**3.1 Dimensionless modeling of FPLA**

The dimension of a physical quantity is associated with combinations of mass, length, time, temperature and heat quantity, represented by symbols [M], [L], [T], [θ] and [H] respectively, each raised to rational powers. Since heat quantity is also a kind of energy, its dimension is replaced with [ML2T-2].

The basic variables that involved were defined based on the dynamic and thermodynamic modeling of FPLA. The variables and their dimensions are listed in Tab.1.


Table 1. Basic variables and their dimensions

The effective stroke length is the distance between the upper edge of the exhaust port and the cylinder and the total stroke length is distance that the translator can travel from cylinder head to cylinder head [21].

Since the FPLA is controlled based on the displacement feedback of displacement sensor, the ignition timing is defined by the position of the translator where the controller gives out the ignition signals.

According to Buckingham's theory [22], four fundamental reference variables which are independent from each other were chosen, and the other variables are described with these reference variables in their index form and every redefined variable is dimensionless. The four reference variables chosen were bore, mass of the translator, scavenging pressure and scavenging temperature.

Dividing the basic variables by the reference variables results in the dimensionless variables:

Dimensionless Parametric Analysis of Spark Ignited Free-Piston Linear Alternator 281

2 2 2 111 <sup>0</sup> <sup>0</sup> <sup>222</sup>

2 \* \* \* \* \*\* \* \* \*2 *LR f* \*

\* <sup>3</sup> <sup>3</sup> <sup>0</sup> <sup>0</sup>

*<sup>p</sup> <sup>p</sup> <sup>V</sup> <sup>Q</sup> <sup>d</sup> <sup>d</sup> <sup>d</sup> dp p p D p D dt t tt V V*

3 3

and equations (28) (30) are the dimensionless form of the translator's

\* \* 3 3

*m T mT*

0 0 0 0

(31)

*m R* (32)

*d d*

*r rr*

*t tt D D*

\* \* \* \* \* \*\* \* \* *dp p dV dQ* 1 *dt V dt V dt*

*r r*

*LR f*

*x x d d m A D D <sup>M</sup> pp F*

*m p <sup>D</sup> p D <sup>t</sup> <sup>t</sup> p Dm <sup>d</sup> <sup>d</sup> <sup>t</sup> <sup>t</sup>* 

0

*d x dx m p pAF M dt dt* (28)

1 *<sup>o</sup>*

(30)

(27)

(29)

(33)

\* \* <sup>3</sup> \* <sup>0</sup> <sup>0</sup>

*p V pV p <sup>D</sup> mT pV <sup>R</sup> <sup>R</sup> m T m T pD pD m T*

*in in in*

Since the in-cylinder temperature is strongly transient, the dimensionless in-cylinder

\* \* \* \* \* *in <sup>p</sup> <sup>V</sup> <sup>T</sup>*

\*= *t*c/*tr*

The combustion model and heat transfer equations can also be transferred to their

 <sup>1</sup> \* \* \* \* \* \* 0 0 \* \*\* \*

dimensionless form using the dimensionless parameters we have already obtained:

*<sup>Q</sup> a b tt tt Q a t tt t* 

1

\*=*Ff*/(*p*0·*D*2)

exp

*c c c*

*b b*

The dimensionless gas constant was deduced based on the ideal gas law:

So the dimensionless variables defined in equations (28) (30) and (31) are:

temperature is acquired using the dimensionless ideal gas law:

Dimensionless time *t*\*=*t*/*tr* Dimensionless combustion duration *tc*

Dimensionless frequency *f*\*=*f*·*tr* Dimensionless friction force *Ff*

Dimensionless pressure *p*\*= *p*/*p*<sup>0</sup> Dimensionless energy *Q*\*=*Q*/(*p*0·*D*3) Dimensionless gas constant *R*\*=*R*/(*p*0·*D*3/*m*/*T*0)

*<sup>c</sup> in*

1 1

2

\*

1 11 2 22 *<sup>r</sup>* <sup>0</sup> *t mp D*

dynamic and thermodynamic equations.

