A Chemical Kinetics Model of Current Signatures in an Ionization Sensor
by
S. M. Aithal, A. R. White, and V. V. Subramaniam,
Center for Advanced Plasma Engineering
Non-Equilibrium Thermodynamics Laboratories
Department of Mechanical Engineering
The Ohio State University
Columbus, Ohio 43210, U.S.A.
V. Babu,
Department of Mechanical Engineering
Indian Institute of Technology
Chennai (Madras), India
and
G. Rizzoni
Powertrain Controls and Diagnostics Laboratory
Department of Mechanical Engineering
The Ohio State University
Columbus, Ohio 43210, U.S.A.
I. Introduction
The presence of ions in combustion systems due to chemi-ionization processes has long been recognized[1-3]. Recently, there has been considerable interest in using these ions as a diagnostic tool for monitoring the combustion process in spark-ignition internal combustion engines[4-10]. Following the initial generation of a spark to ignite a combustible fuel/oxidizer mixture, a D.C. (after-spark) sub-breakdown voltage on the order of a hundred volts is applied across the spark gap. Subsequently, a current is drawn as a function of time. A typical current signature obtained from an internal combustion engine using this method is shown in Fig. 1. The current signature versus crank-angle (which can be converted to time, given the rpm) displays several peaks, resembling a spectrum. The temporal occurrences of the peaks and their magnitudes have been found to vary with the air-to-fuel ratio (A/F), intake pressure, engine speed, and ignition timing. Of significance is the fact that this ionization current is found to closely follow the variation of pressure in the engine cylinder. In combustion bomb experiments, it is found to follow the variation of pressure in the bomb. These observations suggest that there is a strong correlation between the ionization current and the cylinder/bomb pressure. Despite the obvious utility of this approach for monitoring and control of combustion processes, there is at present little understanding of the processes that yield such current spectra, in the published literature[4-10]. In this work, we examine important reaction channels that are expected to occur in an ignited pre-mixed methane/air mixture. Temporal variations of important species are obtained using finite-rate kinetics. A simple model is then used to predict current signatures. The predicted current signature is then compared with existing experimental measurements reported for a constant-volume bomb[11].
Monitoring of the current flow by applying an after-spark bias on an ordinary spark plug can be used to detect misfires and knock. These are important in complying with more stringent air quality standards such as California's On-Board Diagnostics Regulation (OBD II)[7]. Some automotive manufacturers have begun implementing such sensing strategies despite the lack of fundamental understanding of the mechanisms that lead to the specific, observed current signatures[7]. This interest is due in part to the fact that little modification to existing internal combustion engines is needed since the spark plug is an integral part of the automotive engine. The spark plug can be used as an inexpensive substitute for pressure sensors, and can thus be used to predict the temporal occurrence (or position with respect to crank angle) of the piston's top-dead-center (TDC). Therefore, in addition to detecting misfires and knock, monitoring of the ionization current enables determination of the appropriate time for fuel injection and ignition. Better understanding of the identities of the charged species responsible for the observed current signatures can be potentially even more beneficial. For instance, it can lead to improved monitoring and control of combustion processes and perhaps to additional uses for this device in other systems such as gas turbine engines as a sensor for detecting certain gas species.
The identities of the charge carriers responsible for generating the observed ionization current are presently unknown. However, it is clear that the ionization cannot be produced by electron-impact processes since gas temperatures as well as the applied field, are too low to cause breakdown. The only remaining process that can conceivably produce ionization in lean CH4/air mixtures, is chemi-ionization, and this requires temperatures in excess of approximately 2600 K. Earlier work on the origins of the observed ionization current has alluded either to chemi-ions such as CHO+ and H3O+[6] or NO+[10] as the charge carriers responsible for current flow. However, no detail regarding mechanisms or models are provided. For instance, ref.[10] assumes values for the number densities of N2, H2O, and CHO instead of actually calculating these from first principles. More recently, this work has been modified to incorporate chemi-ionization processes, and actually calculates the species compositions from a set of assumed reaction channels[12]. However, important electron attachment processes leading to formation of and are ignored in the revised work[12]. Moreover, these works ignore the contribution to the current that comes from the electrons themselves. The contribution of electrons produced by chemi-ionization to the total current flow is shown to be significant in this work.
The focus of this paper is on determining the mechanisms responsible for the observed current flow in pre-mixed methane/air mixtures ignited in a constant volume combustion bomb, following application of an after-spark bias. This is accomplished by constructing a simple model that incorporates the finite-rate chemical kinetics associated with combustion and chemi-ionization reactions. The temporal variations of the concentrations of relevant charged species and current signatures are then predicted for three methane/air ratios, and compared with measurements for two air/fuel ratios.
This paper is organized as follows. Model formulation of the problem and the method of solution are discussed in the following section. Current signatures predicted by the model, and comparisons with experiments reported in ref.[11] are discussed in Section III. This paper then concludes with a summary of the most important findings in Section IV.
II. Problem Formulation & Solution
Consider a constant volume chamber such as that in ref.[11], containing a mixture of CH4 and dry air, at a given initial pressure, at room temperature. Suppose that a spark is set off in order to ignite this combustible mixture. Typical duration of a spark is on the order of tens to hundreds of nanoseconds. Combustion typically begins many hundreds of microseconds to a millisecond later. The small flame then expands and propagates through the chamber, finally extinguishing at the colder walls. This process takes on the order of a hundred milliseconds. Hundreds of microseconds after the spark event is over and at the onset of combustion, a constant DC bias of 250 Volts is applied across the spark gap and a current is drawn from the spark gap.
