*2.1. Introduction*

Modeling is the initial stage of the formal approach to the issues related to the analysis of the operation and the synthesis of technical objects. Modeling allows approximating the principles of organization, and principles of operation of the research object, enabling obtaining information about the modeled system. Modeling also allows reducing the cost and time of testing, especially for such complex and miniature elements as injector nozzles.

Models at the evaluation stage, design and production of objects are created for the needs of inference in simulation or experimental studies [20,21]. The model does not reflect a real object but is only a reflection of the knowledge currently possessed, cannot be treated as something permanent and is not subject to change.

An object model is a tool that allows describing an object and its behavior under various conditions through relations on a set of input and output quantities. The purpose of modeling is to obtain a reliable mathematical model that allows tracing the behavior of an object under various conditions. When building a model, the laws of physics are mainly used, which express the balance of forces, moments, flows, continuity equations and geometric relationships, as well as the results of experimental research [20–22].

The structural model of the object shows the connections and the geometric location of the distinguished elements of the object. It is a graphical representation of elements for structural analysis. These models are usually descriptive and graphic [21]. A functional model of the device is a set of functional blocks denoted by, e.g., rectangles, blocks each containing several inputs and outputs, where the injection pump output may be the injection pipe input [21,23]. Appropriate selection of the design features of this subsystem, its control system and supervision of its operation is one of the fundamental problems faced by modern designers, producers and users of engines and injection apparatus.

#### *2.2. Influence of Geometrical Features on the Functioning of the Injector*

The nozzle-hole geometry is one of the important parameters considered in the engine to alter engine's performance, combustion and emission courses. To improve the entire airfuel mixture in self-ignition engines, swirl and turbulence characteristics need modification in the combustion chamber shape and injector nozzle geometry.

The flow characteristics at the orifice outlet for the four investigated nozzles were analyzed in terms of their stationary mass flow, effective outlet velocity and area coefficient [14]. The results showed that divergent nozzles exhibit high cavitation intensity. The geometry of injector nozzles significantly affected the characteristic spray behavior and emissions formation of diesel engines. In paper [14], a nozzle concept consisting of orifices with convergen<sup>t</sup> -divergent shapes was investigated. Three nozzles, characterized by different degrees of conicity, were compared to a nozzle with cylindrical orifices, which acts as a baseline.

In the article [24], a method of calculation of the hydraulic working process of a diesel fuel feed system having an injector nozzle with different positions of its spray holes was tested. Research in self-ignition engine injector nozzle design for two groups of holes was carried out. Inlet edges of the first group of flow were in the sack volume and inlet edges of the second group were on the locking taper surface of the nozzle body. The coefficients of flow in both groups differ considerably and depend on the nozzle needle position. This makes it possible to distribute the injection quantity rationally by spray holes considering the operating conditions of the self-ignition engine and the combustion chamber zones.

Karra and Kong 2010 [11] observed the 10-hole nozzle geometry hole nozzle had the best performance in the engine at full-load conditions. The number and cone of hole variations in the nozzle injector lead to better engine performance and reduced emissions in work by Lahane and Subramanian 2014 [12]. The combined effect of cylindrical shape with the 5-hole nozzle geometry reduced NOx emissions up to 45% and slightly reduced CO emissions of the engine as compared to that of a standard shape by Shivashimpi et al. 2018 [7].

#### *2.3. Hitherto Existing Methods of Optimizing the Geometry of Injector Nozzles*

The static flow rate of high-pressure injector nozzles was characterized through measurements and computations as an optimization process of nozzle design [25]. Prototype injectors were fabricated by modifying multi-zone injection nozzles. Measurements of flow rate were compared with the computations with respect to the nozzle geometry and the needle lift. The initial cone angle was calculated and compared with measurements by the direct photographic imaging method. A parametric study was carried out to terms the flow rate and the initial cone angle for selected design parameters.