Where


Since the dimensionless translator ignition position is changing with different stroke length and fixed translator ignition position, in the dimensionless calculation, its actual dimensionless value taken is defined by the compression ratio the engine has already achieved when the spark plug ignites, as is described in Fig.6.

$$\varepsilon\_{\rm ign} = \frac{V\_{\rm eff}}{V\_{\rm ign}} = \frac{D^2 L\_{\rm eff}}{D^2 \left(L\_{\rm tot} / 2 - x\_{\rm ign}\right)} = \frac{L\_{\rm eff} \, ^\circ D}{\left(L\_{\rm tot} / 2 - x\_{\rm ign}\right) / D} = \frac{L\_{\rm eff} \, ^\circ}{L\_{\rm tot} \, ^\circ / 2 - x\_{\rm ign} \, ^\circ} \tag{25}$$

Then the dimensionless translator ignition position can be deduced:

$$\left(\propto\_{ign}\right)^{\*} = \frac{L\_{tot}}{\mathfrak{D}} - \frac{L\_{eff}}{\mathfrak{E}\_{ign}}\tag{26}$$

Fig. 6. Diagram of translator ignition position

Substituting all the dimensionless variables into equations (1) and (18), the following equations can be derived:

$$\frac{m}{m}\frac{d^2\left(\frac{\mathbf{x}}{D}\right)}{d\left(\frac{\mathbf{t}}{t\_r}\right)^2} = \frac{1}{p\_0}\frac{A}{D^2}(p\_L - p\_R) - \frac{1}{p\_0 D^2}F\_f - \frac{M}{p\_0 \frac{1}{2}\frac{1}{D^2}\frac{1}{m^2}}\frac{d\left(\frac{\mathbf{x}}{D}\right)}{d\left(\frac{\mathbf{t}}{t\_r}\right)}\tag{27}$$

$$m^\* \frac{d^2 \mathbf{x}^\*}{dt^{\*2}} = \left(p\_L \stackrel{\*}{\ -} p\_R \stackrel{\*}{\ }\right) \mathbf{A}^\* - \mathbf{F}\_f^\* - \mathbf{M}^\* \frac{d\mathbf{x}^\*}{dt} \tag{28}$$

$$\frac{dp^\*}{dt^\*} = \frac{d\left(\frac{p}{p\_0}\right)}{\left(\frac{t}{t\_r}\right)} = -\gamma \frac{\frac{p}{p\_0}}{\frac{V}{D^3}} \frac{d\left(\frac{V}{D^3}\right)}{d\left(\frac{t}{t\_r}\right)} + \frac{\gamma - 1}{\frac{V}{D^3}} \frac{d\left(\frac{Q}{p\_o D^3}\right)}{d\left(\frac{t}{t\_r}\right)}\tag{29}$$

$$\frac{dp^\*}{dt^\*} = -\gamma \frac{p^\*}{V^\*} \frac{dV^\*}{dt^\*} + \frac{\gamma - 1}{V^\*} \frac{dQ^\*}{dt^\*}\tag{30}$$

1 11

280 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

Since the dimensionless translator ignition position is changing with different stroke length and fixed translator ignition position, in the dimensionless calculation, its actual dimensionless value taken is defined by the compression ratio the engine has already

*eff eff eff eff*

\* \* \* 2 *eff tot ign*

Substituting all the dimensionless variables into equations (1) and (18), the following

*L L*

*V DL L D L V DL x L x D L x*

*ign tot ign tot ign tot ign*

2 \* 2 \* \* / / 2 /2 / / 2

(25)

*ign*

(26)

Dimensionless bore *D*\*=*D*/*D*=1 Dimensionless piston area *A*\*=*A*/*D*<sup>2</sup> Dimensionless cylinder volume *V*\*=*V*/*D*<sup>3</sup> Dimensionless total stroke length *Ltot* \*= *Ltot* /*D* Dimensionless mass of the translator *m*\*=*m*/*m*=1 Dimensionless scavenge air pressure *p*0\*= *p*0/*p*0=1 Dimensionless scavenging temperature *T*0\*=*T*0/*T*0=1 Dimensionless temperature *T*\*=*T*/*T*<sup>0</sup> Dimensionless wall temperature *T*w\*=*T*w/*T*<sup>0</sup>

Dimensionless load coefficient *M*\*=*M*/(*p*00.5*D*0.5*m*0.5)

achieved when the spark plug ignites, as is described in Fig.6.

