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A thermodynamic model for the plasma kernel volume and temperature resulting from spark discharge at high pressures

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In today’s internal combustion engines assisted ignition is predominantly achieved by conventional spark ignition. The spark ignition starts with the breakdown phase after which complex flow dynamics govern the energy deposition into the gas. The condition of the activated plasma kernel resulting from spark breakdown, among others, is important for the development of the flame kernel to a self-sustained flame. A simple thermodynamic model is formulated based on insight gained from available detailed plasma simulations and experiments. Under the assumption that the problem is cylinder symmetrical and that the breakdown energy is supplied quasi-instantaneously as a line source, the extent of the hot plasma kernel and the time for pressure equalization with the surrounding can be given by exploiting the properties of radially expanding blast waves. The so-defined control volume is treated as an open system, and a simple energy balance leads to a relation for the mean kernel temperature. The plasma volumes and mean temperatures predicted by this simple model are shown to correspond well with published results in the open literature based on full plasma simulations of the spark discharge available for pressures ranging from 1 to 8 bar. The model allows drawing some conclusions about the requirements for experimental methods, such as calorimetry, frequently used to characterize the efficacy of the spark discharge produced by ignition systems and the influence of pressure and breakdown energy on kernel volume and temperature.

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\(B\) :

Energy parameter

\(c\) :

Speed of sound

\(d\) :

Electrode gap

\(e\) :

Internal energy

\(E\) :


\(h\) :

Specific enthalpy

\(m\) :

Mass of plasma kernel

\(M\) :

Mach number

\(p\) :


\(\dot{Q}\) :

Heat flux

\(r\) :

Dimensional radius

\(t\) :

Dimensional time

\(T\) :


\(v\) :

Flow velocity

\(V\) :


\(0\) :

Undisturbed initial state, environment

\({\text{a}}\) :

Activated volume (hot plasma kernel)

\({\text{br}}\) :


\({\text{c}}\) :

Characteristic (inner) scale

\({\text{e}}\) :

Energy exchange with the environment

\({\text{ht}}\) :

Heat transfer

\({\text{rad}}\) :


\(\gamma\) :

Heat capacity ratio

\(\delta\) :

Dirac Delta distribution

\(\lambda\) :

Non-dimensional radius

\(\rho\) :


\(\tau\) :

Non-dimensional time

\(\Omega\) :


\(\partial\Omega\) :

Boundary of domain


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The authors would like to acknowledge the financial support of the “COMET—Competence Centers for Excellent Technologies Programs” of the Austrian Federal Ministry for Transport, Innovation and Technology (BMVIT), the Austrian Federal Ministry of Science, Research and Economy (BMWFW) and the Provinces of Styria, Tyrol and Vienna for the K1-Centre LEC EvoLET. The COMET Program is managed by the Austrian Research Promotion Agency (FFG).

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Correspondence to Georg Meyer.

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Meyer, G., Wimmer, A. A thermodynamic model for the plasma kernel volume and temperature resulting from spark discharge at high pressures. J Therm Anal Calorim 133, 1195–1205 (2018).

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  • Plasma
  • Spark discharge
  • Spark ignition
  • Internal combustion engines
  • Blast wave