Negative Samples

**4. Conclusions**

satisfactory results.

Table 6. Datasets description and averaged results.

some actual Fuzzy based approaches are considered.

optimal tool outperforming other traditional techniques.

SPECT WBCD FNA THERM

212 212 235 5

55 357 457 65

In CLFNN the behaviour is characterized by a linguistic term derived by fuzzy set *A*. *A* is a mapping of *X* to the linguistic model of CLFNN; each set *A* identifies a set of locations in the input space characterizing by their membership functions. CLFNN is less subjected to data imbalance problem because it builds its knowledge from positive and negative classes separately and the influence of each class on the other one is minimized. Moreover in CLFNN the inference system uses the lateral inhibition which improves the system performance treating with imbalanced dataset.

In order to demonstrate the efficiency of CLFNN method, four imbalanced datasets are used: Single Photon Emission Computed Tomography (SPECT) (Blake & Merz, 1998), Wisconsin Breast Cancer diagnosis (WBCD) (Blake & Merz, 1998), Fine Needle Aspiration (FNA) (Cross & Harrison, 2006) and Thermogram (THERM).(Ng &Fok, 2003).

The dataset is divided in training set, testing set and validation set maintaining the class distribution. The averaged performance of CLFNN, which is calculated by the F-Measure over three cross-validation sets, is compared with other popular methods: Multilayer Perceptron (MLP)(Rosenblattx, 1958)., Radial Basis Function (RBF) (Powell, 1985), Linear Discriminant Analysis (LDA) (McLachlan, 2004), Decision tree C4.5 (Brodley & Utgoff,P.E.) and Support Vector Machine (SVM).

Table 6 illustrates the description of datasets and the averaged results obtained over the different cross validations with the several approaches. The acronym IR indicates the imbalanced ratio, i.e. the ratio between the number of positive samples and the number of negative samples.

Table 6 shows that CLFNN provides better results than the other approaches. In thermogram dataset none of the system can give satisfactory results because of its very high imbalance ratio but, also in this case, CLFNN outperforms the other approaches. This work confirms that CLFNN provides more consistent results over different data distributions coming a promising tool for handling imbalanced dataset.

Table 6. Datasets description and averaged results.


**5. Skin cell cultures for psoriasis**

platelets are activated and migrated to the site of injury [57]. The second phase, inflammation, begins about 1 day postinjury and inflammatory mediators such as histamine are released, providing the typical traits of inflammation such as heat and swelling [57]. Proliferation is the phase in which granulation tissue forms at the site of injury and after which re-epithelization occurs [58]. In the last phase, remodeling of the tissue occurs to improve the strength of the skin tissue, which is only ever within 80% of the original tissue's strength [58]. The normal wound healing process could be affected, however, leading to chronic ulcers or excessive wound healing resulting in hypertrophic scars [59]. For this reason, topical agents to improve wound healing or reduce scarring may be of interest and as such *in vitro* testing models for wound healing will

Both 2D and 3D skin cell cultures are available as wound healing models [60]. 2D wound healing models involve the creation of a site of injury in a monolayer of skin cells, either through mechanical or chemical means, in which cells then migrate to the site of injury [60, 61]. Cells in 2D monolayer cultures are thought to adhere to the flat environments, such as a Petri dish, in which they are cultured and will therefore migrate to areas of free space within the dish, an activity thought to mimic *in vivo* migration involved in cell differentiation [62]. One method of mechanical introduction of a wound to a 2D cell culture is through the scratch assay that utilizes materials such as pipette tips or needles to introduce a wound or scratch into the monolayer cell culture [63, 64]. Images of the wound are taken within set time frames to assess the migration of cells [63]. Typically, however, it is difficult to ensure wounds that are equal in size using this method [63], and thus for this reason, testing of pharmaceutical products on these types of cultures are not ideal. 2D skin cell cultures also lack essential functions that could mimic *in vivo* processes, such as immune functionality and blood perfusion [65]. 3D wound healing models have thus attempted to more closely mimic *in vivo* wound healing processes and are available