Experimental study and optimization were carried out in the layouts and the structure of the high-pressure common rail fuel injection system for a marine diesel engine [26]. A novel optimization solution to improve the steady-state performance of the common rail fuel injection system designed for a ship engine retrofitting was proposed. The tests concentrate on optimizing the hydraulic layouts and the structure parameters to manage

and use the model. In tests, the modified multi-objective genetic algorithm was employed to the reduction of rail pressure fluctuation.

The nozzle needle design concept which was considered in the research project was focused on maximizing the contact area between the needle and its cartridge to reduce needle wear [27]. The injector feeding part was realized using two series of holes. The design was replenished by 3D numerical simulations which indicated the optimal feedinghole number and geometry to obtain a maximum mass flow rate. The study's results indicated that the hole number substantially influences the flow losses along the internal flow path and the global mass flow rate.

Cavitation inside the injector nozzle was observed as a critical phenomenon, which induces a decrease in the flow capacity of the nozzle, but also an increase in the spray outlet effective velocity. Some authors [14] have also performed visualization tests on transparent nozzles, but most of these studies are performed in scaled-up or simplified injector nozzles due to difficulties in production and optical measurements in manufactured geometries.

An experimental research investigation was carried out to analyze the influence of conical and cylindrical orifice geometries on a common rail fuel injection subsystem [1]. This behavior of the injection rate in the different nozzles was characterized by using the non-dimensional parameters of cavitation number, discharge coefficient and Reynolds number. The results evidenced essential differences in the permeability of both nozzle geometries and resistance of the conical nozzle to cavitation.

In the paper, a nozzle flow model was used to design an injector nozzle and obtain initial spray conditions for the common rail injection system [28]. The injector for dimethyl ether needs nozzle holes with larger diameters and a higher sack volume for the same injection duration. Additionally, the needle lift and needle seat diameter should be increased to achieve a minimum flow area ratio.

In the conducted experimental investigation, an effort has been made to enhance the performance of a dual fuel combustion engine utilizing different nozzle orifices [29]. In work, the injector nozzle with 3, 4 and 5 holes, each having 0.2, 0.25 and 0.3 mm hole diameter, respectively, and injection pressure in the injector subsystem (varied from 210 to 240 bar in steps of 10 bar) were optimized. The configurations of re-entrant type combustion chamber and 230 bar pressure in injection subsystem, 4 hole and 0.25 mm nozzle have contributed to maximum performance.

Increasing the injection pressure course in injector space band downsizing the spray holes' orifice diameter has been the principal measures for self-ignition engine to influenced fuel—ambient gas mixture formation and combustion processes [30]. Additionally, the combination significantly accelerates the mixing of fuel and ambient gas and greatly decrease the soot formation.

#### *2.4. Numerical Simulations Description*

In paper [15], simulated performance at different load conditions for a constant speed self-ignition engine was determined using GT-Power software. The critical parameters were optimized for achieving the desired fuel nozzle hole diameter. A very close validation with less than 1% difference is achieved at full load. In comparison, a less than 10% difference is achieved at part loads. The simulation model predicts the power, torque, brake specific fuel consumption, in-cylinder pressure, outlet temperature and NOx with very good accuracy at different loads. The optimized spray hole diameter of the testing engine was validated with the experimental results and compared with the baseline model.

The work in [6] studied a new sub model of simulation model that correlated the discharge coefficient of the injector nozzle with the design variables of the nozzle and injection conditions. This solution enables the prediction of the effect of microvariations in the nozzle hole geometry as well as the variations in the main design parameters on self-ignition evolution and combustion processes. The working cycle simulation model has predicted changes in engine performance and toxic emissions due to injector nozzle design changes.

