**2. Materials and methods**

KIVA-3 V [7] is a code developed by Los Alamos National Laboratory for numerical calculation of transient, two- and three-dimensional chemically reactive fluid flows with sprays [1]. It is used to perform numerical simulations which were then used to compare and investigate the variation against the experimental data for a premixed case in a spark ignition engine. To perform the studies, iso-octane is used as a gasoline surrogate.

CHEMKIN [16] has many modules; one of which is SENKIN [17] that performs sensitivity analysis and subsequent reduction of skeletal reaction mechanism. Extraction of the sensitivity data using SENKIN leads to the next step of performing mechanism reduction using KINALC [18] which uses the computational singular perturbation (CSP) method. Sensitivity analysis for spark ignition (SI) engines is performed for both the compression and power strokes since combustion occurs in some portion of each stroke. CSP analysis utilizes to set the size of the time criterion for the characteristic chemical reaction in such a way that fast reactions are discarded from the reduced chemical reaction mechanism while rate limiting reactions are

included into the reduced reaction mechanism since the slower reactions are the rate-controlling reactions [19]. On the other hand, fast reactions cause magnification of the inaccuracies which makes the numerical methods to be unstable. The best way to set the time criterion for CSP is such that it encompasses the combustion process to the end of power stroke. This in effect creates a reduced mechanism that encompasses for the full duration of combustion process, therefore, the selected reactions selected are sensitive and important to the whole combustion process in the engine. For performing the numerical simulation using the reduced reaction mechanism, a mixture of 90% iso-octane and 10% n-heptane is recommended to be used as a gasoline surrogate.

The comprehensive analysis discussed above, helps in construction of a reduced reaction mechanism which mitigates experimental results as well as promotes a greater understanding of the chemical kinetics. Experimental results obtained from [20, 21] for premixed case, at equivalence ratio of 0.98 and 1.3 are mitigated to prove the accuracy of the above-mentioned process. For this, an engine geometry used 85.96 mm bore, 94.6 mm stroke, compression ratio of 11.97 running at 2100 rpm. Engine performance parameters of in-cylinder pressure and heat release rate showed a comparative analysis.

Due to chemical stiffness and large size of reaction mechanisms, computations based on detailed reaction mechanisms are complex. Therefore, it is required to reduce chemical mechanisms. This can be performed using the two levels of reductions mentioned below. As a result, the reduced mechanism is extracted which is a subset of the detailed reaction mechanism.

Level I - Skeletal Reduction: Methods used to eliminate unimportant species and reactions: Directed relation graph (DRG), DRG with error propagation (DRGEP), path flux analysis (PFA), revised DRG (DRGMAX), and computational singular perturbation (CSP) [22].

Level II – Global Reduction Methods: Methods used for analysis of timescale impact on reaction mechanism: Computational singular perturbation (CSP), and quasi-steady-state approximation (QSSA) [22].

The first step to reduce the detailed reaction mechanism to a skeletal reaction mechanism is the elimination of unimportant species using the DRG method which identifies the species closely coupled with major species, such as fuel and oxidizer [19]. This is achieved from a sensitivity analysis that uses Jacobian matrix or sensitivity matrix that can be normalized. The Jacobian matrix is the matrix of all first-order partial derivatives of a vector valued function [19]. This method is important for reduction of a large reaction mechanism. The reduced skeletal mechanisms obtained from DRG are not minimal in size due to assumption of the upper-bound error propagation. A more straightforward definition of DRG is,

$$r\_{A,B} = \frac{\sum\_{j=1,I} |\nu\_{A,i}\alpha\_i \delta\_{Bi}|}{\sum\_{j=1,I} |\nu\_{A,i}\alpha\_i|} \tag{1}$$

where, δBi ¼ 1, if the ith elementary reaction involves species B, otherwise, 0; rAB is the relative error induced to species A by elimination of B, subscript 'i' represents the ith elementary reaction and 'j' represents the jth species, ν<sup>A</sup> is the net stoichiometric coefficient of species A, ω<sup>i</sup> ¼ ωf,i � ωb,i where ωf,i, ωb,i and ω<sup>i</sup> is the forward, backward and net reaction rates respectively, that can be calculated from the already *Numerical Simulations and Validation of Engine Performance Parameters Using Chemical… DOI: http://dx.doi.org/10.5772/intechopen.106536*

given coefficients and activation energy in the CHEMKIN input file. If rA,B < ϵ for all species, then the relation between B and A is considered to be negligible. Species B is selected when rA,B ≥ ϵ. Here ϵ is a user-defined small threshold value. In most cases, ϵ = 0.1 is used [23, 24]. Since ϵ is a user-defined small threshold value, it depends on the application and user experience.

