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19.2.2 Ignition Model Theory

Both the knock and the ignition delay models are treated similarly in FLUENT, in that they share the same infrastructure. These models belong to the family of single equation autoignition models and use correlations to account for complex chemical kinetics. They differ from the eight step reaction models, such as Halstead's "Shell'' model [ 133], in that only a single transport equation is solved. The source term in the transport equation is typically not stiff, thus making the equation relatively inexpensive to solve.

This approach is appropriate for large simulations where geometric accuracy is more important than fully resolved chemical kinetics. The model can be used on less resolved meshes to explore a range of designs quickly, and to obtain trends before utilizing more expensive and presumably more accurate chemical mechanisms in multidimensional simulations.



Transport of Ignition Species


Autoignition is modeled using the transport equation for an Ignition Species, $Y_{ig}$, which is given by


 \frac{\partial {\rho Y_{ig}}}{\partial t} + \nabla \cdot ( ... ... \frac{\mu_t}{{\rm Sc}_t} \nabla Y_{ig} \right) + \rho S_{ig} (19.2-1)

where $Y_{ig}$ is a "mass fraction'' of a passive species representing radicals which form when the fuel in the domain breaks down. $Sc_t$ is the turbulent Schmidt number. The term $S_{ig}$ is the source term for the ignition species which has a form


 S_{ig} = \int_{t=t_0}^{t} \frac{dt}{\tau_{ig}}

where $t_0$ corresponds to the time at which fuel is introduced into the domain. The $\tau_{ig}$ term is a correlation of ignition delay with the units of time. Ignition has occurred when the ignition species reaches a value of 1 in the domain. It is assumed that all the radical species represented by $Y_{ig}$ diffuse at the same rate as the mean flow.

Note that the source term for these radical species is treated differently for knock and ignition delay. Furthermore, the form of the correlation of ignition delay differs between the two models. Details of how the source term is treated are covered in the following sections.



Knock Modeling


When modeling knock or ignition delay, chemical energy in the fuel is released when the ignition species reaches a value of 1 in the domain. For the knock model, two correlations are built into FLUENT. One is given by Douaud [ 86], while the other is a generalized model which reproduces several correlations, given by Heywood [ 141].

Modeling of the Source Term

In order to model knock in a physically realistic manner, the source term is accumulated under appropriate conditions in a cell. Consider the one dimensional flame in Figure  19.2.1. Here, the flame is propagating from left to right, and the temperature is relatively low in front of the flame and high behind the flame. In this figure, $T_b$ and $T_u$ represent the temperatures at the burnt and unburned states, respectively. The ignition species accumulates only when there is fuel. In the premixed model, the fuel is defined as $fuel = 1-c$, where $c$ is the progress variable. If the progress variable has a value of zero, the mixture is considered unburned. If the progress variable is 1, then the mixture is considered burned.

Figure 19.2.1: Flame Front Showing Accumulation of Source Terms for the Knock Model
\begin{figure}\begin{center} \begin{picture}(300,150)(10,10) \put(0,20){\vecto... ...\put(180,65){\makebox(0,0){$S_{ig} > 0$}} \end{picture} \end{center}\end{figure}

When the ignition species reaches a value of 1 in the domain, knock has occurred at that point. The value of the ignition species can exceed unity. In fact, values well above that can be obtained in a short time. The ignition species will continue to accumulate until there is no more fuel present.

Correlations

An extensively tested correlation for knock in SI engines is given by Douaud and Eyzat [ 86]:


 \tau = 0.01768 \left( \frac{ON}{100} \right)^{3.402} p^{-1.7} \exp \left( \frac{3800}{T} \right) (19.2-2)

where $ON$ is the octane number of the fuel, $p$ is the absolute pressure in atmospheres and $T$ is the temperature in Kelvin.

A generalized expression for $\tau$ is also available which can reproduce many existing Arrhenius correlations. The form of the correlation is


 \tau = A \left( \frac{ON}{100} \right)^{a} p^b T^c \mbox{RPM}^d \Phi^d \exp \left( \frac{-E_a}{RT} \right) (19.2-3)

where $A$ is the pre-exponential (with units in seconds), RPM is the engine speed in cycles per minute and $\Phi$ is the fuel/air equivalence ratio.