The evolution of the flame from ignition and propagation to extinction at the walls is complicated, and therefore several simplifying assumptions are made in this work. First, the model presented in this paper considers temporal variations only since our aim is to explain the origin of the observed current signatures at the spark plug. Experimentally, it is known that the ionization current closely follows the pressure variation, rising slowly on time scales of tens of milliseconds in the bomb[11]. This strongly suggests that the concentrations of charged species (which are temperature dependent) depend on some average of burned gas and unburned gas temperatures, and not on the burned gas temperature alone. The colder, unburned gases diffuse and mix with the burned gases, causing the temperature at the location of the spark plug to be lower and to rise more slowly than the burned gas temperature. This has indeed been observed (see ref.[13] for example), and it is found that the temperature behind the propagating front drops quickly on time scales of 1 to 100 microseconds following the expansion resulting from the initial ignition or energy addition event, and then rises slowly over many milliseconds following heating of the gas behind the combustion front.
We assume at the outset that the average temperature T at the spark plug where the current is measured, is some average of the burned gas and unburned gas temperatures:
(1)
where xb is the fraction of mixture burned, is the burned gas temperature, and is the unburned gas temperature which is taken to be 350 K in this work. The burned gas temperature is calculated from the ideal gas equation, where nT is the total number density (, ni is the number density of species i, and N is the total number of species considered in the model), P is the pressure measured experimentally, and kB is Boltzmann’s constant. Next, we assume that the burned gas fraction xb varies linearly with the burned gas temperature , with xb = 0 when , and xb = xbmax when :
(2)
Substituting equation (2) into equation (1) yields an expression for the average temperature at the spark plug for 0 £ xb £ xbmax:
(3)
or,
(4)
In this work, xbmax is a parameter that is set to 0.81 for the geometry of the combustion bomb in ref.[11], its value based on the fixed distance between the spark plug and the pressure transducer and independent of the air-fuel ratio. Equation (4) can then be used to determine the temperature governing the chemical kinetics, from the measured instantaneous pressure[11]. It is important to emphasize that equation (4) is only valid over the interval 0 £ xb £ xbmax. Therefore, after xb attains its maximum value of xbmax (which occurs when T is maximum), xb is maintained at the value xbmax for the remainder of the transient and the temperature is still calculated using equation (4). The temperature history determined from this simple model is dependent on the air/fuel ratio, and is shown in Fig. 2 for the calculations reported in this work. By its nature, this model is expected to predict the rising portion of the temperature transient reasonably well, but not the decaying portion of the transient. This is because in reality, xb gradually approaches unity after the temperature reaches a maximum value whereas in this model it is limited to xbmax.
The validity of this model for estimating the average instantaneous temperature at the spark plug can only be assessed a posteriori, and for now, is based on the knowledge that the ionization current history follows the temporal variation of the pressure. Since the ionization current is dependent on the concentrations of charged species, which in turn are dependent on the local temperature, it is clear that the temporal variation of this local average temperature must follow the pressure variation. The initial pressure , and initial temperature are taken to be specified (Pi = 103.4 kPa or 15 psia, and Ti= 350 K for the results reported here), as is the initial ratio of air to CH4 (i.e. air/fuel ratio). It is important to emphasize that some ions such as , , , , and are produced by the spark and may be present until the constant DC bias is applied, hundreds of microseconds or a millisecond later. Nevertheless, their initial presence is ignored in this work, and the initial gas composition is taken to be the original mixture comprising CH4, N2, and O2 only. This is done because we are interested in determining those mechanisms that lead to the dominant peak in the ionization current signature. A set of 65 elementary reactions incorporating 29 species deemed important in describing the combustion process and subsequent generation of ions by chemi-ionization reactions, is considered in this work. These are summarized in Table I. Electron-impact ionization processes have been excluded from this list of reaction processes since the problem under consideration involves a low, sub-breakdown electric field on the order of a kV/cm as well as temperatures too low for thermal ionization to occur. Moreover, the presence of tightly bound diatomic species such as N2 and O2 lower average electron energies considerably via inelastic collisions which strongly couple electron energies to the vibrational modes of these molecules.
Given the set of reactions and species as described in Table I, it is possible to generalize an elementary process labeled r as:
(5)
where air and bir are the stoichiometric coefficients of species i in reaction r, Si is the identity of species i, N is the total number of species, and kr is the rate constant for reaction r. Then, the evolution of concentration of species i is described by the following rate equation:
(6)
The rate constants kr are functions of temperature[1], and are expressed here in the form of modified Arrhenius rates:
(7)
The quantities Ar, br, and Ear are listed in Table I. Rg in equation (7) is the universal gas constant (Rg = 8.314 J/mol/K or 1.98726 cal/g-mol/K). Equation (6) is modified for the fuel, CH4, in order to reflect mixing between burned and unburned gases as discussed earlier in this section:
(8)
where is the initial concentration of CH4, and tmix is the characteristic time for mixing between burned and unburned gases. In this work, tmix is taken to be constant at 0.1 seconds as estimated from the dimensions of the combustion bomb to the flame speed.
Given a pressure history P(t), the temperature history T(t) can be calculated from equation (4). Given the temperature history, the rate constants can be evaluated from equation (7). The set of non-linear equations (6) and (8) can then be integrated to determine the evolution of the number densities (i.e. concentrations) of species i. Here, we are specifically interested in the instantaneous concentration of the charged species: electrons (e-), H3O+, , CHO+, C3H3+, and . Once the species number densities are determined, the mobility of charged species j can be evaluated using mean free path theory[14]: