2%) are included. We used the realizable k-ε model for turbulence. The ignition delay has been computed for different εISAT while keeping the other parameters constant. For the simplified mechanism, the ignition delay converges to around 0.7 ms and for the detailed mechanism, to around 0.5 ms. As can be seen on Figure 14, through the use of a detailed mechanism, the experimental ignition delay is very well predicted. The corresponding pressure traces are shown in Figure 15. 1

Ignition delay [ms]

0.9 0.8 0.7 0.6 0.5 0.4 0.3

ï6

ï5

10

ï4

10

10

ε IS AT Figure 14 - When the tolerance on the ISAT method is below 10-5, the ignition delay (maximum pressure rise rate) using the detailed mechanism (square symbols) converges to a value that better predicts the experimental ignition delay (dashed line) than using the simplified mechanism (circle symbols). 42.3

Pressure [bar]

42.25

42.2

42.15

42.1

42.05 0

0.2

0.4

0.6

Time [ms]

0.8

Figure 15 - Pressure trace for the Sandia combustion chamber case. Experimental data (symbols) and computed results with TDAC (tolerance 10-6) for the simplified mechanism (dash-dotted) and the detailed mechanism (dashed). Page 15 of 19

1

From 0 to 1 ms, the speed-up factor has been evaluated to 5 with the detailed mechanism and to 2 with the simplified mechanism. Using the same case with the detailed mechanism, we have explored a range of temperature from 900 K to 1300 K (with oxygen at 21%vol) and oxygen concentration from 8%vol to 21%vol (with temperature at 1000 K). As can be seen on Figure 16, the effect of temperature is well predicted except at 900K. The trend of the oxygen effect is in good agreement with experimental data but the accuracy decreases significantly below 15%vol. This might be explained by a lower precision of the mechanism for longer ignition delays. The predicted spray combustion after 2 ms is shown in Figure 17. a)

1.5 TDAC Experiments

Ignition delay [ms]

1 0.5 0 900

1000

1100

T [K]

1200

1300

b)

2 1.5 1 0.5 8

10

12

15

% vo lO 2

21

Figure 16 - Ignition delays computed in the Sandia vessel with TDAC (circle symbols) compared to experimental data (square symbols) for various temperature (a) and oxygen volume fraction (b).

Figure 17 - Predicted temperature field after 2 ms of the spray combustion in the Sandia combustion chamber at 42.1 bar and 1000 K.

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CONCLUSIONS In this paper, a new version of the tabulation of dynamic adaptive method (TDAC) is presented. TDAC couples a tabulation method with a mechanism reduction method to significantly reduce the computational cost associated with the use of detailed kinetic mechanisms in numerical simulations of internal combustion engines. We have applied this method to HCCI and Diesel cases. First, we have shown that the results with and without TDAC on simplified cases were in very good agreement. Then, on more detailed cases, we have validated the method with experimental data for a HCCI engine and for spray injection in a closed vessel at high pressure and temperature. Using the detailed mechanism, the results obtained with TDAC agreed very well with the experimental ignition delays and pressure traces. The speed-up factors were evaluated to around 500 for the HCCI cases and up to around 9 for the Diesel cases. The Diesel cases are more challenging than HCCI cases since they involve a very large range of thermochemical conditions. During mixing controlled combustion, the proportion of retrieve decreases to around 70% of the queries, which is significantly lower than in HCCI cases (on average more than 95% of the queries are retrieved) and is the main limitation of the method. Further modification of the tabulation algorithm to increase this proportion will remain the focus of future work.

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CONTACT INFORMATION Dr. Francesco Contino SST/IMMC/TFL Stévin Place du Levant 2, bte L5.04.03 1348 Louvain-la-Neuve Belgium Tél.: +32 (0)10/479278 Fax: +32 (0)10/452692

ACKNOWLEDGMENTS The funding of the F.R.S. - FNRS in Belgium is gratefully acknowledged.

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