Research Papers: Gas Turbines: Combustion, Fuels, and Emissions

A Linear Least-Squares Algorithm for Double-Wiebe Functions Applied to Spark-Assisted Compression Ignition

[+] Author and Article Information
Erik Hellström

University of Michigan,
Ann Arbor, MI 48109
e-mail: erikhe@umich.edu

Anna Stefanopoulou

University of Michigan,
Ann Arbor, MI 48109
e-mail: annastef@umich.edu

Li Jiang

Robert Bosch LLC,
Farmington Hills, MI 48331
e-mail: li.jiang@us.bosch.com

The solution is obtained by differentiating the criterion in Eq. (3) with respect to p, setting the result to zero, and solve for p.

Contributed by the Combustion and Fuels Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received February 15, 2014; final manuscript received February 16, 2014; published online May 5, 2014. Editor: David Wisler.

J. Eng. Gas Turbines Power 136(9), 091514 (May 05, 2014) (7 pages) Paper No: GTP-14-1096; doi: 10.1115/1.4027277 History: Received February 15, 2014; Revised February 16, 2014

An algorithm for determining the four tuning parameters in a double-Wiebe description of the combustion process in spark-assisted compression ignition engines is presented where the novelty is that the tuning problem is posed as a weighted linear least-squares problem. The approach is applied and shown to describe well an extensive data set from a light-duty gasoline engine for various engine speeds and loads. Correlations are suggested for the four parameters based on the results, which illustrates how the double-Wiebe approach can also be utilized in a predictive simulation. The effectiveness of the methodology is quantified by the accuracy for describing and predicting the heat release rate and predicting the cylinder pressure. The root-mean square errors between the measured and predicted cylinder pressures are 1bar or less, which corresponds to 2% or less of the peak cylinder pressure.

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Grahic Jump Location
Fig. 1

SACI combustion heat release data transformed using Eq. (2) to show the linear regions

Grahic Jump Location
Fig. 2

The heat release rate Q' (top panel) and the first three derivatives of the burn fraction xb (bottom panel). The transition angle θ1 is defined at the peak value of xb''' between θ0 and θx, the peak of xb'.

Grahic Jump Location
Fig. 3

Using Eq. (2) to determine the parameters for the flame propagation (FP) Wiebe function x0 and the autoignition (AI) Wiebe function x1, respectively

Grahic Jump Location
Fig. 4

Construction of the composite Wiebe function from Eqs. (11) and (14), by smoothly joining the two Wiebe functions

Grahic Jump Location
Fig. 5

Sweep of the eEGR at 5 bar BMEP and 2000 rpm. Data are shown with gray thick lines and fits with thin black lines. The dots mark the locations of θ0, θ1, and θ2.

Grahic Jump Location
Fig. 6

Simultaneous sweep of the spark and eEGR at 5 bar BMEP and 2000 rpm. Data are shown with thick gray lines and fits with thin black lines. The dots mark the locations of θ0, θ1, and θ2.

Grahic Jump Location
Fig. 7

Double-Wiebe function fits for a grid of engine speed, in krpm, and load, BMEP in bar. The different combustion characteristics at each operating point correspond to a multitude of conditions for various actuator settings (eEGR valve, start of injection, intake and exhaust cam timing, and spark timing) based on the chosen design of experiments. The root-mean-square (rms err) and maximum (max err) errors between the fitted curves (thin black lines) and measured data (thick gray lines) are computed for the interval (θsoc, θeoc).

Grahic Jump Location
Fig. 8

Predicted heat release rate (top three rows) and predicted cylinder pressure (bottom three rows) are compared with data for varying engine speed, in krpm, and load, BMEP in bar. The root-mean-square (rms err) and maximum (max err) errors between the predicted curves (thin black lines) and measured data (thick gray lines) are computed for the interval (θsoc, θeoc). The coefficients for the Wiebe functions are calculated from the regressors in Table 1 and the pressures are then simulated using the predicted heat release rate.




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