The authors of the work in [31] showed that separate measurement of these single jet flow injection quantities is necessary to evaluate the injection holes. A measuring adapter for the determination of the injection quantities for individual spray holes was elaborated here. The influence of variation in shape and geometry of the individual nozzle components on singlejet flow injection quantities can be measured and simulated with a measuring adapter. It was demonstrated that the smaller the spray hole diameter, the larger the mass flow difference between the individual nozzle holes due to the increase in variation in the manufacturing process. The macro- and micro-geometrical state of the nozzle hole surface and the transient areas of the nozzle interior into the spray hole have a strong influence on the flow conditions within the range in the space of the nozzle needle seat and within the spray holes and thus on flow stream values and flow symmetry. Design features such as the angle in which the inclination-angle of individual nozzle hole [32], and manufacturing processdriven parameters, such as the shape of the grinding radius of spray hole inlet edge substantially, influence the quantity of fuel mass flux through the individual nozzle hole.

Fuel pressurization up to 300 MPa, as required by modern common rail self-ignition engines, resulted in significant variation of the physical fuel properties (density, viscosity, heat capacity and thermal conductivity) relative to those at atmospheric pressure and room temperature conditions [33,34]. The high acceleration of the fuel at velocities reaching 700 m/s as it is flowed through the nozzle hole orifices to induce cavitation was found. The simulations were assuming variable properties reveal two processes strongly affecting the fuel injection quantity and its temperature. The intense pressure and density gradients at the central part of the spray hole induce fuel temperatures even lower than that of the inlet fuel temperature. The local values can exceed the fuel boiling point and induce reverse heat transfer from the liquid to the nozzle's metal nozzle body. Local values of the thermal conductivity and heat capacity affect the transfer of heat produced at the nozzle surface to the flowing fuel. That creates large temperature gradients within the flowing fuel, which must be considered for accurate simulations of the flow through injector nozzles.

Two different injector designs have been studied: one with sharp-inlet cylindrical holes and one with tapered holes with inlet rounding [33]. The findings indicated considerable changes in the flow development but also bulk flow values such as the volumetric efficiency of the injectors and the mean fuel injection temperature.

In this section, the main achievements of simulation modeling are described, together with a description of specific nozzle geometries studied, are presented. The analyzed state of knowledge shows that individual authors achieved partial results for various design solutions and under different conditions. The problems cannot be considered as solved but some steps have been achieved. Only a few authors have dealt with fuel flow losses resulting in pressure losses and temperature increases. The authors also took up this problem.

## **3. Research Results**

#### *3.1. Simulation of the Fuel Mass Stream through the Spray Holes*

The fuel flow through the multi-hole injector nozzle can be compared to the flow rate when measured with a measuring orifice, e.g., a small orifice (Figure 1). The fuel mass flow *qmi* through the spraying hole, similarly to the measuring orifice or the nozzle, is determined by the relationship [9,22,23]:

$$q\_{\rm mi} = \mu\_r \varepsilon A\_r \sqrt{2\Delta p\rho} \tag{1}$$

where *Ar*—spray hole cross-section, *μr*—flow number, Δ*p*—pressure difference in the injector nozzle space (sack) *pi* and in the combustion chamber *pc*, *ε*—expansion number and *ρ*—density of the flowing fuel, determined for the conditions directly in front of the orifice.

**Figure 1.** Cross-section of a measuring nozzle compliant with ISO 5167-1:1991 (**a**) and a cross-section of an injector nozzle (**b**): *d—*diameter of the throat flow opening, *D—*diameter of the pipeline before the measuring nozzle, *rk—*radius of the nozzle edge rounding on the inlet side, *dr—*diameter of the nozzle hole, *lr—*nozzle hole length, <sup>1</sup>*—*spherical surface of the injector nozzle.

The area of the spray hole can be calculated from the formula:

$$A\_{\rm r} = \frac{\pi d\_{\rm rt}^2}{4} \tag{2}$$

where *drt—*spray hole diameter at the temperature *t*.

Both cylindricity and circularity errors occur during production [35] (Figure 2). In addition, there are material des decrements operation, as well as the deposit formation. Moreover, this diameter changes with temperature; therefore, the hole diameter *drt* at temperature *t* is:

$$d\_{rt}^2 = k\_t d\_r^2 \tag{3}$$

**Figure 2.** The solid model of the injector nozzle (**a**) and image of cross-section of the nozzle hole with rounding (**b**).

