Numerical Evaluation of the Influence of the Outlet Nozzle Diameter of a Piezoelectric Gas Injector on Its Flow Properties

The paper presents the results of flow analyses of a piezoelectric gas injector with pulse action. The conceptual solution, due to the use of a piezoelectric actuator, is characterized by shorter opening and closing times. This promotes the precision of fuel dosing. It also meets current trends in environmental protection. The innovation in the injector under study is the use of a beam actuator comprising three layers, two active and one passive. The electromechanical characteristics of the converter were determined based on the method in which the actuator is treated as a bent beam with a locally implemented piezoelectric segment. Based on such assumptions, the deformation shape of the actuator and, as a result, the opening degree of the injector valve were determined. Flow studies were carried out using Computational Fluid Dynamics (CFD) in the ANSYS Fluent environment. The RANS (Reynolds-averaged Navier–Stokes) approach and the k - ω SST turbulence model were adopted. The solution was sought based on the standard SIMPLE scheme and polyhedral mesh. It allowed the determination of the flow characteristics of the analysed injector, with variable valve lift and outlet nozzle diameter.


Introduction
Further use of fossil fuels (petrol and diesel) signi cantly a ects the volume of oil reserves in the ground. erefore, it is justi ed to work on the use of alternative fuels as well as diversi cation of fuel production and its components. e most widely used alternative fuels in transport are lique ed petroleum gas (LPG) [1] and compressed natural gas (CNG) [2]. ese fuels, depending on the complexity of the classical engine supply system, are supplied using new-build fuel systems, which in most cases are universal [3], although dedicated installations for the type and model of the engine are required for the latest designs [4] LPG and CNG are also used to power o -highway machinery [5,6]. ere is also growing interest in hydrogen (H2) power in transport [7].
Some hopes are also placed on further research into the use of vegetable oil, where pure vegetable oil (PVO) [8] and fatty acid methyl esters biodiesel (FAME) [9] can be distinguished. In the case of the recently presented fuels, the costs incurred for ancillary charges like excise tax are decisive, as the production process itself has already been mastered and implemented. In energy applications where the aim is to generate electricity (generators), biogas is largely used [10]. An important aspect of biogas, just like with vegetable oil, is the puri cation process. is process generates additional costs, but the use of fuel without adequate puri cation compromises the combustion process [11], which can eventually lead to damage to the power unit.
Piezoelectric transducers have found widespread use in recent years in the automotive industry, where great emphasis is being placed on reducing CO 2 emissions. In transport vehicles, this can be achieved, among other things, by reducing vehicle weight, engine displacement, turbocharging, and the use of lower carbon fuels. In gas supply systems based on LPG in the vapour phase or CNG, sequential injection systems are most commonly used. eir integral parts are low-pressure gas injectors whose task is to deliver a speci c dose of fuel to the combustion chamber at the right time. e existing design solutions of injectors in fuel metering systems usually use electromagnetic circuits that induce movement of the valve's working element, such as a piston, disc, diaphragm, or ap. e aim of the study is to determine numerically the in uence of the outlet nozzle diameter of the conceptual piezoelectric gas injector on its ow properties. e knowledge of such characteristics is essential for the appropriate con guration of the fuel system. It also becomes useful in engine energy calculations. As a result, the appropriate con guration of the fuel system favours the reduction of exhaust gas toxicity. e introduction should be succinct, with no subheadings. Limited gures may be included only if they are truly introductory and contain no new results.

Object of the Analysis.
e piezoelectric injector analysed ( Figure 1) is a conceptual solution developed on the basis of a work of Szpica et al. [12]. A piezoelectric converter is xed in the rear part of the injector, between upper and bottom casings. is type of xing identi es the converter with a beam element xed on one side. e injector housings are sealed between each other and connected using a screw connector. In the state without electrical supply, the gas owing through the inlet lls the injector internally. e appearance of electric power in the electrical connection results in the deformation of the piezoelectric converter and the opening of the valve. Gas ows in the direction of the outlet. At this stage, the gas ow can be adjusted by the degree of opening of the valve and the inner diameter of the nozzle. e degree of opening of the valve results from the limiter setting, which can be changed. e limiter setting is determined for the maximum diameter of the injector outlet (without nozzle). en, in order to adjust the injector output to the engine demand, it is necessary to use a nozzle with an appropriate internal diameter. is type of injector belongs to the low-pressure gas-phase injector group. Its characteristic feature in relation to solutions with the electromagnetic drive is, in addition to the speed of operation, the fact that the piezoelectric converter works not only during the opening but also during the closing of the injector valve.

