**2.3 Thermodynamic modeling**

276 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

 *<sup>L</sup> s LL di t t R Rit L*

> 

*dt*

*s L*

*R R <sup>t</sup> <sup>L</sup>*

*R R dt*

(12)

3 3

*dt* (14)

(13)

*s L*

*R R s L <sup>t</sup> <sup>L</sup>*

(9)

(10)

The induced current from the load circuit can be derived in the following equations:

<sup>1</sup>

*s L <sup>t</sup> i t <sup>e</sup> R R* 

The magnetic force has the opposite direction to the direction of the translator's movement.

22 2 <sup>2</sup>

*x dx F N BxiH HN B*

When it comes to three-phase linear alternator, the third phase is derived from another two

2 4 sin

*coil*

2 2 sin sin sin

22 2 2 2 2

*dx x x x F HN B*

2 2 <sup>4</sup> sin sin sin

22 2 <sup>1</sup> <sup>6</sup> *coil m*

*R R*

In order to develop a first approach analysis, it's assumed that the alternator circuit works in resonance condition, which means that the current and voltage have the same phase, so that it is possible to consider the circuit as only a resistance [18]. Thus, the electromagnetic force

*M HN B*

*e dx F M* *s l*

So the total electromagnetic force produced by a three-phase linear alternator is:

*s l*

*R R dt*

1

*e*

3 3

  

 (11)

*L*

According to Ampere's law, it is described in the following equation:

*e coil L m*

*s L*

*R R <sup>t</sup> <sup>L</sup>*

*s L*

*R R dt*

*s L R R <sup>t</sup> <sup>L</sup>*

 

*dx HN B e*

*dt*

according to the following equation [17]:

22 2

 

*coil m*

*dx M e*

*coil*

*e m*

1

Where

1

*e*

1 61

*s L*

*R R <sup>t</sup> <sup>L</sup>*

is proportional to the speed of the translator:

The thermodynamic model is derived based on the first law of thermodynamics and ideal gas law. It consists of the calculation of the process of scavenging, compression, combustion, expansion and exhaust. The zero-dimensional, single zone model is used to describe the thermodynamic process.

Appling the first law of thermodynamics and ideal gas law on the cylinder as an open thermodynamic system, shown in Fig.5, and assuming that the specific heat *c*V and the gas constant *R* are constant:

$$\frac{d\mathbf{L}I}{dt} = -p\frac{dV}{dt} + \frac{dQ}{dt} + \dot{H}\_i - \dot{H}\_e \tag{15}$$

Fig. 5. Thermodynamic system of FPLA

For the case of compression and expansion process neglecting the crevice flow and the leakage, the first law of thermodynamics applied to the cylinder content becomes:

$$m\_{in}\frac{d\left(c\_V T\right)}{dt} = -p\frac{dV}{dt} + \frac{dQ}{dt} \tag{16}$$

Considering the cylinder content is ideal gas, and then at every instant the ideal gas law is satisfied:

$$pV = m\_{in}RT\tag{17}$$

Substitution and mathematical manipulation yield the following equation which is used to calculate the in-cylinder pressure at each time step.

Dimensionless Parametric Analysis of Spark Ignited Free-Piston Linear Alternator 279

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

The basic variables that involved were defined based on the dynamic and thermodynamic

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

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

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

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

modeling of FPLA. The variables and their dimensions are listed in Tab.1.

Bore *D* [L] Piston area *A* [L2] Volume of the cylinder *V* [L3] Effective stroke length *Leff* [L] Total stroke length *Ltot* [L] Translator ignition position *xign* [L] Mass of the translator *m* [M] Load coefficient *M* [MT-1] Friction force *Ff* [MLT-2] Energy *Q* [ML2T-2] Scavenge pressure *p0* [ML-1T-2] Scavenge temperature *T0* [θ] Pressure *p* [ML-1T-2] Gas constant *R* [θL-1T-2] Temperature *T* [θ] Wall temperature *Tw* [θ] Combustion duration *tc* [T] Compression ratio *ε* [1] indicated efficiency *η<sup>i</sup>* [1] Effective efficiency *η<sup>e</sup>* [1] Time *t* [T] Frequency *f* [T-1]

Variables Symbol Dimensions

**3. Dimensionless analysis** 

**3.1 Dimensionless modeling of FPLA** 

dimension is replaced with [ML2T-2].

Table 1. Basic variables and their dimensions

cylinder head to cylinder head [21].

the ignition signals.

scavenging temperature.

$$\frac{dp}{dt} = \frac{\gamma - 1}{V} \frac{dQ}{dt} - \gamma \frac{p}{V} \frac{dV}{dt} \tag{18}$$

In the combustion model, since the engine is crankless, a time-based Wiebe functions (as opposed to a conventional crank angle-based approach) is used to express the mass fraction burned in combustion process [4]:

$$\mathcal{X}(t) = 1 - \exp\left(-a\left(\frac{t - t\_0}{t\_c}\right)^{1+b}\right) \tag{19}$$

and the heat rate released during combustion is:

$$\frac{\partial \mathcal{Q}\_c}{\partial t} = \mathcal{Q}\_{in} \frac{\partial \mathcal{X}(t)}{\partial t} = \mathcal{Q}\_{in} \frac{1}{t\_c} a(1+b) \left(\frac{t-t\_0}{t\_c}\right)^b \exp\left[-a\left(\frac{t-t\_0}{t\_c}\right)^{1+b}\right] \tag{20}$$

The in-cylinder heat transfer effect is modeled according to Hohenberg [19]:

$$\frac{\partial Q\_{ht}}{\partial t} = hA\_{cyl} \left( T - T\_w \right) \tag{21}$$

$$h = 130V^{-0.06} \left(\frac{p}{10^5}\right)^{0.8} T^{-0.4} \left(\bar{\mathcal{U}} + \mathbf{1.4}\right)^{0.8} \tag{22}$$

Since most heat transfer models, like the ones proposed by Woschni and Hohenberg, are made for Diesel engines. This means that they take radiative heat transfer effect into account, which is hardly present in premixed combustion. Hence, in the numerical simulations a factor of 0.5 is introduced to reduce the heat transfer coefficient [9].

So the total energy that is used to increase the in-cylinder pressure in equation (18) can be expressed in the following equation:

$$\frac{dQ}{dt} = \frac{\partial Q\_c}{\partial t} - \frac{\partial Q\_{ht}}{\partial t} \tag{23}$$

Exhaust blown down is modeled to be a polytrophic expansion process while the exhaust port is opening and the scavenging ports are still covered by the piston [11].

$$\frac{dp}{dt} = \frac{n-1}{V}\frac{dQ}{dt} - n\frac{p}{V}\frac{dV}{dt} \tag{24}$$

For two-stroke spark ignition engines with under piston or crankcase scavenging, the scavenging efficiency is about 0.7~0.9 [20], , a scavenging efficiency of 0.8 is introduced to evaluate the effects of incomplete scavenging effect. The moment the scavenging ports are open, the pressure and temperature are assumed to be the same with the scavenging conditions and the incoming gases mix entirely with the burned gases.