Then the dimensionless translator ignition position can be deduced:

*x*

*ign*

Fig. 6. Diagram of translator ignition position

equations can be derived:

Where 2 22 *<sup>r</sup>* <sup>0</sup> *t mp D* and equations (28) (30) are the dimensionless form of the translator's dynamic and thermodynamic equations.

The dimensionless gas constant was deduced based on the ideal gas law:

$$\mathcal{R}^\* = \frac{p^\* V^\*}{m\_{in} \, ^\* T} = \frac{\frac{p}{p\_0} \frac{V}{D^3}}{\frac{m\_{in}}{m} \frac{T}{T\_0}} = \frac{m T\_0}{p\_0 D^3} \frac{p \, V}{m\_{in} T} = \frac{R}{\frac{p\_0 D^3}{m T\_0}}\tag{31}$$

Since the in-cylinder temperature is strongly transient, the dimensionless in-cylinder temperature is acquired using the dimensionless ideal gas law:

$$T^\* = \frac{p^\* V^\*}{m\_{in} \, ^\* \mathcal{R}^\*} \tag{32}$$

So the dimensionless variables defined in equations (28) (30) and (31) are:


The combustion model and heat transfer equations can also be transferred to their dimensionless form using the dimensionless parameters we have already obtained:

$$\frac{\partial \mathcal{Q}\_c^{\*}}{\partial t^\*} = \mathcal{Q}\_{in} \frac{\ast \, a(b+1)}{t\_c^\*} \left( \frac{t^\*-t\_0^\*}{t\_c^\*} \right)^b \exp\left[-a \left(\frac{t^\*-t\_0^\*}{t\_c^\*}\right)^{b+1}\right] \tag{33}$$

Dimensionless Parametric Analysis of Spark Ignited Free-Piston Linear Alternator 283

energy (opening proportion of the throttle) were the variable input factors of the parametric

case1 case2 case3 case4 case5 case6 case7

*Leff*\* 0.5 0.6 0.7 0.8 0.9 1.0 1.1 *RL*/Ω 2 2.5 3 4 × × × *M*/N·(m·s-1)-1 62.2320 55.3174 49.7856 41.4880 × × × *M*\* 0.8714 0.7746 0.6971 0.5809 × × × *εign* 3 4 5 6 × × × *tc*/ms 3 4 5 6 × × ×

*\** 0.1326 0.1768 0.2210 0.2652 × × × *α* 25% 30% 35% × × × ×

The dimensionless effective stroke length was defined as the first independent variable while the dimensionless load coefficient, dimensionless translator ignition position, dimensionless combustion duration and dimensionless input energy were taken as second independent variables. When analyzing the effects of each second variable, the other second variables would be set to equal to the base case which was decided by the FPLA prototype.

Since the combustion model used in the numerical program is zero dimensional Wiebe function and some parameters like combustion duration are of great uncertainty, the accuracy of the numerical calculated results is suspectable. Nowadays, multi-dimensional CFD computational tools have become an integral part of the engine design process due mainly to advances in computing capabilities and improvements in the modeling techniques utilized. In this study, in order to validate the results of dimensionless analysis, a multi-

study. The dimensionless operating matrixes of each parameter are listed in Tab.3.

Basic parameters Value Dimensionless value

*D*/mm 34 1 *L*eff/mm 20 0.5882 *Ltot*/mm 36 1.0588 *Rl*/Ω 2.5 × *R*s/Ω 2 × *M*/N·(m·s-1)-1 55.3 0.7746 *m*/kg 1.74 1 *p0*/bar 1 1 *T0*/K 293 1 *Tw*/K 453 1.5461 *tc*/ms 5 0.2210 *ε*ign 4 × *α* 30% ×

Table 2. Parameters of the FPLA prototype

Table 3. Dimensionless operating matrix of FPLA

**4.1 Combustion modeling of FPLA** 

**4. CFD calculation of combustion process for validation** 

Parameters Value

*tc*

$$h^\* = 130V^{\*-0.06} \left(\frac{p^\*}{10^5}\right)^{0.8} T^{\*-0.4} \left(\stackrel{-}{\mathcal{U}}^\* + 1.4\right)^{0.8} \tag{34}$$

$$\frac{\left\|\mathcal{Q}\right\|\_{\text{ht}}^{\*}}{\left\|\mathcal{T}\right\|^{\*}} = \hbar^{\*} A\_{cyl} \left(T^{\*} - T\_{w}{}^{\*}\right) \tag{35}$$

