**2. Analysis of the fuel fluid mechanics for the designed smart sensor**

In this chapter, the behavior of fluid is analyzed by classic interpretation, such as "the second Newton law," where a differential of mass is stored over a differential of

**Figure 1.** *Smart sensor scheme.*

*Optimal Analysis for the Enhancement in the Thermal Variables Measurement by Smart… DOI: http://dx.doi.org/10.5772/intechopen.112676*

**Figure 2.** *Theoretical representation of a differential of mass and volume as part of a fluid.*

volume by density "ρ", which is represented in **Figure 2**. This differential of mass is under gravity force (as a consequence of the gravity field "G"); moreover, the force is a consequence of the gradient of the pressure around the differential volume, since it must be interpreted in the fluid dynamic analysis from the point 1 to the point 2, as well as it is depicted in **Figure 2**.

The Eq. (1) gives the information over a mass differential "*δm*", which is supported by the Newton's second law due to get its dynamic analysis as a consequence of the speed "v", Cartesian space derivative over the pressure "P", differential volume "*δV*", volumetric forces "f and F", in which "f" is a consequence of an external cause that could be an electro valve, furthermore "F" is caused by gradient of "G" effects [8, 9]. Hence, the dynamic of the fluid described above is represented in the Eq. (1).

$$
\delta m \frac{dv}{dt} + \left(\frac{\partial P}{\partial \mathbf{x}}, \frac{\partial P}{\partial \mathbf{y}}, \frac{\partial P}{\partial \mathbf{z}}\right) \delta V = (f + F)\delta V \tag{1}
$$

Eq. (1) was organized considering the fluid density "*ρ*" and the gradient of the pressure "P" to get Eq. (2) [8, 10, 11]:

$$
\rho \delta V \frac{dv}{dt} + \nabla P \delta V = (f + F) \delta V \tag{2}
$$

In order to get the gravity effect analysis over the differential of mass, F force is replaced with gradient G [8, 10, 11]:

$$F = \rho \left(\frac{\partial G}{\partial \mathbf{x}}, \frac{\partial G}{\partial \mathbf{y}}, \frac{\partial G}{\partial \mathbf{z}}\right) \tag{3}$$

Hence, replacing Eq. (3) in Eq. (2), it was achieved Eq. (4) [8, 10, 11]:

$$
\rho \rho \delta V \frac{dv}{dt} + \nabla P \delta V = (f + \rho \nabla G) \delta V \tag{4}
$$

In spite of, the previous Eq. (4), it was possible to obtain the external force "f," which can be useful for the simulation analysis according to get an understanding and applications of smart electro valves, as it is described by Eq. (5). Notwithstanding, this research is focused on smart sensor analysis [8, 10, 11]:

$$
\rho \frac{dv}{dt} + \nabla P - \rho \nabla G = f \tag{5}
$$

Therefore, in Eq. (2), for "f" equal to zero, it is obtained the Eq. (6) [8, 10, 11]:

$$
\rho \delta V \frac{dv}{dt} = (F - \nabla P) \delta V \tag{6}
$$

In addition, Eq. (6) is multiplied by the speed "v" in both of its members, owing to get an energy analysis model, which is given by Eq. (7) [8, 10, 11]:

$$
\rho \, \delta V \frac{d\left(\frac{1}{2}v^2\right)}{dt} = v.(F - \nabla P) \delta V \tag{7}
$$

Eqs. (8) and (9) are quite necessary to get reduction over Eq. (7), from which their demonstration is written in the appendix of this chapter [8, 10, 11]:

$$\frac{d}{dt} = \frac{\partial}{\partial t} + v.\nabla \tag{8}$$

$$\frac{d}{dt}\delta V = \delta V \nabla.\nu\tag{9}$$

From Eq. (8), it is analyzed its modeling over Eq. (7) reduction, which is given by Eq. (10) [8, 10, 11]:

$$
\rho \rho \delta V \frac{1}{2} \frac{dv^2}{dt} = \upsilon.F \delta V + \frac{\partial (P \delta V)}{\partial t} - \frac{d(P \delta V)}{dt} \tag{10}
$$

