Simplified model for combustion reactions in diesel engine.

Sabau, Adrian ; Barhalescu, Mihaela ; Oanta, Emil 等

1. INTRODUCTION

The model involves twelve species and the consequent equations are:

* A stoichiometric, irreversible, kinetic equation (1), a single-step hypothetical fuel combustion ([C.sub.n] [H.sub.m] [O.sub.r,] given by elemental analysis where m, n and r may be not integers);

* Three partial equilibrium reversible equations expressing the extended Zeldowich mechanism (2) for NO evaluation;

* Six reversible equilibrium equations for main combustion products dissociation (3) (Poinsot & Veynante, 2005).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

2. MODEL FORMULATION

In engines, the cylinder pressure rises during the combustion process, so earlier burnt gases are compressed to a higher temperature level after their combustion. Hence, the thermal NO formation in the burnt gases always dominates (Chung 2006) in comparison to the NO formed in the flame front and represents the main source of the nitric oxide in engines whose reaction paths are effective at high temperatures (T > 1600[degrees]K). The reaction mechanism can be expressed in terms of the extended Zeldovich mechanism (2).

In the combustion model software, an irreversible single-step reaction mechanism is used for the conversion of fuel, involving only stable molecules such as [C.sub.n] [H.sub.m] [O.sub.r] (as fuel), [O.sub.2], C[O.sub.2], [H.sub.2]O and [N.sub.2]. The maximum of NO appears at an equivalence ratio of about ~ 0.9, i.e. slightly fuel-lean. In most stoichiometric and fuel-lean flames, the occurring of the OH concentration is very small. Using this fact, the third reaction of the Zeldovich mechanism can be neglected. In addition, the combustion and the NO formation processes can be assumed to be decoupled and therefore, the concentrations of [O.sub.2], N, O, OH and H can be approximated by an equilibrium assumption.

An analysis of experiments and simulations indicates that at high temperatures (T > 1600[degrees] K) the reaction rates of the forward and reverse reactions are equal.

The state of the considered reaction is in a so-called partial equilibrium when the reaction couples are in equilibrium. Using this assumption, the concentrations of the radicals can be expressed in terms of concentrations of stable molecules, which are in far larger concentrations than the radicals. The assumption of partial equilibrium provides satisfactory results only at considerably high temperatures, because at temperatures less than 1600[degrees] K a partial equilibrium is not established.

For the formation of thermal NO, the partial equilibrium approach can be used and so the equilibrium of the first two reactions can be expressed as follows:

k1[[N.sub.2]][O] = k2[ NO][N], k3[N][[O.sub.2]] = k4[NO][O] (4)

Using these expressions, the system of equations can be solved. The results in a global reaction approach for the thermal nitric oxide formation can be expressed as:

[N.sub.2] + [O.sub.2] = 2NO, (5)

with kf = k1 x k3 as the forward and kb = k2 x k4 as the reverse reaction rate.

However, an analysis of the stoichiometric coefficients of the six equilibrium reactions (3) shows that we actually use the modified forms. The rate coefficients for these modified reactions are easily obtained by combining those for the original reactions with the appropriate equilibrium constants. The reason for the modification is to remove the trace species from the left-hand sides of the reactions. If this is not done, artificially small effective reaction rates may result. The modified reactions above are not subject to this difficulty. It should be noted that the form of any such modified reaction is constrained by the requirement that every species participating in a reaction may appear only one side of the reaction.

3. NUMERICAL TECHNIQUE

The chemical source term in continuity equation is given by equation (6), and the chemical heat release term in the energy equation is given by equation (7) (Stiesch, 2003).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

where [a.sub.kr] and [b.sub.kr] are stoichiometric coefficients, [W.sub.k] is the molecular weight, [[??].sub.r] is the rate of progress of r-th reaction, [q.sub.r] is the heat of reaction at 0[degrees] K. If r is a kinetic reaction, then [[??].sub.r] is computed using equation (8), which is an equilibrium reaction and [[??].sub.r] is implicitly determined by the condition expressed by equation (9):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

where: [k.sub.fr] and [k.sub.br] are the rate coefficients for reaction (generalized Arrhenius form) r, [a'.sub.kr] and [b'.sub.kr] are orders of the reaction and [K.sup.r.sub.C] (T) is the concentration equilibrium constant.

It was suposed that the equilibrium constant is given by an expression, such as:

[K'.sub.c] = exp([A.sub.r] 1n [T.sub.A] + [B.sub.r] / [T.sub.A] + [C.sub.r] + [D.sub.r] [T.sub.A] + [[E.sub.r] [T.sup.2.sub.A]), (10)

where [A.sub.r], [B.sub.r], [C.sub.r], [D.sub.r] and [E.sub.r] are constants for each reaction, and [T.sub.A] = T /1000 .

The calculus is done under the hypotheses that the reactios are de-coupled and the pressure is constant. The reaction speed for the equilibrium conditions, [[??].sub.r], (where r is the number of chemical reactions in equilibrium conditions) is computed by the use of an iterative algorithm. Thus, every reaction is relaxed until the equilibrium constant given by equation (9) becomes equal to the value resulted from equation (10), with an acceptable error. The code is written in Matlab (Oanta 2007).

4. NUMERICAL SIMULATION

The model presented in the paper was used for numerical simulation of the T684 engine manufactured by "Tractorul" Plant of Brasov, a four stroke automotive engine.

For this case study were considered NO emissions, (fig. 1 and fig. 2), C[O.sub.2] (fig. 3), and [O.sub.2] (fig. 4) measured and calculated with the Wave 5 computer code, for many cases, detailed presented in (Sabau 2007). Data for NO, C[O.sub.2,] and [O.sub.2] is available only averaged for a cycle, for this reasons it is difficult to appreciate the performance of chemical models. Wave 5 uses equilibrium approaches for the chemical reactions.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

5. CONCLUSION

The software is able to estimate the NO emissions (2-12% error from average measured data). Results are in good compliance with the experiment, mainly for regimes of full speed and load where the error is about 5%.

The results largely depend on the constants of the models and for this reason they have to be carefully analysed.

The performances of the program are limited by the models used, few of them requiring improvements, such as:

* only thermal NO formation were implemented, more chemical reactions are need to increase the accuracy;

* models for shoots formation and non burn hydrocarbon need to be implemented, because CO2 and non burn O2 is strongly affected (7-15% error);

* numerical algorithms should be redesigned in order to have an increased accuracy and lower run times.

6. ACKNOWLEDGEMENT

Several ideas presented in this paper use the accomplishments of the "Computer Aided Advanced Studies in Applied Elasticity from an Interdisciplinary Perspective" ID1223 scientific research project (Oanta et al. 2007).

7. REFERENCES

Chung, K. L. (2006). Combustion Physics, Cambridge University Press, ISBN 0521870526, New York

Oanta, E. (2007). Numerical methods and models applied in economy, PhD Thesis, Academy of Economical Studies of Bucharest, Promoter Prof. Mat. Ec. Ioan Odagescu

Oanta, E. (2007-2010). Computer Aided Advanced Studies in Applied Elasticity from an Interdisciplinary Perspective, ID1223 Scientific Research Project, under the supervision (CNCSIS), Romania

Poinsot, T. & Veynante D. (2005). Theoretical and Numerical Combustion, R.T. Edwards Inc., ISBN 1930217102, Paris

Sabau, A. (2007). Studies regarding the combustion process in marine diesel engines in order to reduce the pollutant emissions, PhD Thesis, 'Transilvania' University of Brasov

Stiesch, G. (2003). Modeling Engine Spray and Combustion Processes, Springer; ISBN 3540006826, Berlin