All the equations were solved with a numerical simulating program in Matlab. The program starts calculation using a group of mathematic equations describing the dimensionless dynamic and dimensionless thermodynamic processes of FPLA, including the heat transfer rate, the in-cylinder pressure, the in-cylinder temperature, the fraction of fuel that burnt, the load and the work done, etc. At each time step, the program calls the dynamic subroutine and updates the displacement, the velocity, the averaged velocity and the acceleration. After the engine stabilizes, the program calculates the engine frequency, the compression ratio, the indicated power, the frictional power, the effective power output and the effective efficiency.

The dimensionless form of the output parameters are:


The dimensionless compression ratio and efficiency have just the same value with their dimensional forms, which are used to validate the correctness of the dimensionless process.

#### **3.2 Operating ranges**

The variables of the dimensionless analysis were chosen to cover the normal operating ranges of two-stroke free-piston engine. The base case represented the parameters of the experimental spark ignited, two-stroke FPLA prototype that was built in Beijing Institute of Technology. The basic parameters are listed in Tab.2.

The effective stroke length to bore ratio, dimensionless load coefficient, dimensionless translator ignition position, dimensionless combustion duration and dimensionless input


energy (opening proportion of the throttle) were the variable input factors of the parametric study. The dimensionless operating matrixes of each parameter are listed in Tab.3.

Table 2. Parameters of the FPLA prototype

282 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

<sup>5</sup> 130 1.4 10 *<sup>p</sup> h V TU*

\* \* \*\* \*

*<sup>Q</sup> hA T T*

All the equations were solved with a numerical simulating program in Matlab. The program starts calculation using a group of mathematic equations describing the dimensionless dynamic and dimensionless thermodynamic processes of FPLA, including the heat transfer rate, the in-cylinder pressure, the in-cylinder temperature, the fraction of fuel that burnt, the load and the work done, etc. At each time step, the program calls the dynamic subroutine and updates the displacement, the velocity, the averaged velocity and the acceleration. After the engine stabilizes, the program calculates the engine frequency, the compression ratio, the indicated power, the frictional power, the effective power output and the effective

\*

\* \* \* 4 *U xf <sup>s</sup>*

 

 \*

The dimensionless compression ratio and efficiency have just the same value with their dimensional forms, which are used to validate the correctness of the dimensionless process.

The variables of the dimensionless analysis were chosen to cover the normal operating ranges of two-stroke free-piston engine. The base case represented the parameters of the experimental spark ignited, two-stroke FPLA prototype that was built in Beijing Institute of

The effective stroke length to bore ratio, dimensionless load coefficient, dimensionless translator ignition position, dimensionless combustion duration and dimensionless input

*cyl w*

\* \* 0.06 \* 0.4

\* *ht*

*t*

The dimensionless form of the output parameters are:

Dimensionless frictional work \* \*\* 4 *W Fx f f <sup>s</sup>*

Dimensionless indicated work \*

Dimensionless effective power output \* \*\* *P W e e f* Dimensionless frictional power \* \*\* *P W f f f*

Technology. The basic parameters are listed in Tab.2.

Dimensionless effective work \*\* \* *W WW e i <sup>f</sup>*

Dimensionless compression ratio

Dimensionless averaged speed

Dimensionless indicated efficiency

Dimensionless effective efficiency

**3.2 Operating ranges** 

efficiency.

0.8 0.8 \* \*

\*

\* \*\* *<sup>i</sup> <sup>V</sup> <sup>W</sup> <sup>p</sup> dV*

\* \*

\*

\* \* \* *i i i i in in W W Q Q*

\* \* / 2 / 2 *eff eff tot s tot s*

*i f if e e in in*

*W W WW Q Q*

*L L L x L x*

(35)

(34)


Table 3. Dimensionless operating matrix of FPLA

The dimensionless effective stroke length was defined as the first independent variable while the dimensionless load coefficient, dimensionless translator ignition position, dimensionless combustion duration and dimensionless input energy were taken as second independent variables. When analyzing the effects of each second variable, the other second variables would be set to equal to the base case which was decided by the FPLA prototype.
