Chemical kinetics

From the mathematical point of view, the analysis of the reacting flows during the combustion process is a difficult and complicated procedure that is affected mainly by the complex chemical kinetics that requires the solution of different differential equations related with the mass fractions of the species that are coupled non-linearly via the reaction rate laws. Pre-tabulated kinetically controlled reaction models implemented in CFD have several limitations such as the number of allowable reversible and irreversible reactions or the simplicity of the reaction rate expressions. These models are applicable for the simulation of relatively simple reaction systems. However, in reality the combustion chemistry is not a simple procedure and includes a high number of reactions and species. Therefore, a robust and accurate chemical kinetics mechanism should include a relatively high number of species and reactions in order to be accurate. During this research, the complex chemistry model incorporated in STAR-CD software was used to simulate the combustion chemistry.

Complex chemistry model

By using the complex chemistry model, chemical kinetics mechanisms which have a high number of reactions and species can be used in the simulations. One limitation of the complex chemistry model is that no other sub-model, for example flamelet model, ignition model or NOx model can be coupled. This is because the combustion process is driven mostly by the reaction rate of the chemical reactions and the species conservation equations.

The transport equations for the mass fraction of the species included in the mechanism is given by:

𝜕

𝜕𝑡(𝑝𝑌_𝑖) +_𝜕𝑥^𝜕

𝑗(𝑝 𝑢_𝑗𝑌_𝑖 − 𝐹_𝑖,𝑗 ) = 𝑆_𝑖 (3.26)

72 where 𝐹_𝑖,𝑗 is the diffusion flux component calculated by:

𝐹_𝑖,𝑗 = _𝜎^𝑚𝑡

𝑖,𝑗

𝜕𝑌_𝑖

𝜕𝑥_𝑗+ 𝑝𝐷_{𝑖𝑚 𝜕𝑥}^𝜕𝑌𝑖

𝑗+ ^𝐷 _𝑇^𝑖𝑇_𝜕𝑥^𝜕𝑇

𝑗 (3.27) in which 𝐷_𝑖^𝑇 is the thermal diffusion coefficient, 𝑆_𝑖 is the production rate, 𝐷_𝑖𝑚 the molecular diffusivity of species i in the mixture and i=1…N(or N-1), where N is the total number of species. It is important to be mentioned here that the molecular diffusivity is different for each individual species included in the mechanism. By resolving the transport equations for N-1 species, the mass fraction for the Nth species is calculated from:

𝑌_𝑁 = 1 − ∑^𝑁−1_𝑖−1 𝑌_𝑖 (3.28) For a reversible reaction containing N chemical species, the general form is expressed by:

∑^𝑁_𝑘=1 (𝑛_𝑅𝑘^′𝑅_𝑘 𝑌_𝑅𝑘′) = ∑^𝑁_𝑘=1(𝑛_𝑅𝑘^′′ 𝑅_𝑘𝑇_𝑅𝑘^′′ ) , 𝑅 = 1, … , 𝑁_𝑅 (3.29) And the production rate is calculated by:

𝑆_𝑖 = 𝑀_𝑖∑^𝑁_𝑅=1 ^𝑅 [(𝑛_𝑅𝑖^′′ − 𝑛_𝑅𝑖^′ )(𝑘_𝑓𝑅∏^𝑁_𝑘=1[𝑅_𝑘]^𝑣𝑅𝑘′− 𝑘_𝑟𝑅∏^𝑁_𝑘=1[𝑅_𝑘]^{𝑣𝑅𝑘′′})] (3.30) where the total number of reactions in the system is 𝑁_𝑅, the species concentration in moles is [𝑅_𝑘] , the stoichiometric coefficients are 𝑛_𝑅𝑘′ and 𝑛_𝑅𝑘′′ , the concentration exponential factors are 𝑣_𝑅𝑘′, and 𝑣_𝑅𝑘′′ and finally the forward rate constant 𝑘_𝑓𝑅 and the backward rate constant 𝑘_𝑟𝑅 [107].

Chemical reactions and combustion

In the chemical kinetics mechanism, different types of reactions may be found. Although all of the reaction rate calculations are based on the standard Arrhenius rate equation, each type of reaction requires a different modified Arrhenius rate expression for its reaction rate calculation.