Histocultures are cultures of intact tissues, consisting of more than one type of skin cell, such as neutrophils and other cells involved in wound healing, and are thus able to better mimic *in vivo* skin responses of wound healing [42, 43]. HSEs, on the other hand, are 3D cell culture models created from various human skin cells and materials that mimic the extracellular matrix [45] and are created as either epidermal equivalents, dermal equivalents or skin equivalents consisting of both layers [8, 45]. Some examples of 3D wound healing skin cell

A 3D wound healing skin equivalent developed by Herman et al. [66] included capillary endothelial cells which were capable of creating a similar formation to *in vivo* capillaries in the skin. This wound healing model involved several different cells, such as keratinocytes and epithelial cells, and used 3D matrices composed of Matrigel™ and collagen [66]. As angiogenesis is involved in the process of wound healing [66], this model is an excellent example of improved 3D wound healing cell culturing protocols to more closely mimic *in vivo* wound

Another example of a 3D skin cell culture model that could be used for wound healing was reported by Sidgwick et al. [67]. This model allows for the immersion of biopsies in Williams E culture media with the epidermal layer of the skin uncovered and uses whole tissue biopsies

be discussed.

10 Cell Culture

as histocultures or HSEs.

healing processes.

cultures and HSEs are described below.

As an inflammatory skin condition that involves the immune system, psoriasis has the characteristic appearance of silver scales that arise from increased proliferation of the keratinocytes in the epidermal layer [68, 69]. Psoriasis is typically treated with topical steroid medications or with vitamin D analogue medications such as calcipotriol as well as with moisturizing agents [68]. The treatment of psoriasis also includes systemic medications that suppress the immune system such as methotrexate and cyclosporine as well as biologic drugs [68, 70].

Psoriasis has been linked to various mental health illnesses such as anxiety and depression which are thought to result from having a chronic visible skin condition [71], and therefore the need for developing new pharmaceutical products and testing of these products is evident. As psoriasis is primarily treated with topical medications [70], having *in vitro* cell cultures or models for testing the safety and efficacy of these topical medications is essential.

Skin cell cultures that are 2D for dermatological conditions such as psoriasis and other autoimmune disorders exist [39]. 2D cell cultures of psoriasis initially consisted of psoriatic keratinocytes; however, this proved to be an ineffective model for psoriasis as the cells were not able to grow and lost their psoriatic genes with time [72]. As a result, alternative methods such as adding cytokines to normal human keratinocytes to induce psoriatic features were developed [73]. As stated previously, however, testing of medications on 2D cell culture models does not always translate to *in vivo* responses [7], and thus, 3D models to better mimic *in vivo* responses for testing of topical medications for psoriasis have been developed.

Barker et al. [74] introduce an *in vitro* model of psoriatic human skin, whereby a monolayer cell culture of psoriatic keratinocytes on top of collagen and fibroblasts from the dermis was created, onto which an epidermal layer was formed approximately 3 weeks later. Similarly, a 3D skin substitute using psoriatic skin cells was developed by Jean et al. [75] with either both psoriatic keratinocytes and fibroblasts or only one type of psoriatic cells (either keratinocytes or fibroblasts were psoriatic). Skin biopsies were obtained from patients with plaque psoriasis, and keratinocytes were extracted and seeded on to mouse fibroblasts and incubated at a temperature of 37°C [75]. For skin substitutes, ascorbic acid was used to culture the fibroblasts which formed dermal sheets that were altered to create a dermal layer onto which keratinocytes were then seeded to form an epidermal layer [75]. This method is the self-assembly method as the fibroblasts release their own extracellular matrix to maintain their growth [75]. A serum-free 3D psoriatic skin cell culture was similarly developed by Duque-Fernandez et al. [66]. This method is similar to the one presented by Jean et al. [75] and may be due to the fact that some authors are the same in both articles. The model by Duque-Fernandez et al. [76] was even used for testing percutaneous permeation of benzoic acid, caffeine, and hydrocortisone, using a Franz-diffusion cell method. This was compared to healthy skin substitutes and revealed that the psoriatic skin substitute had a greater permeability response to the three compounds [76].