The second step is to further reduce the skeletal mechanism by using the technique called directed relation graph-aided sensitivity analysis (DRGASA). This method further reduces the species set by performing the sensitivity analysis on the already obtained species data from the previous DRG method. The parameters that are focused for the DRGASA method are ignition delays, extinction times, and laminar flame speeds. The reduction with this method is carried out using for a range of pressure, temperature and equivalence ratio. These two steps provide the researchers with a skeletal mechanism.

Skeletal mechanisms are still too large to be used in the computational work and it is important to reduce the skeletal mechanism into a reduced mechanism. A major advancement in the CSP technique has been introduction of a new concept of using a vector G that represents the rate of change of species mass fraction (Y) and temperature (T). Numerical computations are used to monitor any contributions to vector G. Identification of various terms becomes straightforward with this method as it can help identify the reactions that are controlling the reaction, constant terms and chemical species that have depleted [24]. The ODEs given for a reactive system,

$$\mathbf{G} = \frac{d\mathbf{y}}{dt} = \mathbf{S\_r}\mathbf{F\_r}, \text{ where } \mathbf{y} = \mathbf{y\_i} \text{ and } \mathbf{y\_i} = \mathbf{Y\_i} \text{ or } \mathbf{T} \tag{2}$$

where, Sr is the stoichiometric vector and Fr is the reaction rate of rth reaction [25].

$$\mathbf{G} = \frac{d\mathbf{y}}{dt} = \mathbf{S\_r}\mathbf{F\_r} = \nu\_\mathbf{r}\mathbf{q\_r} \tag{3}$$

$$\boldsymbol{\nu}\_{r} = \begin{bmatrix} \boldsymbol{\nu}\_{1} \boldsymbol{\nu}\_{2} \dots \boldsymbol{\dots} \dots \boldsymbol{\nu}\_{n} \end{bmatrix}^{T} \tag{4}$$

where, r = 1, … .., N and n = 1, … , s. Here 'r' is number of reactions and 'n' is number of species.

All the reactions in the reactive system form the vector G.

$$\frac{dG}{dt} = \text{J.G. J} = \frac{dG}{dy} \tag{5}$$

where, J is a Jacobian. G is divided into fast and slow subspace. The following steps are performed to solve a CSP problem [25]:


4.An eigenvalue is considered large if, ∣λ Δt∣> 1*:*0, where Δt is the time step.

5.Determine the total number of large eigenvalues (m) from step 4.


Using the CSP method, a term is only discarded or eliminated when it becomes numerically too small that it does not make any difference. This is determined by the importance index Eq. (6). CSP, as a reduction method, has been used by other researchers successfully [19, 26, 27].

Lu and Law [19] developed the CSP method for the removal of the unimportant reactions. The method used an importance index that eliminates the unimportant species. For the above-mentioned method, the importance index of the reaction is defined as

$$I\_{A,i} = \frac{|\nu\_{A,i} q\_i|}{\sum\_{j=1, N\_R} |\nu\_{A,j} q\_j|} \tag{6}$$

where, 'A' is species, 'i' is reaction and 'q' is overall reaction rate of the ith reaction. Also, νA,j is the stoichiometric coefficient of species A in the jth reaction. It must be noted that a reversible reaction must be treated as a single reaction [19].

If IA,I < ϵReac for all species, then the reaction is an unimportant reaction where ϵReac is a user-defined small threshold.

The parameter ϵReac is a dimensionless parameter and can be defined as [26]:

$$\varepsilon\_{Reac} = |\frac{\mathbf{r}\_{fast}}{\mathbf{r}\_{slow}}|\tag{7}$$


Simplification of detailed model is achieved by eliminating the following [5]:


The above procedure results in a smaller kinetic mechanism. This smaller mechanism is formed from the detailed mechanism and is a subset of the detailed mechanism. The accuracy of the skeletal mechanism is defined with respect to the species that are declared of interest for the problem and the domain of applicability defined for the problem. The domain of applicability is defined on the following criterion [5]:

*Numerical Simulations and Validation of Engine Performance Parameters Using Chemical… DOI: http://dx.doi.org/10.5772/intechopen.106536*


Reaction flow analysis can prove to be very helpful with complex reactions. If the reaction rate satisfies the following condition for all times t, then the reaction rate is unimportant [25]:

$$|RR\_{t,r,\boldsymbol{\zeta}}| < \varepsilon |Max\_{r=1,2,\dots,M} RR\_{t,r,\boldsymbol{\zeta}}|\varepsilon = 1,2,\ \dots \\ Nt = 0,\ \dots,t\_{\text{total}} \tag{8}$$

where, ε is a very small value which is arbitrarily specified, 'r' is elementary reaction number, and 's' is number of species.