The standard Arrhenius rate equation has been described in detail in Chapter 2, Section 2.5 of this thesis. Therefore, all of the different modified Arrhenius rate expressions for the reactions included in the chemical kinetics mechanisms are described in this section.

Three-body reaction

This type of reaction is included in the mechanism if “third body” species are needed in a reaction. By adding third body species in the reaction mechanism the rate of production 𝑆_𝑖, Equation 3.26, has to be multiplied by the concentration factor given by:

∑^𝑁_𝑘=1𝛼_𝑘𝑅|𝑅_𝑘| (3.31)

73 Where 𝛼_𝑘𝑅 is the third body efficiency of species 𝑘 in reaction 𝑅.

For three body reactions in the mechanisms, the third body inert molecule that is required to stabilize the reaction’s excited product by collision is refer as M. For example, reaction H2O2+M=OH+OH+M. The inert molecule (M) actually removes the excess energy from the excited product and dissipates it as heat.

Pressure-dependent reaction

The in-cylinder pressure is one of the main factors affecting the combustion process and therefore, is directly related with the combustion chemistry. Chemical kinetics reactions that are activated during the combustion process react differently at different pressure conditions.

In order to express this pressure dependence, chemical kinetic data for both low and high pressure conditions should be included into the mechanism for each pressure-depended reaction. Then the reaction rate of the chemical kinetics reaction, at a pressure between the low and high pressure limits, is calculated based on the two limiting factors. Three different types of reaction formulation can be used in the mechanism; a) the Liendemann form, b) the TROE form and c) the SRI form.

During the Lindemann form the low and high pressure limit values, 𝑘_𝑙 and 𝑘_ℎ, are given by using the standard Arrhenius rate equations:

𝑘_𝑙= 𝐴_𝑙𝑇^𝛽𝑙exp (−_𝐸𝑅^𝐸𝑙) (3.32)

𝑘_ℎ = 𝐴_ℎ𝑇^𝛽ℎexp (−^𝐸_𝐸𝑅^ℎ) (3.33) Where 𝐴_𝑙 and 𝐴_ℎ are the exponential factors used for the low and high pressure limits respectively and 𝛽_𝑙 and 𝛽_ℎ are the temperature exponent factors at low and high pressure limit respectively.

For each reaction, the rate constant at any pressure is then calculated by:

𝑘 = 𝑘_ℎ(_1+𝑃^𝑃𝑟_𝑟) (3.34) Where 𝑃^𝑟 is the reduced pressure which is given by:

𝑃^𝑟 =^𝑘𝑙_𝑘^|𝑀|

ℎ (3.35)

74 Where |𝑀| is the mixture concentration. It is important to mention here that if the pressure depended reaction is also a third body reaction, then the third body efficiency effect will be included in the calculation of the reaction rate.

For pressure depended reactions that use the TROE form, the reaction rate at any pressure is given by:

𝑘_ℎ = 𝑘_ℎ(_1+𝑃^𝑃𝑟

𝑟) 𝐹 (3.36) Where 𝑃^𝑟 is calculated similarly to Lindemann form from Equation 3.35, while 𝐹 is

calculated by

log 𝐹 = [1 + ( _𝑡 ^{𝑙𝑜𝑔𝑃𝑟+𝑡1}

2−0.14(𝑙𝑜𝑔𝑃_𝑟+𝑡₁)²]⁻¹log 𝐹_z (3.37)

Where 𝑡₁ and 𝑡₂ are the characteristic times and 𝐹_𝑧= (1 − z) exp (−^𝑇_𝑧) + z exp (−^𝑇_z) + exp (−^𝑧_𝑇²). z is the characteristic coefficient of the pressure dependent reactions.

The final form of pressure-depended reaction is called SRI form. During SRI form the reaction rate constant of each pressure depended reaction is calculated by [107]:

𝑘 = 𝑘_ℎ(_1+𝑃^𝑃𝑟

𝑟) 𝐹 (3.38) Where 𝐹 is calculated by :

𝐹 = [g 𝑒𝑥𝑜 (−^𝑔_𝑡) + exp (−^𝑇_𝑔)] ^𝐹𝑔𝑑𝑡^𝑔 (3.39)

And 𝐹_𝑔 by

𝐹_𝑔 =_{1+(𝑙𝑜𝑔𝑃}¹

𝑟)² (3.40) Where 𝑔 is the characteristic coefficient of the SRI form reaction and is defined in the reaction mechanism by the user based on the in-cylinder pressure conditions.