Calculation Methods.
e electromechanical characteristic of the actuator was determined based on the method proposed in works [13,14]. In this method, the actuator is treated as a bending beam, in which the so-called piezoelectric segment PS is locally implemented (Figure 2). PS consists of three components-two piezoelectric (active) layers and one passive layer. e heights and thicknesses of the individual layers are identical, and their values are t and b, respectively. In the transducer (Figure 2), the left side is xed while the right side can move freely. In the actuator, two characteristic ranges can be determined, related to the change of load and sti ness. In the range 0 < x < x 1 , there is a PS (generating an electrical load Me, caused by an applied voltage v), with a exural sti ness e second interval is a homogeneous beam with sti ness E b J b . In the latter, the force F due to the gas-intake manifold pressure di erence is applied (at a distance of x 2 from the transducer xed side). e force F causes reactions in the xed support, which are, respectively: In order to simplify the mathematical model of the actuator, the following assumptions were made: (i) Component bending occurs according to Euler's hypothesis, and the radii of curvature of the deformed components are equal (ii) In the plane of connection of the components, there is no intermediate layer and there is no slipping (iii) ere is a transverse piezoelectric e ect 1-3 in the piezoelectric layer, causing pure bending Taking into account the above assumptions, the di erential equation for actuator de ection was written as follows: where -a coe cient to take account of the change in sti ness at the location of the PS, E p , E b -Young's moduli of piezoelectric and passive elements, p )-bending moment due to electric load [14], and d 31 -piezoelectric constant.
After double integration of formula (1), taking into account the following boundary conditions: (zy/zx)(0) 0, y(0) 0, a close form solution describing the electromechanical characteristics of the transducer has been obtained as the following formula: In the electromechanical tests, the geometrical dimensions of the piezoelectric converter were assumed to be constant and were, respectively: passive layer length: L 46 mm, passive/active layers width: b 15 mm, active layer length: x 1 41 mm, layers thickness: t 0.25 mm, and coordinate of the point at which force F is applied: As for the materials used for the individual transducer layers, it was assumed that the active component is made of PTZ5H soft piezoelectric ceramics (E p 62.1 GPa and d 31 − 320 pC/N, [16]), whereas the passive layer is silicon oxide (E b 73 GPa [17]). e use of such materials guarantees the lowest energy consumption of the fuel dosing process [18]. Using the obtained analytical solution formula (2), the electromechanical characteristics of the transducer were developed for di erent values of the e ective valve opening h eff (Figure 3(b)).
In order to obtain characteristics with the desired h eff parameter, we determined at a distance x 2 from the xed end of the transducer (Figure 2), and the value of the applied voltage v and the force F had to be changed in formula (2). e applied electric voltages v for di erent h eff levels are shown in Table 1.
Force F, generated by the pressure di erence between gas and intake manifold, occurs only in the closed state of the injector (at the moment of valve unsealing h eff ≤ 0.01 mm [19]). us, at the assumed opening levels of the e ective valve, it was always assumed that F 0 N. e obtained electromechanical characteristics (Figure 4) were used in ow tests to accurately reproduce the shape of the transducer.
In the numerical analysis of the ow, the Ansys Fluent software based on the RANS (Reynolds-averaged Navier-Stokes) approach was used [20]. e Navier-Stokes equations were supplemented in this case with a Reynolds distribution model. Finally, the continuity and momentum formulas (3) and (4) were obtained for incompressible ow: where u i and u j -mean velocities, u i ′ and u j ′ -the uctuating part of the velocity, p-dynamic pressure, δ ij -Kronecker delta, and ρ-uid density. e member (− ρu i ′ u j ′ ) of formula (4) represents the Reynolds stress. Due to the openness of the equations, it was necessary to use the Boussinesq approximation formula: where μ t -turbulent viscosity and k-kinetic energy of turbulence.
b y Mathematical Problems in Engineering e study uses the k-ω SST turbulence model combining the k-ε and k-ω models [21], for which the basic transport equations are of the form formulas as follows: where G k and G ω -production terms of k and ω, Y k and Y ω -dissipation terms of k and ω, Γ k and Γ ω -e ective di usivity of k and ω, and D ω -cross-di usion term. For the numerical analysis, a solid model of the uid inside the injector was created in Ansys Geometry (Figure 3(a)). Ultimately, the model was restricted to the part located in the area of the injector valve (Figure 3(b)), omitting remote parts, which should not fundamentally a ect the ow process. e opening degree of the injector valve was determined as the value of h eff in the axis of the outlet hole. is corresponded to the α angle formed between the valve seat and the seal tted to the part of the piezoelectric converter located outside the converter's operating area. e value of the α angle was determined on the basis of the characteristics shown in Figure 4. Table 2 shows the valve swing angles determined from Figure 4. A linear approximation of a 12 mm section from the valve centre was used (Figure 3). It was assumed that the valve is always on axis because the swing angles are small.
In further steps, a tetrahedral nite element mesh was created in Ansys mesh for each case analysed.
e main parameters adopted were as follows: normal curvature angle 10 deg, min size 8 × 10 − 4 mm, max face size 0.2 mm, max element size 0.4 mm and growth rate 1.2, and in ation with   following parameters: max layers 4, growth rate 1.2, and noslip variant on the walls. Comparing the obtained values of the mesh quality parameters such as skewness oscillating around 0.21 and orthogonal quality close to 0.88 with the declarations contained in [22], they should be considered satisfactory. Prior to Fluent calculations, the tetrahedral mesh was transformed into a polyhedral mesh, which essentially reduces calculation time with acceptable results [23][24][25][26]. A standard SIMPLE scheme was used in the calculations with control residuals speci ed at 1 × 10 − 4 . e discretization of the continuity and momentum equations was carried out using a pressure-based solver. Air was used as the uid because the low-pressure gas-phase injectors are experimentally studied in this way.