The physics of the process of reaction of complex hydrocarbons follows the three basic reaction steps: chain-initiating reactions, chain-branching/carrying reactions and chain-terminating reactions. Chain-initiating reactions are those elementary reactions that produce free radicals. Similarly, free radicals are destroyed in chain-terminating reactions. Chain-propagating reactions or chain-carrying reactions are defined as the elementary reactions where the ratio of free radicals in products to the free radicals in reactants is equal to 1. If this ratio is greater than 1, then the reactions are called chainbranching reactions [28]. Concentrations of free radicals are treated as constant as they remain essentially constant throughout the reaction, except for the short initial and final periods which is minimal as compared to the entire reaction period.

High temperature oxidation of paraffins (CnH2n + 2) that are larger and more complex than the methane is much more complicated as they include many complex hydrocarbons in the mixture. Over the years, the evolution of combustion science has developed detailed combustion mechanisms for those smaller hydrocarbons that are now part of various combustion libraries. Therefore, it is possible to develop a general framework for the complex combustion process [28].

The first step in the combustion process of larger paraffins (CnH2n + 2) is that rather than directly breaking into CH3, it first breaks down into hydrocarbon radicals of lower order, CnH2n + 1. The hydrocarbon radicals of higher order are highly unstable and are further broken down to CH3 and a lower order olefinic compound, Cn-1H2n-2. For hydrocarbons larger than C3H8, the process of fission takes place between the olefinic compound and a lower order radical. Further reactions of those radicals include intermediate steps that eventually form the methyl (CH3) radical. Also, formaldehyde formed during the reactions is rapidly attacked in flames by the O, H, and OH atoms. Therefore, formaldehyde in found only in trace amounts in flames. The situation is more complex for fuel rich hydrocarbon flames [28].

Hydrocarbons follow a similar set of steps for combustion and oxidation. The difference between various hydrocarbons requires further intermediate steps but they go through the same oxidation procedure. The process of complex hydrocarbon oxidation occurs in the following manner [29]:

1.Carbondcarbon (CdC) bond is broken: The CdC bonds are primarily broken over HdC bond because CdC bonds are much weaker than HdC bonds.


A skeletal mechanism [9] was selected and used for further reduction. The skeletal mechanism was enhanced with extended Zeldovich mechanism for the formation of NOx emissions which resulted in 299 reactions and 75 species. Reduction was performed using the CSP technique to achieve 53 reactions and 44 species in the reduced mechanism.

The reduced mechanism was used in the KIVA-CHEMKIN interface to make it part of the KIVA input file. Simulations were performed for the same conditions performed in the engine study for premixed engine [20, 21]. This reaction mechanism has proved to predict and validate the results for both stoichiometric and fuel-rich conditions. Sensitivity analysis using SENKIN was performed for both compression and power stroke as the combustion takes place during some part of both strokes.

#### **2.1 Premixed case**

Experimental results obtained from [20, 21] for premixed case, at equivalence ratio of 0.98 and 1.3 show validation and improved prediction of the engine performance parameters of in-cylinder pressure and heat release rate (HRR). **Figure 1** and **Table 1** show the pentroof engine geometry with no moving valves.

A mesh independent study performed for the reduced mechanism utilized three meshes shown in **Table 2**. Validation studies are performed to compare the general trend of the engine performance parameters. **Figure 1** shows the isometric view of mesh # 3.

#### **2.2 Direct-injection case**

Similar analysis was performed for injection case [31, 32] for the engine geometry shown in **Table 3**, where validation of peak in-cylinder pressure was performed against the experimental data. In this case, mass fractions were calculated based on equivalence ratio of 1.

The mixture is injected at 30<sup>o</sup> BTDC for 1.75 ms or an equivalent of 15.75<sup>o</sup> . The mixture is ignited at 14<sup>o</sup> BTDC. The initial temperature is 302 K and the pressure is 76 1 kPa [31, 32]. The cylinder wall temperature, cylinder fire deck temperature, and the piston temperature are set at 400 K. Mass fractions for fuel and oxidizer are *Numerical Simulations and Validation of Engine Performance Parameters Using Chemical… DOI: http://dx.doi.org/10.5772/intechopen.106536*

#### **Figure 1.**

*Pentroof engine geometry [1, 30] – isometric view and mesh.*


#### **Table 1.**

*Engine geometry [1, 20, 21].*


#### **Table 2.**

*mesh independent study.*


**Table 3.**

*Engine geometry – direct injection.*

**Figure 2.** *Pentroof engine geometry [1, 30] – isometric view with mesh.*

calculated. The fuel is 90% iso-octane and 10% n-heptane [33] given the complexity of a multi-component detailed chemistry model.

The analysis results again showed better agreement of in-cylinder pressure data with the reduced reaction mechanism for the direct injection case (**Figure 2**).
