**5. Results and discussion**

### **5.1 Zero dimensional models validation**

In order to make sure most part of the numerical simulation program is correct, the numerical simulated in-cylinder pressure is compared with experimental derived pressure in Fig.9. The thermodynamic model used in the numerical analysis is a single zone model. In single zone models the cylinder mixture composition, pressure and temperature are considered to be uniform and the energy release rate is modeled using experiential model. Matching the experimentally and numerically derived pressure profiles was a complicated task due to the simplification and unknown variables, such as the actual heat addition, the combustion duration and the actual load when the engine was running. Seen in Fig.9, the numerical simulation model used to simulate the operation of FPLA proved to be in agreement with the experimental data with a certain combination of the variables.

### **5.2 Effects of dimensionless parameters**

### **5.2.1 Effects of dimensionless effective stroke length**

The stroke to bore ratio is one of the core design variables in internal combustion engines, relating combustion chamber surface area to its volume and piston area to stroke length [11]. In order to predict the performance of FPLA with various sizes and dimensions, seven cases of dimensionless effective stroke length were chosen as the basic variable to investigate the performance of the FPLA in wide operating ranges.

In order to make sure most part of the numerical simulation program is correct, the numerical simulated in-cylinder pressure is compared with experimental derived pressure in Fig.9. The thermodynamic model used in the numerical analysis is a single zone model. In single zone models the cylinder mixture composition, pressure and temperature are considered to be uniform and the energy release rate is modeled using experiential model. Matching the experimentally and numerically derived pressure profiles was a complicated task due to the simplification and unknown variables, such as the actual heat addition, the combustion duration and the actual load when the engine was running. Seen in Fig.9, the numerical simulation model used to simulate the operation of FPLA proved to be in

agreement with the experimental data with a certain combination of the variables.

The stroke to bore ratio is one of the core design variables in internal combustion engines, relating combustion chamber surface area to its volume and piston area to stroke length [11]. In order to predict the performance of FPLA with various sizes and dimensions, seven cases of dimensionless effective stroke length were chosen as the basic variable to

Fig. 8. Piston dynamics with different operating conditions

**5. Results and discussion** 

**5.1 Zero dimensional models validation** 

**5.2 Effects of dimensionless parameters** 

**5.2.1 Effects of dimensionless effective stroke length** 

investigate the performance of the FPLA in wide operating ranges.

Fig. 9. Comparation of experimental and numerical simulated pressure data

Fig. 10. Dimensionless velocity vs. dimensionless displacement with different effective stroke length to bore ratio

Dimensionless Parametric Analysis of Spark Ignited Free-Piston Linear Alternator 289

Fig. 12. Effects of effective stroke length to bore ratio to dimensionless frequency and

Fig. 13. Effects of effective stroke length to bore ratio to dimensionless peak pressure and

dimensionless effective power output

dimensionless frictional power

Fig.10 illustrates the change of dimensionless velocity versus dimensionless displacement for different effective stroke length to bore ratios while the other parameters remain the same with the base case. It can be seen that as the dimensionless effective stroke length increases, the dimensionless velocity increases. However, the profile of all the curves is similar to each other, which means this kind of free-piston engine has its own specific characteristics.

As is shown in Figs.11~13, the dimensionless compression ratio, dimensionless frequency and dimensionless in-cylinder peak pressure keep decreasing as the effective stroke length to bore ratio increases. These are because as the dimensionless effective stroke length increases, the translator has to travel longer strokes and more energy is wasted overcoming friction. And as the effective stroke length to bore ratio increases, the cylinder contains more charge and the charge would contain more energy during compression stoke; therefore the dimensionless compression ratio would decrease.

The highest dimensionless effective efficiency is achieved while *Leff*\* is 0.8 under the basic working conditions, and the peak point is mainly affected by the other four dimensionless parameters, which will be discussed in the following sections.

As the effective stroke length to bore ratio increases, the dimensionless energy input to the engine every cycle increases as a result of increasing the volume of the cylinder. Since the dimensionless effective power output is also strongly determined by the dimensionless frequency and dimensionless effective efficiency, the highest effective power output is achieved while *Leff*\* is 0.9 under the basic working conditions.

Fig. 11. Effects of effective stroke length to bore ratio to dimensionless compression ratio and dimensionless effective efficiency

Fig.10 illustrates the change of dimensionless velocity versus dimensionless displacement for different effective stroke length to bore ratios while the other parameters remain the same with the base case. It can be seen that as the dimensionless effective stroke length increases, the dimensionless velocity increases. However, the profile of all the curves is similar to each other, which means this kind of free-piston engine has its own specific

As is shown in Figs.11~13, the dimensionless compression ratio, dimensionless frequency and dimensionless in-cylinder peak pressure keep decreasing as the effective stroke length to bore ratio increases. These are because as the dimensionless effective stroke length increases, the translator has to travel longer strokes and more energy is wasted overcoming friction. And as the effective stroke length to bore ratio increases, the cylinder contains more charge and the charge would contain more energy during compression stoke; therefore the

The highest dimensionless effective efficiency is achieved while *Leff*\* is 0.8 under the basic working conditions, and the peak point is mainly affected by the other four dimensionless

As the effective stroke length to bore ratio increases, the dimensionless energy input to the engine every cycle increases as a result of increasing the volume of the cylinder. Since the dimensionless effective power output is also strongly determined by the dimensionless frequency and dimensionless effective efficiency, the highest effective power output is

Fig. 11. Effects of effective stroke length to bore ratio to dimensionless compression ratio

characteristics.

dimensionless compression ratio would decrease.

and dimensionless effective efficiency

parameters, which will be discussed in the following sections.

achieved while *Leff*\* is 0.9 under the basic working conditions.

