Internal Combustion Engines. Allan T. Kirkpatrick

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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. The parameter is a nondimensional constant, and are the burning durations for each phase, and are the integrated energy release for each phase, and and are the nondimensional shape factors for each phase. The and parameters are determined empirically from engine performance data. The dual Wiebe function is described in more detail and applied to a fuel–air compression ignition cycle in Chapter 4.

Figure 2.18 Dual Wiebe function for diesel energy release. (Adapted from Miyamoto 1985.)

      Energy Equation

We now develop a simple spark‐ignition finite energy release model by incorporating the Wiebe function equation, Equation (2.71), into the differential energy equation. We assume that the energy release begins with spark ignition at and has a combustion duration during the compression and expansion strokes, and solve for the resulting cylinder pressure as a function of crank angle. The simple model assumes the inlet and exhaust valves are closed at the start of integration at , so it does not account for flow into and out of the combustion chamber.

      As shown in the following derivation, the differential form of the energy equation does not have a simple analytical solution due to the finite energy release term. It is integrated numerically, starting at bottom dead center, compressing to top dead center, and then expanding back to bottom dead center.

      The closed‐system differential energy equation (note

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