cultures, [8] or the lack of interactions in 2D cell cultures of cell-to-cell or cell-to-extracellular matrix may be possible causes [5, 80]. However, advantages to using a 2D cell culture include

2D vs. 3D Cell Culture Models for *In Vitro* Topical (Dermatological) Medication Testing

3D cell cultures, on the other hand, provide a better imitation of *in vivo* processes as they allow for better understanding of the interactions that take place between cell-to-cell and cell-to-extracellular matrix [7]. Cell cultures that are 3D have also revealed to possess gene expression abilities, thus mimicking *in vivo* processes [83]. However, regardless of their improvements vs. the 2D cell cultures, typically 3D cell cultures still lack in their ability to provide key functions such as cellular waste removal [7]. Microfabricated cell cultures or on-chip cell cultures are advantageous in this instance as they provide microfluidic channels that enable the flow of nutrients and waste removal as well as the ability to mimic vasculature [51, 52]. Thus microfabricated 3D cell cultures are arguably the *in vitro* method of cell cultures that most closely mimic *in vivo* processes [50]. With respect to skin cell cultures, those that are 3D have the advantages of containing a stratum corneum and thus the ability of testing pharmaceutical products on the stratum corneum

ease of use, cost, and abundant scientific literature surrounding its use [81, 82].

**Advantages Disadvantages**

• Do not always provide an accurate representation of *in vivo* processes [7] • Use of a single cell type, [80] for example with skin cell cultures, the use of keratino-

http://dx.doi.org/10.5772/intechopen.79868

13

• No cell-to-cell or cell-to-extracellular matrix interactions [5, 80]

• Typically lack in their ability to provide key functions such as cellular waste

• Cost associated with developing cultures

cytes only [8]

removal [7]

[40, 54]

• Ease of use, cost and abundant scientific literature

• Better understanding of the interactions that take place between cell-to-cell and cell-to-extracellular

• Possess gene expression abilities, thus mimicking *in*

• Microfabricated cell cultures or on-chip cell cultures provide microfluidic channels that enable the flow of nutrients and waste removal as well as the ability

• 3D skin cultures have a stratum corneum and thus the ability of testing pharmaceutical products on the

• Longer skin cell culture use between 10 and 30 days [40]; more recent human skin equivalent (HSE) cultures can even be used for up to 20 weeks [41] • 3D bioprinting provides greater accuracy in placement of cells and extracellular matrix, potential of imbedding vasculature in the skin construct and

• 3D bioprinted skin models provide more uniform models vs. manually developed skin models [55]

**Table 1.** Summary of the advantages and disadvantages of 2D vs. 3D cell cultures.

stratum corneum as a skin barrier [40]

surrounding use [81, 82]

matrix [7]

*vivo* processes [83]

to mimic vasculature [51, 52]

have great plasticity [54]

• Better imitation of *in vivo* processes [7]

2D cell cultures

3D cell cultures

For safety and efficacy testing of medications for psoriasis, commercially available 3D *in vitro* models also exist. Two examples of such models are the Psoriasis Skin Model by Creative Bioarray [77] and the model by MatTek Corporation [78]. According to Creative Bioarray [77], their skin model more closely mimics *in vivo* psoriatic skin vs. typical cell cultures as it allows for cell differentiation and can be used for testing of pharmaceutical products. The psoriasis model by MatTek Corporation can also be used for testing of dermatological products as it a 3D model composed of psoriatic fibroblasts while using normal human epidermal keratinocytes and is capable of mimicking *in vivo* psoriatic responses such as basal cell proliferation [78]. These examples were presented for a better understanding of the methods used in developing psoriatic human skin equivalents as well as the commercially available models for testing the safety and efficacy of dermatological products used for psoriasis.


*5.2.3 Home care for a healthy preterm baby: neonatal discharge planning*

The infants who are born preterm are generally cared for in neonatal intensive care units (NICU), and a discharge planning is necessary for a comprehensive method. Because of the rate of mortality and morbidity, they are considered newborns with high risk after discharge.

