Linear Engine / L2V4bGlicmlzL2R0bC9kM18xL2FwYWNoZV9tZWRpYS80Nzg0

With these definitions of a and b, stroke length, constant pressure expansion length, frictional force, and compression ratio can be calculated. By substituting equations 4 and

5 into Equation 2, a becomes:

By substituting into Equation 3 the constant pressure expansion end coordinate can be solved for by:

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3.6 Pressure Balance

During a left to right stroke of the slider the cylinder pressures will be governed by several equations because of the constant pressure heat addition, the placement of the port openings and closings, and the scavenging process. Based on the prototype engine, (described in the following chapter) it was concluded that the pressure force Fp had four

distinct regimes. These regimes come from the equations governing the cylinder pressure. The regimes can best be seen from a plot of the left and right cylinder pressures versus the slider displacement x, shown in Figure 3.4 below. The compression cylinder undergoes a constant pressure gas exchange while the exhaust port is open. Once the exhaust port closes the compression cylinder undergoes an adiabatic compression. Simultaneously, the expansion cylinder undergoes a constant pressure expansion during the constant pressure heat addition, then the pressure decreases aidabaticly until the exhaust port opens and the pressure drops to the intake pressure. While the exhaust port

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is open the pressure remains constant. The governing equations for the cylinder pressures

and pressure force Fp are listed in Table 3.1.

Figure 3.4. Four regions can be seen for the pressure balance due to the limitedpressure cycle of operation.

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Table 3.1. In-cylinder pressures and pressure force as a function of the slider displacement.

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From analysis of Figure 3.4, it can be seen that regime A corresponds to the point

when x ³ -xs and x < xa, regime B corresponds to the point when x ³ xa and x < xepr,

regime C corresponds to the point when x ³ xepr, and x < xepl, and regime D corresponds

to the point when x ³ xepl and x £ xs. P0 is the pressure in the cylinder caused by the

constant volume heat addition. P0 is related to the pressure before the constant volume

heat addition and a by:

P0 =αP2

where P2 is the pressure in the cylinder after the adiabatic compression (Pr at x=xs), this

pressure can be found with knowledge of the intake pressure Pin by

3.7 Numerical Integration

Once all of the above relationships were derived a computer program to solve Equation 1 for the acceleration was developed. This program first solved the thermodynamic model for the half stroke xs, the constant pressure expansion end

coordinate xa, and other variables. With the calculations from the thermodynamic model

complete the program then calculated the in-cylinder pressures, pressure force and

The numerical integration worked by looping through the displacement steps calculating the time step between data points. With the time known the velocity can be

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calculated and the position can be related to the calculated time. The form of the equations to calculate the time and the velocity are as follows

and

These equations can be solved with the knowledge of the boundary condition

v0 = 0 . The MATLAB program can be seen in appendix A.

3.8 Results

The analysis provided relationships which made it possible to obtain the velocity, and position of the piston with respect to time. It also provided the calculation of the stroke length, time required for one stroke, and the compression ratio for a given value of heat input into the system. The time required for one stroke was calculated numerically by integrating the time intervals constituting one stroke. The thermodynamic analysis yielded relationships between the heat input and the achieved stroke length, constant pressure expansion end coordinate, and compression ratio.

The graphs showing the relationships of achieved half stroke, compression ratio, and stroke time with respect to the heat input were plotted for different values of the total heat input and φ. The geometric parameters of the engine were specified to be consistent with the prototype. The bore, b, of the engine cylinders was assumed to be 75mm, while the maximum half stroke, xm, was assumed to be 35.5mm. By specifying these

parameters it was possible to generate plots for the achieved half stroke, the constant volume expansion end coordinate, and the operating compression ratio versus the heat

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input into the system. These plot can be seen below in Figures 3.5, 3.6, and 3.7 respectively. According to Heywood [17], the typical operating range for a conventional compression ignition engine with Diesel fuel is 18 to 70. The upper limit to this range can be seen by the vertical line in Figures 3.5 to 3.10.

Figure 3.5. Achieved half stroke can be seen to be a function of the amount of heat input and the percentage of heat input at constant volume.

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Figure 3.6. The constant pressure expansion coordinate defines how much heat is input at constant pressure. It can be seen to be a function of the heat input and φ.

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Figure 3.7. The compression ratio is defined by the geometry of the engine and the achieved half stroke.

From the numerical integration the stroke time, average frequency, and midstroke slider velocity can be shown relative to the specific heat input and the value of φ. These plots can be seen in Figures 3.8, 3.9, and 3.10 respectively.

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Figure 3.8. The period of the operating cycle is seen to be the smallest for a Diesel cycle of operation. The cycle period increases as the amount of heat input at constant volume increases.

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