Internal Combustion Engines. Allan T. Kirkpatrick

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 that work and heat interaction terms are not true differentials) for a small crank angle change, , is

      (2.73)

      since , and

      (2.74)

      Assuming ideal gas behavior,

      (2.75)

      which in differential form is

      (2.76)

      The energy equation is therefore

      (2.77)

      differentiating with respect to crank angle, and introducing ,

      (2.78)

      Solving for the pressure, ,

(2.79)

In practice, it is convenient to normalize the equation with the pressure and volume at bottom dead center

      (2.80)

      in which case we obtain

(2.81)

      The differential equation for the work is

(2.82)

      where

      (2.83)

      In order to integrate Equations (2.81) and (2.82), an equation for the cylinder volume as a function of crank angle is needed. By reference to Chapter 1, the dimensionless cylinder volume for is

      (2.84)

      upon differentiation,

      (2.85)

      Equations (2.81) and (2.82) are linear first‐order differential equations of the form , and are easily solved by numerical integration. Solution yields and , which once determined, allows computation of the net work of the cycle, the thermal efficiency, and the indicated mean effective pressure. Note that in this analysis, we have neglected heat and mass transfer losses, and will consider them in the next section.

      The thermal efficiency is computed directly from its definition

      (2.86)

      The imep is then computed using Equation ( Скачать книгу