Internal Combustion Engines. Allan T. Kirkpatrick

rel="nofollow" href="#fb3_img_img_df2a3830-41f3-5e46-9c16-f0a9f70a7029.png" alt="images"/> versus the crank angle. The characteristic features of the mass fraction burned curve are an initial small slope region beginning with spark ignition and the start of energy release at , followed by a region of rapid growth, and then a more gradual decay. The three regions correspond to the initial ignition development, a rapid burning region, and a burning completion region. This S‐shaped curve can be represented analytically by a trigonometric function, as indicated by Equation (2.69):

(2.69)

      or an exponential relation, known as a Wiebe function, as given in Equation (2.70):

(2.70)

      where

      The Wiebe function is named after Ivan Wiebe (1902–1969), a Russian engineer who developed a energy release model based on analysis of combustion chain reaction events (Ghojel 2010). The Wiebe function can be used for modeling the energy release in a wide variety of combustion systems. For example, as shown in the next section, diesel engine combustion, which has a premixed phase and a diffusion phase, can be modeled using a combined double Wiebe function. The energy release curve for the diesel engine is double peaked due to the two combustion phases.

Figure 2.15 Cumulative mass fraction burned function.

Since the cumulative energy release curve asymptotically approaches a value of 1, the end of combustion needs to defined by an arbitrary limit, such as 90%, 99%, or 99.9% complete combustion; i.e., 0.90, 0.99, or 0.999, respectively. Corresponding values of the Wiebe efficiency factor are 2.302, 4.605, and 6.908 respectively. The value of the efficiency factor was used by Wiebe in his engine modeling calculations.

      The values of the form factor and burn duration depend on the particular type of engine, and on some degree on the engine load and speed. These parameters can be deduced using experimental burn rate data, which in turn is obtained from the cylinder pressure profile as a function of crank angle, discussed in more detail in the combustion analysis section of Chapter 12. Values of and have been reported to fit well with experimental data (Heywood 1988). For further general information about energy release models the reader is referred to Foster (1985).

      The rate of energy release for the Wiebe function as a function of crank angle, Equation (2.71), is obtained by differentiation of the cumulative energy release function:

(2.71)

The computer program BurnFraction.m is listed in Appendix F and can be used to plot the Wiebe function cumulative burn fraction and the rate of energy release for different engine conditions. The use of the program is detailed in the following example.

      Example 2.4 Rate of Energy Release

      Using the Wiebe function, plot the cumulative burn fraction and the rate of energy release for a combustion event with the start of energy release at and the duration of energy release . Assume the Wiebe efficiency factor , i.e., = 0.9933, and the Wiebe form factor .

      Solution

      The above parameters are entered into the computer program BurnFraction.m as shown below, and the resulting plots are shown in Figures 2.16 and 2.17.

      Comment: Note the asymmetry of the burn rate, as a result of the form factor value, and the peak value of the burn rate at 18 atdc. As discussed in more detail in the next example, optimal work from an engine usually occurs with a peak burn rate a few degrees after top dead center, so a significant fraction of the combustion will occur during the expansion process.

       function [ ]=BurnFraction( ) This program computes and plots the cumulative burn fraction and the instantaneous burn rate. a = 5; Wiebe efficiency factor n = 4; Wiebe form factor thetas = -20; start of combustion thetad = 60; duration of combustion ....Burn fraction curve for Example 2.4. Rate of energy release curve for Example 2.4.

      Compression Ignition Energy Release

Diesel combustion energy release is characterized by a double peak energy release, resulting from the two types of combustion that occur during the diesel fuel injection process. The first type is premixed combustion resulting from the leading edge of the fuel jet rapidly mixing and then reacting with the cylinder air. The second phase is a diffusion flame in which the remaining injected fuel mixes and reacts with the cylinder air more slowly. The rate of combustion in a diffusion flame is limited by the rate at which the fuel can be mixed with the cylinder air.

A dual Wiebe function (see Figure 2.18), has been used to fit diesel combustion energy release data (Miyamoto et al. 1985). The dual equation, Equation (2.72) with seven parameters is

(2.72)

      The subscripts and refer to the premixed and mixing controlled combustion portions, respectively.