Results and Discussion
e analyses took into account the following (according to Figure 3 Figure 5(a), is re ected in the polyhedral mesh ( Figure 5(b)). e de ection of the valve plate results in a higher concentration of the stream owing through the valve gap in the part without the piezoelectric converter (A in Figure 6(a)). e signi cant in uence of the injector nozzle on the ow velocity and jet shape (B in Figure 6(b)), which determines the ow properties, becomes visible.
For individual variants, the ow value at the given boundary and initial conditions was read from the injector outlet surface as volumetric ow rate Q and converted to L/ min. As a result of the performed calculations, the ow characteristic presented in Figure 7 was obtained. It shows a gradual increase in Q value as the injector opening degree increases. In the case of the smallest of the analysed nozzle diameters, d 1.5 mm, the maximum Q value stabilises at h eff 0.2 mm, while at d 2 mm, at about h eff 0.35 mm. In other cases, it increases as the degree of opening increases, although above h eff 0.4 mm, the gradient is smaller. Also shown is the di erence in the values obtained from those presented in [18], where the plate seal was positioned parallel to the valve seat.  Additionally, the ow characteristics are presented in the "Tanaka" coordinate axes, i.e., as a function of (Q/Q max ) f(h eff /d) [27]. Flow characteristics of the injector according to "Tanaka' is given in Figure 8. In this way, the ow capacity of the injector is inferred with respect to the maximum ow value obtained from the calculations at the maximum degree of opening and no outlet nozzle. e nozzle grading adopted in the calculations yielded a ow variation of 0.2 (Q/Q max ).

Conclusions
e numerical calculations presented in the paper allowed the ow properties of the piezoelectric gas injector to be assessed. e methodology adopted for creating the uid model, mesh, and the method of solving uid mechanics problems using ANSYS software allowed, apart from a quantitative evaluation, also a qualitative assessment. e ow characteristics obtained as a result of the calculations indicated, apart from the maximum values of the volumetric ow rate, also the limits of the injector valve opening degree, at which the values determining the ow rate do not change or the gradient of changes is smaller. By taking the ow characteristic in the form of the "Tanaka" characteristic, the percentage di erence of the volumetric ow rate corresponding to the jump in the opening degree of the injector valve was determined. On this basis, in addition to adjusting the nozzle diameter, a maximum valve opening degree can be suggested. Changing the diameter of the injector nozzle requires its replacement while varying the maximum opening degree is easier to implement. is can be done by a mechatronic system controlling the position of the maximum opening degree limiter. All measures aimed at increasing the precision of fuel dosage are conducive to reducing exhaust gas toxicity.
Data Availability e data used to support the ndings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no con icts of interest.  Mathematical Problems in Engineering 7