Energy Release

Once ignition has occurred in the domain, the knock event is modeled by releasing the remaining fuel energy with a single-step Arrhenius reaction. An additional source term, which burns the remaining fuel in that cell, is added to the rate term in the premixed model. The reaction rate is given by


 \dot{\omega} = A_0 \exp{\frac{-E_a}{RT}} (19.2-4)

where $A_0 = 8.6 \times 10^{9}$, and $E_a = -15078$. These values are chosen to reflect single-step reaction rates appropriate for propane as described in Amsden [ 8]. The rate at which the fuel is consumed is limited such that a completely unburned cell will burn during three of the current time steps. Limiting the reaction rate is done purely for numerical stability.



Ignition Delay Modeling


When modeling ignition delay in diesel engines, chemical reactions are allowed to occur when the ignition species reaches a value of 1 in the domain. For the ignition delay model, two correlations are built into FLUENT, one given by Hardenburg and Hase  [ 137] and the other, a generalized model which reproduces several Arrhenius correlations from the literature.

If the ignition species is less than 1 when using the ignition delay model, the chemical source term is suppressed by not activating the combustion model at that particular time step; thus, the energy release is delayed. This approach is reasonable if the user has a good high-temperature chemical model, but does not wish to solve for typically expensive low temperature chemistry.

Modeling of the Source Term

In order to model ignition in a physically realistic manner, the source term is accumulated under appropriate conditions in a cell. Consider the one dimensional spray in Figure 19.2.2.

Figure 19.2.2: Propagating Fuel Cloud Showing Accumulation of Source Terms for the Ignition Delay Model
\begin{figure}\begin{center} \begin{picture}(300,150)(10,10) \put(0,20){\vecto... ...\put(180,65){\makebox(0,0){$S_{ig} = 0$}} \end{picture} \end{center}\end{figure}

Here, the spray is propagating from left to right and the fuel mass fraction is relatively low in front of the spray and high behind the spray. If there is no fuel in the cell, the model will set the local source term to zero, nevertheless, the value of $Y_{ig}$ can be nonzero due to convection and diffusion.

Correlations

If fuel is present in the cell, there are two built-in options in FLUENT to calculate the local source term. The first correlation was done by Hardenburg and Hase and was developed at Daimler Chrysler for heavy duty diesel engines. The correlation works over a reasonably wide range of conditions and is given by


 \tau_{id} = \left( \frac{C_1 + 0.22 \overline{S}_p}{6 N} \r... ...0} \right) + \left( \frac{21.2}{p-12.4} \right)^{e_p} \right] (19.2-5)

where $\tau_{id}$ is in seconds, $C_1$ is 0.36, $N$ is engine speed in revolutions per minute, $E_a$ is the effective activation energy and $e_p$ is the pressure exponent. The expression for the effective activation energy is given by


 E_a = \frac{E_{hh}}{CN + 25} (19.2-6)

where $CN$ is the cetane number. The activation energy, $E_{hh}$, pre-exponential, $C_1$, pressure exponent, $e_p$, and cetane number, $CN$, are accessible from the GUI. The default values of these variables are listed in the table below.


Table 19.2.1: Default Values of the Variables in the Hardenburg Correlation
Variable $E_{hh}$ $CN$ $C_1$ $e_p$
Default 618,840 25 0.36 0.63

The second correlation, which is the generalized correlation, is given by Equation  19.2-3 and is available for ignition delay calculations.

Energy Release

If the ignition species is greater than or equal to 1 anywhere in the domain, ignition has occurred and combustion is no longer delayed. The ignition species acts as a switch to turn on the volumetric reactions in the domain. Note that the ignition species "mass fraction'' can exceed 1 in the domain, therefore, it is not truly a mass fraction, but rather a passive scalar which represents the integrated correlation as a function of time.


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