Home care for the preterm newborn is coordinated by a complex team of caregivers: physician, paediatric primary care provider/family doctor, nurses, occupational and/or physical therapists, dieticians, pharmacists, parents and social workers. Neonatal discharge planning is developed in four major aspects:

**129**

*The Family as Recipient and Provider of Home Care: A Primary Care Perspective*

Respiratory control, maintaining normal temperature (axillar 36.5–37.5°C and rectal temperature 36.6–38°C) in an ambient temperature of 20–22°C, oral feeding skills (breast and/or bottle) and an ascendant parallel growth curve are mandatory

A complete routine screening must be performed before discharge, and ophthalmological, hearing and prematurity anaemia screening must be included in the follow-up programme. The schedule of vaccine is planned according to the chronological age of preterm infant and follows the same schedule of a full-term newborn. Before discharge it is mandatory to observe the preterm infants for a minimum period of 90–120 minutes while sitting in a car safety seat because of frequent oxygen desaturation and episodes of apnoea, hypotonia or bradycardia. Families/caregivers should be trained by hospital staff regarding the proper position in the car safety seat [37, 38]. The parents/primary care providers of preterm infants planned to be discharged from NICU must prove competency in daily care regarding breastfeeding/bottle feeding, care for the infant's skin and genitalia and the way the baby is bathed and clothed. It is necessary for the parents/caregivers to demonstrate some basic medical knowledge on the possible signs of illness, like changes in pattern of feeding or sleep, significance of dry diapers or modifications of the stool aspect. The caregivers must be able to monitor the temperature of preterm baby, evaluate the respiratory pattern and skin colour and assess the muscular tonus. It is important to educate the parents/caregivers about the safe sleep position for preterm infants as they must sleep alone on a plan surface and in a supine position [39]. The readiness of parents to care for newborns discharged from NICU should be assessed in order to improve the discharge preparation process. A parental and nurse survey demonstrated that iterative improvements of the discharge preparation process lead to an increase in family self-assessment of discharge readiness and in nurse assessment of

*DOI: http://dx.doi.org/10.5772/intechopen.91926*

for planning the discharge of the preterm infant.

the family's emotional discharge readiness [40].

drug and alcohol use, smoking and poverty [42].

**5.3 Home care for children with special healthcare needs**

*5.3.1 Children and youth with special healthcare needs (CYSHCN)*

The definition of this category of children has become more comprehensive during time, and now it is accepted that CYSHCN are those "who have or are at increased risk of developing a chronic physical, developmental, behavioral, or emotional condition and require health and related services of a type or amount beyond that usually required by children" [41]. More than a quarter of families have at least one child with special healthcare needs. Data from the 2016–2017 National Survey of Children's Health (NSCH) indicate that 18.8% of children <18 years of age in the USA have special healthcare needs, with 13.2% of children being medically complex. In Western Europe non-communicable diseases (NCDs) for children under 15 represent almost 75% of the total diseases related with disability-adjusted life years (DALYs). In the first month of life, complications due to prematurity are the leading cause of death and DALYs, the second cause being congenital anomalies, which then holds the leading place until the age of 4 years. Some of these causes of death or DALYs for children are largely preventable as they are especially due to road injury,

The framework of care for CYSCHN comprises the Standards for Systems of Care for Children and Youth with Special Health Care Needs, version 2.0 [43]. It is necessary to coordinate all the parts of care, comprising family professional partnership, medical home, insurance and financing and early and continuous assessment of needs. All these elements have to be culturally and linguistically

#### *The Family as Recipient and Provider of Home Care: A Primary Care Perspective DOI: http://dx.doi.org/10.5772/intechopen.91926*

Respiratory control, maintaining normal temperature (axillar 36.5–37.5°C and rectal temperature 36.6–38°C) in an ambient temperature of 20–22°C, oral feeding skills (breast and/or bottle) and an ascendant parallel growth curve are mandatory for planning the discharge of the preterm infant.

