Simplified Models of Combustion Processes in Internal Combustion Engines

In the pursuit of optimizing thermal efficiency and advancing the performance of internal combustion engines (ICE), the

In the pursuit of optimizing thermal efficiency and advancing the performance of internal combustion engines (ICE), the ability to accurately characterize the combustion process is paramount. Combustion is a profoundly complex multi-scale phenomenon, involving a chaotic interplay of chemical kinetics, turbulent mixing, heat transfer, and multi-phase fluid dynamics.

While high-fidelity Computational Fluid Dynamics (CFD) simulations can capture these intricacies with remarkable precision, their immense computational cost makes them impractical for rapid iterative design or real-time system-level thermodynamic cycle analysis. To bridge this gap, researchers and engineers rely on simplified combustion models. These models abstract the complex chemical reactions into mathematical functions, focusing primarily on the temporal characteristics of energy release. By doing so, they provide a computationally efficient way to simulate pressure-volume (P-V) diagrams and predict engine performance.
The foundation of most simplified models is the zero-dimensional (0D) assumption. In this framework, the combustion gases within the cylinder are treated as a perfectly homogeneous mixture. This implies that at any given instant, the temperature, pressure, and chemical species concentrations are spatially uniform throughout the combustion chamber.

Although this assumption neglects the physical propagation of the flame front and local concentration gradients, it remains highly effective for macro-scale analysis. When the objective is to determine global parameters such as indicated work, thermal efficiency, or mean effective pressure, the 0D approach provides a sufficiently accurate perspective. The primary mathematical challenge in these models is to define the Mass Burning Rate or the Heat Release Rate (HRR) as a function of the crank angle ($\theta$).

The Wiebe Function: The Industry Standard

Within the field of engine modeling, the Wiebe function stands as the most widely adopted empirical model for describing the combustion process. It introduces a dimensionless variable known as the burn fraction ($x_b$), which represents the progress of the combustion process from start to finish.

1. Mathematical Formulation

The standard mathematical expression for the Wiebe function is:

$$x_b(\theta) = 1 - \exp\left[ -a \left( \frac{\theta - \theta_{start}}{\theta_{end} - \theta_{start}} \right)^{m+1} \right]$$

The physical significance of the parameters is defined as follows:

  • $\theta$: The current crank angle.
  • $\theta_{start}$: The Start of Combustion (SOC) angle.
  • $\theta_{end}$: The End of Combustion (EOC) angle.
  • $a$: The shape factor, typically ranging between 5 and 7, which dictates the smoothness of the combustion curve.
  • $m$: The exponential parameter, which characterizes the combustion rate. A higher $m$ value results in a more concentrated and steeper heat release profile.

2. Interpreting the Burn Fraction

The burn fraction $x_b$ serves as a normalized measure of fuel consumption:

  • At $\theta = \theta_{start}$, $x_b = 0$, signifying the onset of combustion.
  • At $\theta = \theta_{end}$, $x_b = 1$, signifying that the fuel has been completely consumed.
  • The first derivative, $\frac{dx_b}{d\theta}$, represents the combustion rate per unit crank angle.

Critical Parameters and Their Physical Implications

To ensure that a simplified model accurately reflects diverse engine operating conditions—such as the spark-ignition (SI) of gasoline engines or the compression-ignition (CI) of diesel engines—several key parameters must be carefully calibrated:

  • Combustion Phasing: This refers to the timing of $\theta_{start}$ and $\theta_{end}$. In SI engines, this is heavily influenced by the spark advance. In CI engines, it is a function of fuel injection timing and injection duration.
  • Burn Duration ($\Delta\theta$): Defined as $\theta_{end} - \theta_{start}$. A shorter duration implies a more intense heat release, leading to rapid pressure rises. While this can enhance work output, it also increases mechanical stress on engine components and elevates the risk of engine knock.
  • Heat Release Profile: By adjusting the $m$ parameter, engineers can simulate different combustion regimes. For instance, premixed combustion is characterized by a high $m$ value (rapid, explosive energy release), whereas diffusion combustion (typical of diesel engines) exhibits a much more gradual heat release.

Implementation in Thermodynamic Cycle Analysis

Integrating a simplified combustion model into a full thermodynamic analysis typically follows a structured numerical workflow:

  1. Geometric Modeling: Calculate the instantaneous cylinder volume $V(\theta)$ based on the piston's displacement, bore, and stroke.
  2. Burn Fraction Calculation: Determine $x_b(\theta)$ using the Wiebe function for the current crank angle.
  3. Energy Release Calculation: Compute the total heat released $Q(\theta)$ at a specific angle:
    $$Q(\theta) = m_{fuel} \cdot LHV \cdot x_b(\theta)$$
    where $m_{fuel}$ is the mass of injected fuel and $LHV$ is the Lower Heating Value.
  4. Pressure Determination: Solve for the cylinder pressure $P(\theta)$ by combining the Ideal Gas Law ($PV=mRT$) with the First Law of Thermodynamics through numerical integration.
  5. Indicated Work Extraction: Calculate the work produced by integrating the area within the P-V diagram:
    $$W_{ind} = \int P , dV$$

Practical Application: A Numerical Example

Consider a single-cylinder engine analysis with the following known values:

  • $LHV = 44,000 \text{ kJ/kg}$
  • $m_{fuel} = 0.0005 \text{ kg}$
  • $\theta_{start} = 350^\circ$
  • $\theta_{end} = 380^\circ$
  • $a = 5, m = 5$

Objective: Calculate the combustion progress and total heat released at $\theta = 365^\circ$.

Step 1: Calculate the Burn Fraction ($x_b$)
First, determine the normalized position within the combustion window:
$$\frac{365 - 350}{380 - 350} = \frac{15}{30} = 0.5$$

Applying the Wiebe formula:
$$x_b(365^\circ) = 1 - \exp\left[ -5 \cdot (0.5)^{5+1} \right]$$
$$x_b(365^\circ) = 1 - \exp\left[ -5 \cdot 0.015625 \right] = 1 - \exp(-0.078125)$$
$$x_b(365^\circ) \approx 1 - 0.9248 = 0.0752 \text{ (or } 7.52% \text{)}$$

Step 2: Calculate the Heat Released ($Q$)
$$Q = 0.0005 \text{ kg} \times 44,000 \text{ kJ/kg} \times 0.0752$$
$$Q = 22 \text{ kJ} \times 0.0752 = 1.6544 \text{ kJ}$$

Observation: Even though the crank angle is exactly at the midpoint of the combustion duration, the burn fraction is only $7.52%$. This illustrates the non-linear nature of the Wiebe model, reflecting how combustion often starts slowly before accelerating toward its peak rate.

Conclusion

Simplified combustion models serve as a vital bridge between microscopic chemical kinetics and macroscopic thermodynamic performance. By utilizing empirical tools like the Wiebe function, engineers can capture the essential energy release characteristics of an engine without the prohibitive cost of full-scale CFD. While these models are limited in their ability to predict local combustion instabilities or precise pollutant distributions, they remain indispensable for engine performance prediction, thermal efficiency optimization, and the development of advanced control strategies.