75 Landau-Teller reaction

The last form of reaction that is included in the mechanism is the Landau –Teller reaction. By using that reaction form the reaction rate constant is calculated by:

𝑘_𝑓𝑟− 𝐴_𝑅𝑇^𝛽𝑅exp [−^𝐸_𝑅𝑇^𝑅+^𝐵𝑅1

𝑇¹³ +^𝐵𝑅2

𝑇²³] (3.41) Where 𝐵_𝑅1 and 𝐵_𝑅2 are the Landau-Teller constants. When both constants are zero the reaction rate constant is calculated by the simple Arrhenius rate Equation.

As it can be seen, a comprehensive and detailed list of chemical reactions and their reaction rates must be included in the chemical kinetics mechanism for the accurate prediction of the ignition behaviour, the NOx formation and the combustion characteristics (e.g. pressure, ROHR and flame characteristics). This is the reason why a chemical kinetics mechanism must be developed carefully with specific attention on the reduction procedure so that elimination of species or reactions that may affect the accuracy of the simulations is avoided and to ensure that all of the reactions with the correct form are included. It is important to mention here, that for turbulent combustion, an eddy break up based reaction could be included in the mechanism and the rate constant calculated by the standard eddy break up model. That type of reaction is implemented for the spray modelling and n-heptane chemistry and is analysed in detail in Chapter 3, section 3.2.5.

Thermodynamic and transport properties of the species

For all of the individual species included in the syngas mixtures used during this thesis as well as the individual species included in the developed mechanisms, their decomposition rates as well as the reaction rates are included as thermal and transport files in Appendix A Table A1 and Table A-2 respectively. The thermal file includes the coefficients of each species included in the mechanism that were used for the calculation of specific heats ,standard state enthalpies and standard state entropies as a function of temperature for each species included in the mechanism [112]. Two different temperatures used for each species (min and max) and seven different coefficients for each temperature used. Thus, for each species, 14 coefficients are used. The final specific heat (csp) enthalpy (HEnthalpy) and entropy (SEntropy ) for each species are calculated by:

𝑐_𝑠𝑝(𝑇) = 𝑅[𝛿₁+ 𝛿₂𝑇 + 𝛿₃𝑇²+ 𝛿₄𝑇³+ 𝛿₅𝑇⁴] (3.42)

76 𝐻_{𝐸𝑛𝑡ℎ𝑎𝑙𝑝𝑦}(𝑇) = 𝑅𝑇 [𝛿₁+^𝛿₂²𝑇 +^𝛿₃³𝑇²+^𝛿₄⁴𝑇³+^𝛿₅⁵𝑇⁴+^𝛿_𝑇⁶] (3.43) 𝑆_{𝐸𝑛𝑡𝑟𝑜𝑝𝑦}(𝑇) = 𝑅 [𝛿₁𝑙𝑛𝑇 + 𝛿₂𝑇 +^𝛿₂³𝑇²+^𝛿₃⁴𝑇³+^𝛿₄⁵𝑇⁴+ 𝛿₇ ] (3.44) In which R is the gas constant and T is the temperature. Furthermore, the thermodynamic database includes the name of the species, its elemental makeup and the temperatures in which the fits are valid. For accuracy reasons, all of the thermodynamic properties of the species have been taken from the NASA chemical database [161] and are similar to the thermodynamic data used in CHEMKIN [112].

For the transport properties of each species, a transport data file is presented in Appendix A Table A-2. The transport database includes important molecular properties for each individual species such as [162]:

1) Its geometrical configuration. An index showing if the molecule has a monoatomic, non-linear or non-linear configuration. For monoatomic, an index value of 0 is used. For non-non-linear an index 2 is given. Finally, for linear an index 1 is given.

2) The Lennard-Jones potential well depth ε/kB in Kelvins 3) The Lennard-Jones collision diameter, DLJ in Angstroms

4) The dipole moment, µ in Debye. Note: a Debye is 10^-18 cm^3/2 erg ^½ 5) The polarizability Ppl in cubic Angstroms

6) And the rotational relaxation collision number Zrot.

Similar to the data file the transport properties of each species have been taken directly from NASA chemical database[161].