A complete routine screening must be performed before discharge, and ophthalmological, hearing and prematurity anaemia screening must be included in the follow-up programme. The schedule of vaccine is planned according to the chronological age of preterm infant and follows the same schedule of a full-term newborn. Before discharge it is mandatory to observe the preterm infants for a minimum period of 90–120 minutes while sitting in a car safety seat because of frequent oxygen desaturation and episodes of apnoea, hypotonia or bradycardia. Families/caregivers should be trained by hospital staff regarding the proper position in the car safety seat [37, 38].

The parents/primary care providers of preterm infants planned to be discharged from NICU must prove competency in daily care regarding breastfeeding/bottle feeding, care for the infant's skin and genitalia and the way the baby is bathed and clothed. It is necessary for the parents/caregivers to demonstrate some basic medical knowledge on the possible signs of illness, like changes in pattern of feeding or sleep, significance of dry diapers or modifications of the stool aspect. The caregivers must be able to monitor the temperature of preterm baby, evaluate the respiratory pattern and skin colour and assess the muscular tonus. It is important to educate the parents/caregivers about the safe sleep position for preterm infants as they must sleep alone on a plan surface and in a supine position [39]. The readiness of parents to care for newborns discharged from NICU should be assessed in order to improve the discharge preparation process. A parental and nurse survey demonstrated that iterative improvements of the discharge preparation process lead to an increase in family self-assessment of discharge readiness and in nurse assessment of the family's emotional discharge readiness [40].

#### **5.3 Home care for children with special healthcare needs**


**3.6 Ovarian organoids**

Ovarian cancer is spread among a wide range of populations and the development of patient-specific ovarian organoids was found to be one of the precise approaches to enhance the treatment of ovarian cancer [39]. The ovarian organoid can enumerate the morphological heterogeneity of parent tissue and can aid in personalized therapy to enhance the treatment efficiency [46].


**2.2 Modeling of the linear alternator**

The linear alternator consists of two main components, a stator and a translator. The permanent magnets are mounted on the stator and the translator is the moving portion of the machine which is made up of coils. A schematic of a three-phase, "U" shaped linear alternator with permanent magnet (PM) excitation is shown in Fig.3.

The FPLA operates on the same basic physical principles as conventional rotary alternators. The principle that governs the voltage generating operation of the alternator is Faraday's law expressed as [14]:

$$
\varepsilon\_{ind} = -\frac{d\lambda}{dt} = -N\_{coll}\frac{d\phi}{dt} \tag{2}
$$

The permanent magnets create a magneto motive force (MMF) in the air gap between the stator and the winding coils as shown in Fig.4, and it can be described by the following mathematic equation:

Dimensionless Parametric Analysis of Spark Ignited Free-Piston Linear Alternator 275

2

1 cos 0 *F x a M x dx*

> 2 *p*

.

>

(5)

(7)

(8)

.

(6)

4 sin sin 2

>

  2

1 4 sin sin

4 sin sin

0 0 4 sin sin sin

*x Mx M*

*F p*

2 *p*

2 *p F p m x x Bx M x M B*

0 4 sin

*p B M m p g*

Both experimental measurements and numerical calculation by finite element method showed that the flux in the air gap of the PM exited linear alternator in Fig.4 could be

*d B x dA B x Hdx*

Then the total flux contained in the coil of one phase at random position *x* is described by

2

*p*

*x x N HB x dx N M dx g*

0 8 sin sin 2 *p*

*d dx HN M x dt g dt*

*x x p*

*coil coil p x x*

*HN M x*

*coil p*

  1 0 *F a M x dx*

0 0

1 0

2

*F p*

1 0

So the flux density in the air gap due to PM is:

*g g*

assumed to be sinusoidal supporting the above result [16]. Therefore, the flux contained in the differential element *dx* is:

0

8 sin cos

*coil p*

0 2

Thus, the induced electromotive force produced in the coil of one phase is:

*g*

 2

*x b M x dx M*

Where,

Then

Where

the following equation:

Fig. 3. Schematic of a U shaped three-phrase linear alternator

Fig. 4. Model of the linear alternator

$$M\_F(\mathbf{x}) = \begin{cases} 0 & 0 \le \mathbf{x} < \frac{\tau - \tau\_p}{2} \\ M\_p & \frac{\tau - \tau\_p}{2} < \mathbf{x} < \frac{\tau + \tau\_p}{2} \\ 0 & \frac{\tau + \tau\_p}{2} < \mathbf{x} < \frac{3\tau - \tau\_p}{2} \\ -M\_p & \frac{3\tau - \tau\_p}{2} < \mathbf{x} < \frac{3\tau + \tau\_p}{2} \\ 0 & \frac{3\tau + \tau\_p}{2} < \mathbf{x} < 2\tau \end{cases} \tag{3}$$

Where *Mp*= *Hc*·*hm*.

The mean value of the MMF can be obtained from the single-order truncated Fourier series [15]:

$$M\_F\left(\mathbf{x}\right) = \frac{a\_0}{2} + a\_1 \cos\left(\frac{\pi\chi}{\pi}\right) + b\_1 \sin\left(\frac{\pi\chi}{\pi}\right) \tag{4}$$

Where,

274 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

Fig. 3. Schematic of a U shaped three-phrase linear alternator

*F*

*M x*

0 0<x

*p*

*M*

0

2 *F*

3 0 <x<2 2

The mean value of the MMF can be obtained from the single-order truncated Fourier series

*a x x Mx a b*

*p*

*M*

2

*p p*

*p p*

*p*

2 2

*p p*

(4)

  (3)

 <x< 2 2 3

0 <x< 2 2 3 3 <x<

*p*

1 1 cos sin

Fig. 4. Model of the linear alternator

Where *Mp*= *Hc*·*hm*.

[15]:

$$\begin{aligned} a\_0 &= \frac{1}{\pi} \int\_0^{2\pi} M\_F\left(x\right) dx = 0 \\\\ a\_1 &= \frac{1}{\pi} \int\_0^{2\pi} M\_F\left(x\right) \cos\left(\frac{\pi x}{\pi}\right) dx = 0 \\\\ b\_1 &= \frac{1}{\pi} \int\_0^{2\pi} M\_F\left(x\right) \sin\left(\frac{\pi x}{\pi}\right) dx = \frac{4}{\pi} M\_p \sin\left(\frac{\pi \tau\_p}{2\pi}\right) \end{aligned}$$

Then

h

m

$$M\_F\left(\mathbf{x}\right) = \frac{4}{\pi}M\_p \sin\left(\frac{\pi\tau\_p}{2\tau}\right)\sin\left(\frac{\pi\mathbf{x}}{\tau}\right).$$

So the flux density in the air gap due to PM is:

$$B(\mathbf{x}) = \frac{\mu\_0}{g} M\_F \left(\mathbf{x}\right) = \frac{\mu\_0}{g} \frac{4}{\pi} M\_p \sin\left(\frac{\pi \tau\_p}{2\tau}\right) \sin\left(\frac{\pi \chi}{\tau}\right) = B\_m \sin\left(\frac{\pi \chi}{\tau}\right) \tag{5}$$

Where

$$B\_m = \frac{\mu\_0}{\mathcal{S}} \frac{4}{\pi} M\_p \sin\left(\frac{\pi \tau\_p}{2\tau}\right).$$

Both experimental measurements and numerical calculation by finite element method showed that the flux in the air gap of the PM exited linear alternator in Fig.4 could be assumed to be sinusoidal supporting the above result [16].

Therefore, the flux contained in the differential element *dx* is:

$$d\phi = B(\mathbf{x})dA = B(\mathbf{x})Hdx \tag{6}$$

Then the total flux contained in the coil of one phase at random position *x* is described by the following equation:

$$\begin{split} \mathcal{A}(\mathbf{x}) &= \int\_{\mathbf{x}-\tau}^{\mathbf{x}} N\_{coli} H B(\mathbf{x}) \, d\mathbf{x} = \int\_{\mathbf{x}-\tau}^{\mathbf{x}} N\_{coli} \frac{\mu\_0}{g} \frac{4}{\pi} M\_p \sin\left(\frac{\pi \tau\_p}{2\tau}\right) \sin\left(\frac{\pi \mathbf{x}}{\tau}\right) d\mathbf{x} \\ &= -\tau H N\_{coli} M\_p \frac{\mu\_0}{g} \frac{8}{\pi^2} \sin\left(\frac{\pi \tau\_p}{2\tau}\right) \cos\left(\frac{\pi}{\tau} \mathbf{x}\right) \end{split} \tag{7}$$