Fig. 12. Effects of effective stroke length to bore ratio to dimensionless frequency and dimensionless effective power output

Fig. 13. Effects of effective stroke length to bore ratio to dimensionless peak pressure and dimensionless frictional power

Dimensionless Parametric Analysis of Spark Ignited Free-Piston Linear Alternator 291

lead to higher dimensionless effective efficiency and higher dimensionless frequency. Therefore, we can conclude that the main factor that controls the power output of FPLA is

Fig. 15. Effects of dimensionless load coefficient to dimensionless frequency and

Ignition timing is one of the major parameters that control the engine's operating conditions, such as frequency and compression ratio. Since the dimensionless ignition timing is changing with different dimensionless stroke length, the ignition timing is defined by the compression ratio the engine has already achieved when the spark plug ignites in the calculation, and it means that the lower the ignition compression ratio is the bigger the

According to some literatures [3][5], it's held that an earlier combustion in diesel free-piston engines would lead to more waste of energy to reverse the translator, thus the efficiency and frequency would drop. However, according to the results of spark ignited FPLA obtained in this paper, with different effective stroke length to bore ratio the best ignition advance differs with each other, since an early ignition is associated with negative work in the compression stroke and a late ignition is associated with low peak in-cylinder pressure, as is

As is described in Figs.17~18, with smaller effective stroke length to bore ratio (closer to 0.5), a bigger ignition advance would lead to higher dimensionless compression ratio, higher dimensionless effective efficiency, higher dimensionless frequency and higher dimensionless effective power output. The reason is that with small dimensionless effective

**5.2.3 Effects of dimensionless translator ignition position** 

dimensionless effective power output

ignition advance is.

shown in Fig.16.

its frequency.

### **5.2.2 Effects of dimensionless load coefficient**

Increasing the dimensionless load coefficient means the load demand of the linear alternator is increasing and the electromagnetic force produced by the linear alternator is increasing. Four different dimensionless load coefficients (M\*1>M\*2>M\*3>M\*4) were chosen to investigate the effects of changing the load of the linear alternator. The load coefficient was varied by changing the value of the load resistance. According to the results calculated, the dimensionless load coefficient has large impact on different parameters studied and can affect the operating condition of FPLA.

According to Figs.14~15, as the dimensionless load coefficient increases, the dimensionless compression ratio and dimensionless frequency decrease since bigger electromagnetic force is acting on the translator. The highest dimensionless effective efficiency is changing with different dimensionless load coefficient and effective stroke length to bore ratio. As is shown in Fig.14, when the effective stroke length to bore ratio is less than 0.67, smaller dimensionless load coefficient would lead to a higher dimensionless effective efficiency and when the effective stroke length to bore ratio is more than 1.0, the larger the load coefficient the higher the dimensionless effective efficiency. The reason behind these is believed to be caused by the percentage of heat released before top dead center (TDC), which is strongly determined by the frequency of the translator.

Fig. 14. Effects of dimensionless load coefficient to dimensionless compression ratio and dimensionless effective efficiency

As is shown in Fig.15, smallest dimensionless load coefficient lead to the highest dimensionless power output although the dimensionless effective efficiency is the lowest since the dimensionless frequency with smaller load coefficient is higher. It is more obvious when the effective stroke length to bore ratio is more than 1.0 since smaller load coefficient

Increasing the dimensionless load coefficient means the load demand of the linear alternator is increasing and the electromagnetic force produced by the linear alternator is increasing. Four different dimensionless load coefficients (M\*1>M\*2>M\*3>M\*4) were chosen to investigate the effects of changing the load of the linear alternator. The load coefficient was varied by changing the value of the load resistance. According to the results calculated, the dimensionless load coefficient has large impact on different parameters studied and can

According to Figs.14~15, as the dimensionless load coefficient increases, the dimensionless compression ratio and dimensionless frequency decrease since bigger electromagnetic force is acting on the translator. The highest dimensionless effective efficiency is changing with different dimensionless load coefficient and effective stroke length to bore ratio. As is shown in Fig.14, when the effective stroke length to bore ratio is less than 0.67, smaller dimensionless load coefficient would lead to a higher dimensionless effective efficiency and when the effective stroke length to bore ratio is more than 1.0, the larger the load coefficient the higher the dimensionless effective efficiency. The reason behind these is believed to be caused by the percentage of heat released before top dead center (TDC), which is strongly

Fig. 14. Effects of dimensionless load coefficient to dimensionless compression ratio and

As is shown in Fig.15, smallest dimensionless load coefficient lead to the highest dimensionless power output although the dimensionless effective efficiency is the lowest since the dimensionless frequency with smaller load coefficient is higher. It is more obvious when the effective stroke length to bore ratio is more than 1.0 since smaller load coefficient

**5.2.2 Effects of dimensionless load coefficient** 

affect the operating condition of FPLA.

determined by the frequency of the translator.

dimensionless effective efficiency

lead to higher dimensionless effective efficiency and higher dimensionless frequency. Therefore, we can conclude that the main factor that controls the power output of FPLA is its frequency.

Fig. 15. Effects of dimensionless load coefficient to dimensionless frequency and dimensionless effective power output
