Internal Combustion Engines. Allan T. Kirkpatrick

(2.87)

      For the portions of the compression and expansion strokes before ignition and after combustion, i.e., where and , the energy release term , allowing straightforward integration of the energy equation and recovery of the isentropic pressure‐volume relation:

      (2.88)

      (2.89)

      (2.90)

The differential energy equation, Equation (2.79), is also used to compute energy release curves from experimental measurements of the cylinder pressure. This procedure is discussed in detail in Chapter 12. Commercial combustion analysis software is available to perform such analysis in real time during an engines experiment.

      The computer program FiniteHeatRelease.m is listed in the Appendix, and can be used to compare the performance of two different engines with different combustion and geometric parameters. The program computes gas cycle performance by numerically integrating Equation (2.79) for the pressure as a function of crank angle. The integration starts at bottom dead center = −180), with initial inlet conditions , the gas molecular mass, , and specific heat ratio, , given. The integration proceeds degree by degree to top dead center = 0) and back to bottom dead center. Once the pressure is computed as a function of crank angle, the net work, thermal efficiency, and imep are also computed. The use of the program is detailed in the following example.

      Example 2.5 Finite Energy Release

      A single‐cylinder spark‐ignition cycle engine is operated at full throttle, and its performance is to be predicted using a Wiebe energy release analysis. The engine has a compression ratio of 10. The initial cylinder pressure, , at bottom dead center, is 1.0 atm, with a temperature at bottom dead center of 300 K. The bore and stroke of the engine are mm and mm. The total heat addition, = 1764 J, and the combustion duration is constant at 40 degrees. Assume that the ideal gas specific heat ratio is 1.4, the molecular mass of the gas mixture is 29 kg/kmole, and the Wiebe energy release parameters are and .

      1 Compute the displacement volume , the volume at bottom dead center, , the dimensionless energy addition , and the mass of gas in the cylinder .

      2 Plot the pressure and temperature profiles versus crank angle for −20 (Engine 1) and 0 (Engine 2).

      3 Determine the effect of changing the start of energy release from −50 to +20 atdc on the thermal efficiency, and imep of the engine.

      Solution

      1 The displacement volume isThe volume at bottom dead center isThe dimensionless energy addition isThe mass of gas in the cylinder is

      2 The above engine parameters are entered into the FiniteHeatRelease.m program as shown below. The start of energy release is for Engine 1 and for Engine 2, and all other parameters are the same for both engines. function [ ] = FiniteHeatRelease( ) Gas cycle heat release code for two engines Engine input parameters: thetas(1,1) = -20; Engine 1 start of energy release (deg) thetas(2,1) = 0; Engine 2 start of energy release (deg) thetad(1,1) = 40; Engine 1 duration of energy release (deg) thetad(2,1) = 40; Engine 2 duration of energy release (deg) r =10; Compression ratio gamma = 1.4; Ideal gas const Q = 20.0; Dimensionless total energy addition a = 5; Wiebe efficiency factor a n = 3; Wiebe exponent n ...}The pressure profiles are compared in Figure 2.19. The pressure rise for Engine 1 is more than double that of Engine 2. The maximum pressure of about 8800 kPa occurs at 11 after top dead center for Engine 1, and at about 25 after top dead center for Engine 2. The temperature profiles are shown in Figure 2.20. Engine 1 has a peak temperature of about 2900 K, almost 400 K above that of Engine 2.

      3 The start of energy release is varied from to , as shown in Figures 2.21 and 2.22, and the resulting thermal efficiency and imep are plotted.

      Comment: The results indicate that there is an optimum crank angle for the start of energy release, which will maximize the thermal efficiency and imep. For this computation, the optimum start of energy release is about , resulting in a maximum thermal efficiency of about , and imep/ of about 13.2. At crank angles less than or greater than this optimal angle, the thermal efficiency and imep/ decrease.

      Pressure profiles for Example 2.5. Temperature profiles for Example 2.5.

An explanation for the optimal crank angle is as follows. If the energy release begins too early during the compression stroke, the negative compression work will increase, since the piston is doing work against the increasing combustion gas pressure. Conversely, if the energy release begins too late, the energy release will occur in an increasing cylinder volume, resulting in lower combustion pressure, and lower net work. In practice, the optimum spark timing also depends on the engine load, and is in the range of