It means that it is necessary to find a balance between the total derivative " *<sup>d</sup> dt*" and the partial derivative "*<sup>∂</sup> ∂t* " on the time dependence, such as it is explained through Eq. (11) [8, 10, 11]:

$$
\rho \delta V \frac{1}{2} \frac{dv^2}{dt} = \nu . F \delta V + \frac{\partial (P \delta V)}{\partial t} - \left[ P \frac{d(\delta V)}{dt} + \delta V \frac{d(P)}{dt} \right] \tag{11}
$$

Moreover, it was regrouped Eq. (11), owing to find the components on the dependence of time domain for total derivative as energy balance analysis, which is described by Eq. (12) [8, 10, 11]:

$$
\rho \,\delta V \,\frac{1}{2} \frac{dv^2}{dt} + \delta V \frac{d(P)}{dt} - \nu .F \delta V = \frac{\partial (P \delta V)}{\partial t} - P \frac{d(\delta V)}{dt} \tag{12}
$$

On the other hand, returning again the force F over the previous Eq. (12), Eq. (1) was achieved [8, 10, 11].

$$
\rho \delta V \frac{1}{2} \frac{dv^2}{dt} + \delta V \frac{d(P)}{dt} - v. [-\rho \nabla G] \delta V = \frac{\partial (P \delta V)}{\partial t} - P \frac{d(\delta V)}{dt} \tag{13}
$$

Additionally, using Eq. (8) on the previous Eq. (13), it is able to be obtained Eq. (14) [8, 10, 11]:

*Optimal Analysis for the Enhancement in the Thermal Variables Measurement by Smart… DOI: http://dx.doi.org/10.5772/intechopen.112676*

$$
\rho \delta V \frac{1}{2} \frac{dv^2}{dt} + \delta V \frac{d(P)}{dt} + \rho [v.\nabla G] \delta V = \frac{\partial (P \delta V)}{\partial t} - P \frac{d(\delta V)}{dt} \tag{14}
$$

Hence, by Eq. (8) in Eq. (14), it was proposed, as a target, to find the total derivation as dependent on the time over the gravity effect for the mass differential on the fluid, which is given by Eq. (15) [8, 10, 11]:

$$
\rho \delta V \frac{1}{2} \frac{dv^2}{dt} + \delta V \frac{d(P)}{dt} + \rho [v.\nabla G] \delta V = \frac{\partial (P \delta V)}{\partial t} - P \frac{d(\delta V)}{dt} \tag{15}
$$

Therefore, on Eq. (16) is possible to identify the total derivation as dependent on the time for the pressure and gravity effect over the mass differential of the analyzed fluid [8, 10, 11]:

$$
\rho \delta V \frac{1}{2} \frac{dv^2}{dt} + \delta V \frac{d(P)}{dt} + \rho \left[ \frac{dG}{dt} - \frac{\partial G}{\partial t} \right] \delta V = \frac{\partial (P \delta V)}{\partial t} - P \frac{d(\delta V)}{dt} \tag{16}
$$

It means that Eq. (17) separates in both members the total derivative on the time domain, in comparison to partial derivation on the time domain [8, 10, 11]:

$$
\rho \rho \delta V \frac{1}{2} \frac{dv^2}{dt} + \delta V \frac{d(P)}{dt} + \rho \frac{dG}{dt} \,\delta V = \frac{\partial (P \delta V)}{\partial t} - P \frac{d(\delta V)}{dt} + \rho \frac{\partial G}{\partial t} \,\delta V \tag{17}
$$

In fact, it was obtained Eq. (18), from which it was possible to propose a general differential equation under the time domain with the capability to get information on the pressure in the road of the fluid in order to achieve the correlation of the flow with the pressure difference over two points in the fluid road. Furthermore, the correlation between the flow with the pressure difference can give information on the static curve and coefficients that information on the designed sensor parameters [8, 10, 11]:

$$
\delta V \frac{d}{dt} \left( \rho \frac{1}{2} v^2 + P + \rho G \right) = \frac{\partial (P \delta V)}{\partial t} - P \frac{d(\delta V)}{dt} + \rho \frac{\partial G}{\partial t} \,\delta V \tag{18}
$$

The second member of Eq. (18) is zero, while it is analyzed by conservative force, and it is possible to get Eq. (19), which is quite important for designing the algorithm of the smart sensor because it gives the difference between pressure and the fuel flow by Bernoulli in the theoretical model [8, 10, 11]:

$$
\rho \frac{1}{2} v^2 + P + \rho G = \text{constant} \tag{19}
$$
