**2.1. Spray model**

Two parameters for the modeling of droplet breakup based on liquid properties have been introduced by Eggers [31]; these are time and length parameters. For the calculation of length parameter (LP), we take into account viscosity, density and surface tension of liquid, as:

$$\text{LP} = \frac{\text{v}\_f^2 \rho\_f}{\sigma} \tag{1}$$

We have proposed to use LP for spray parameters calculation including Sauter mean diameter (SMD) and spray penetration [32, 33]. Our analysis shows that the spray penetration of biodiesels will be proportional to LP0.1:

$$\mathcal{S}\_{t\rho} = A\_{1\mathcal{P}} (d\_0^{0.5} \rho\_{\text{iv}}^{0.36} \rho\_{\text{g}}^{-0.29}) \text{ LP}^{0.1} t\_{\text{iv}}^{0.5} \tag{2}$$

where *d*<sup>0</sup> is nozzle diameter, *ρ<sup>g</sup>* is gas density, and *pinj* is injection pressure.

It has been suggested in [34] that the better prediction of SMD for diesel and biodiesel fuels can be predicted as:

$$\text{SMD} = \text{3.08} \,\text{v}\_{/}^{0.385} \,\text{\{\sigma}\rho\} \,\,^{0.737} \rho\_{\varepsilon}^{0.08} \,\Delta p^{-0.54} \tag{3}$$

We have used the following expression to calculate the middle droplet diameter of biodiesel fuels:

$$\text{SMD} = \text{23} \, d\_{\text{p}}^{0.35} \, \text{LP}^{0.1} \, \Delta p^{-0.54} \, \rho\_{\text{g}}^{0.06} \tag{4}$$

where Δ*p* is the pressure difference.

For ethanol-kerosene blends for air-blast atomizer, it was found that the SMD can be calculated as [33]:

$$\text{SMD} = \text{2253 } \mu\_l^{0.633} p\_l^{-0.907} p\_s^{-4.565 \times 10^{-6}} \tag{5}$$

#### **2.2. Evaporation model**

The DC model is based on the analytical solutions to the heat transfer and species diffusion equations via the effective thermal conductivity (ETC) model and effective diffusivity (ED) model. The importance of these models can be attributed to the fact that they take into account the recirculation, temperature gradients and species diffusion inside droplets. The heat conduction equation for the temperature *T* = *T*(*t*, *R*) in the liquid phase in a spherical droplet can be presented as:

$$\frac{\partial T}{\partial t} = \chi\_{\text{eff}} \left( \frac{\partial^2 T}{\partial R^2} + \frac{2}{R} \frac{\partial T}{\partial R} \right) \tag{6}$$

where, *t* is the time, *R* is the distance from the center of the droplet, *T* is the temperature and *<sup>κ</sup>*eff is the effective thermal diffusivity.

The time evolution of species mass fractions at any *R* is described by the following Eq. [19, 35]:

$$\frac{\partial \, Y\_{\bar{\imath}}}{\partial t} = \, \, D\_{\imath \bar{\imath}t} \left( \frac{\partial^2 \, Y\_{\bar{\imath}}}{\partial \, R^2} + \frac{2}{R} \frac{\partial}{\partial R} \frac{\partial \, Y\_{\bar{\imath}}}{\partial R} \right) \tag{7}$$

**Carbon no**

C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 C22 C23 C24 C25 C26 C27 Total% **Table 1.**

The diesel fuel composition (molar fractions) used in our analyses [28].

13.65

26.40

14.88

7.62

1.56

16.17

9.15

8.68

1.22

0.66

0.0052

0.0257

0.0155

—

—

—

—

—

—

—

Atomization of Bio-Fossil Fuel Blends http://dx.doi.org/10.5772/intechopen.73180 63

0.0106

0.0319

0.0214

—

—

—

—

—

—

—

0.0221

0.0776

0.0445

0.0169

—

—

—

—

—

—

0.0461

0.1442

0.0755

0.0234

—

0.0473

—

—

—

—

0.1083

0.2889

0.109

0.0488

—

0.0494

—

—

—

—

0.2328

0.5662

0.304

0.0701

—

0.0969

0.0781

—

—

—

0.3756

1.0674

0.5374

0.0935

—

0.2032

0.1638

—

—

—

0.7054

1.5284

0.9961

0.1965

0.1696

0.2848

0.3227

0.2256

—

—

1.0916

2.438

1.3566

0.9901

0.2159

0.3226

0.3403

0.2151

—

—

1.0596

2.1807

1.2848

0.6554

0.3304

0.5234

0.5357

0.2358

—

0.0917

1.0564

1.8041

1.0162

0.6529

0.4001

0.4204

0.5598

0.2072

—

0.1154

1.0442

1.6137

1.0449

0.4921

0.1773

0.4564

0.6684

0.2854

0.2602

0.1705

1.2051

1.9595

1.2353

0.4314

0.1799

0.4782

0.7051

0.3843

0.2589

0.2033

1.0146

1.6351

1.3633

0.5631

0.0914

0.6469

0.8406

0.4879

0.3217

0.0768

0.9646

1.5793

1.2646

0.6611

—

0.6873

1.0652

0.7692

0.3834

—

0.9866

1.6290

1.6951

0.9753

—

1.9149

1.1951

1.4012

—

—

1.1002

2.0628

1.8745

1.0524

—

0.9492

1.3632

2.5290

—

—

1.2635

3.7906

0.6408

0.6926

—

5.3584

1.3157

1.9366

—

—

1.0513

1.9807

—

—

—

3.2357

—

—

—

—

0.308

—

—

—

—

0.497

—

—

—

—

**N-Alkane**

**Iso-alkane**

**Cycloalkane**

**Bi-cycloalkane**

**Tri-cycloalkane**

**Alkylbenzene**

**Indane & tetraline**

**Naphthalene**

**Diaromatic**

**Phenanthrene**

where, *<sup>i</sup>* <sup>&</sup>gt; 1, *D*eff is the effective liquid species diffusivity, *<sup>D</sup>*eff <sup>=</sup> *<sup>χ</sup><sup>Y</sup> Dl* , *Dl* is the liquid diffusivity and *χ<sup>Y</sup>* is a coefficient that varies between 1 and 2.72 [19, 20]. *χ<sup>Y</sup>* takes into account the recirculation inside droplets.