The spray hole dimensions change during the measurement with the flow of the medium at a higher temperature. The thermal expansion of the hole considers the correction factor for thermal expansion *kt*:

$$k\_l = 1 + 2\alpha\_l (t\_1 - 20^\circ) \tag{4}$$

where *<sup>α</sup>t*10| *t*120 is the average coefficient of thermal expansion per 1 ◦C from the temperatureof20 ◦Cto *t*1.

There is also an increase in fuel temperature by 1 K at *x* for a pressure increase of 10 MPa [23], whereby the formula can by:

$$t\_{\rm px} = t\_d + (p\_{\rm px} - p\_o)/10\tag{5}$$

where *tpx*—fuel temperature in place of *x*, *td*—fuel inlet temperature, *ppx*—fuel pressure at *x* and *po*—ambient pressure. In addition, the injector nozzle and the fuel heat up from the working medium in the combustion chamber by the Clapeyron equation [22].

The flow number *μr* can be an experimentally determined coefficient for similar flows. In general, it is calculated from the formula [9]:

$$
\mu\_r = k\_1 \; k\_2 \; k\_3 \mu\_o \tag{6}
$$

where: *k*1—correction factor of viscosity taking into account the influence of Reynolds number, *k*2—correction factor of roughness taking into account the influence of roughness of the internal surface of the pipeline in which the differential device is installed, *k*3— correction factor of the inlet edge bluntness, *μo*—the computational number of the flow corresponding to the actual value *μ* determined in the smooth bore for the highest Reynolds number (Remax).

The correction factor of viscosity *k*1 depends on the Reynolds number and the quotient of the flow diameter of the spray hole and the inflow channel *Dt*, which can be written as:

$$k\_1 = f(\frac{d\_{rt}^2}{D\_t^2}, R\_t) \tag{7}$$

The correction factor for viscosity can be read from the diagram for the calculated Reynolds number and the quotient *lrt*/*drt* ([23], p. 449) or a relationship can approximate it.

The correction factor *k*2 depends on the quotient of the flow diameter of the nozzle hole and the inlet channel, the quotient of the surface roughness Δ and the channel diameter *Dt*, and the Reynolds number. Correction factor for nozzle holes *k*2 was calculated from the dependence [23]:

$$k\_2 = \frac{1}{\sqrt{1.5 + \frac{l\_{rl}}{d\_{rl} \left(1.14 + 2l \lg \frac{d\_{rl}}{\Lambda}\right)^2}}}\tag{8}$$

where *lrt*—nozzle hole length at temperature *t* and Δ—hole surface roughness.

The roughness of the surface of the injector nozzle sack was determined according to industry requirements and measurements using a roughness gauge for the sections of the spray channels.

The correction factor for the inlet edge defocus *k*3 depends on the quotient of the flow diameter of the nozzle hole and the inlet channel, the channel diameter in front of the hole *Dt* and the edge radius *rk* on the upstream inlet side of the spray hole. The correction multiplier *k*3 was determined based on the quotient of the edge rounding radius and the diameter of the spray hole *rk*/*drt* and the hole module *m*:

$$m = \frac{d\frac{2}{rt}}{D\_t^2} \tag{9}$$

This relationship was approximated by a second-order polynomial in the form:

$$k\_3 = -6 \times 10^{-6} \left(\frac{r\_k}{d\_{rt}}\right)^2 + 0.01 \frac{r\_k}{d\_{rt}} + 10.8 \tag{10}$$

The design number of flow *μo* corresponds to the actual value determined in a smooth pipeline for the highest Reynolds number (*Remax*) and the quotient of the flow diameter of the spray hole and the inflow channel.