Thus, the induced electromotive force produced in the coil of one phase is:

$$\varepsilon = -\frac{d\lambda}{dt} = \text{HN}\_{col} M\_p \frac{\mu\_0}{g} \frac{8}{\pi} \sin\left(\frac{\pi \tau\_p}{2\pi}\right) \sin\left(\frac{\pi}{\tau} x\right) \frac{d\lambda}{dt} \tag{8}$$

Dimensionless Parametric Analysis of Spark Ignited Free-Piston Linear Alternator 277

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

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

*dU dV dQ p H H*

*dV p dt*

For the case of compression and expansion process neglecting the crevice flow and the

*dcT dV dQ m p dt dt dt*

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

Substitution and mathematical manipulation yield the following equation which is used to

leakage, the first law of thermodynamics applied to the cylinder content becomes:

*V*

*in*

*dt dt dt*

. *Hi*

*dQ dt*

. . *i e*

*dU*

(15)

. *H e*

(16)

*in pV m RT* (17)

**2.3 Thermodynamic modeling**

thermodynamic process.

constant *R* are constant:

*dt*

Fig. 5. Thermodynamic system of FPLA

calculate the in-cylinder pressure at each time step.

satisfied:

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

$$
\varepsilon \left( t \right) = \left( R\_s + R\_L \right) i\_L \left( t \right) + L \frac{d i\_L \left( t \right)}{dt} \tag{9}
$$

$$i\_L\left(t\right) = \frac{\varepsilon\left(t\right)}{R\_s + R\_L} \left(1 - e^{-\frac{R\_s + R\_L}{L}t}\right) \tag{10}$$

The magnetic force has the opposite direction to the direction of the translator's movement. According to Ampere's law, it is described in the following equation:

$$F\_e = 2N\_{col}B\left(\mathbf{x}\right)i\_L H = 4H^2 N\_{col}^2 B\_m^2 \frac{\left(1 - e^{-\frac{R\_s + R\_L}{L}t}\right)}{R\_s + R\_L} \sin^2\left(\frac{\pi x}{\tau}\right)\frac{d\mathbf{x}}{dt} \tag{11}$$

When it comes to three-phase linear alternator, the third phase is derived from another two according to the following equation [17]:

$$\sin \phi = -\sin \left(\phi + \frac{2}{3}\pi\right) - \sin \left(\phi - \frac{2}{3}\pi\right) \tag{12}$$

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

$$\begin{split} F\_c &= 4H^2 N\_{col}^2 B\_m^2 \frac{\left(1 - e^{-\frac{R\_s + R\_l}{L}t}\right)}{R\_s + R\_l} \frac{dx}{dt} \Big(\sin^2\left(\frac{\pi x}{\tau} - \frac{2}{3}\pi\right) + \sin^2\left(\frac{\pi x}{\tau}\right) + \sin^2\left(\frac{\pi x}{\tau} + \frac{2}{3}\pi\right)\Big) \\ &= 6H^2 N\_{col}^2 B\_m^2 \Big(1 - e^{-\frac{R\_s + R\_l}{L}t}\Big) \frac{1}{R\_s + R\_l} \frac{dx}{dt} \\ &= M \Big(1 - e^{-\frac{R\_s + R\_l}{L}t}\Big) \frac{dx}{dt} \end{split} \tag{13}$$

Where

$$M = 6H^2 N\_{cal}^2 B\_m^2 \frac{1}{R\_s + R\_I}$$

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 is proportional to the speed of the translator:

$$F\_e = M \frac{d\chi}{dt} \tag{14}$$