The Reynolds number can be calculated for the diameter ahead of the nozzle hole at temperature *t* or for the diameter *drt* (radius *rrt*):

$$Re\_d = \frac{c\_{\mathcal{g}} 4r\_{ht}}{\upsilon\_{p,t}} = \frac{c\_{\mathcal{g}} 4r\_{ht} \rho\_{p,t}}{\eta\_{p,t}} \tag{11}$$

where *cg*—average fuel flow velocity in the minimum cross-section, *rht*—hydraulic radius, *vp*,*<sup>t</sup>*—fuel kinematic viscosity for temperature *t* and pressure *p* and *<sup>η</sup>p*,*<sup>t</sup>*—absolute viscosity for temperature *t* and pressure *p.*

The kinematic viscosity can be calculated from the relationship:

$$\nu\_{p,t} = \frac{\eta\_{p,t}}{\rho\_{p,t}} \tag{12}$$

The expansions number *ε* takes into account the change in the specific volume of the compressible fluid related to the pressure reduction as it flows through the spray hole. Fuel is a compressible liquid subordinated to Hooke's law with the modulus of elasticity [23].

The relationship between temperature and fuel density was also determined by the following relationship [3,23]:

$$
\rho\_{p,t} = \rho\_{15} - \Delta\rho(t - 15) \tag{13}
$$

where Δ*ρ* is the reduction of fuel density after heating by 1 ◦C, Δ*ρ* = 6.5 kg/m3.

The viscosity of the fuel is important due to the use of different liquid fuels of different viscosities for the supply of reciprocating internal combustion engines.

The pressure losses on the way from the injection pump drive to the injector nozzle were calculated for two engine types in [19,20]. The absolute pressure loss Δ *ω* in the spray hole depends on the type of nozzle hole, its constriction, the flow rate and the differential pressure. For the shape compared to the orifice, it is calculated based on the formula [22]:

$$
\Delta\omega = \frac{\sqrt{1-\beta^4} - k\_2 2}{\sqrt{1-\beta} + k\_2 \beta^2} \Delta p \tag{14}
$$

where *β*—spray hole narrowing is equal. The narrowing *β* of the spray opening is defined by:

$$
\beta = \frac{d\_{rt}}{D\_t} \tag{15}
$$

Figure 3a shows the influence of surface roughness on pressure losses when flowing through the spray hole. This relationship has a minimum of 0.05 μm. Figure 3b shows the effect of the spray hole length on pressure losses. As the surface roughness increases and the spray hole length increases, the irreversible pressure loss increases non-linearly with related variables.

Relative pressure losses were calculated from the relationship:

$$
\Delta \overline{\omega} = \frac{\Delta \omega}{\Delta p} \tag{16}
$$

where Δ*p* = *pi* − *pc*—pressure difference in the nozzle sack and the combustion chamber.

The fuel mass flux *qm*, when flowing through the number of *i* spray holes can be determined by the following relationship [9]:

$$q\_m = \mu\_o k\_1 k\_2 k\_3 \varepsilon i \frac{\pi d\_{rt}^2}{4} \sqrt{2(p\_i - p\_c)\rho\_{p,t}} \tag{17}$$

**Figure 3.** An example of the influence of surface roughness (**a**) and the length of the spray hole (**b**) on pressure losses Δ*ω*.

The value of these features is determined at the production stage. The final formula for the fuel mass flux was calculated by the following dependence:

$$
\eta\_m = \mu\_r \varepsilon i \frac{\pi d\_{rt}^2}{4} \sqrt{2(p\_i - p\_c)\rho\_{p,t}} \tag{18}
$$

Experimental measurements of the fuel mass flux and determination of flow characteristics for new and used injector nozzles were carried out on a flow test stand using calibration oil for many years [9,10]. These measurements made it possible to experimentally determine the flow coefficient *μr* and the spray holes' average area. The pressure courses in the combustion chamber were calculated and measured using a resistance sensor with digital processing.

The mass of the injected fuel as a result of the pressure loss is lower:

$$q\_{m} = \mu\_{r} \varepsilon i \frac{\pi d\_{rt}^{2}}{4} \sqrt{2(p\_{i} - p\_{c}) \Delta \overline{\omega} \rho\_{p,t}} \tag{19}$$

Thus, the fuel mass flux, which should be maximized due to flow losses, depends on the following geometrical values and results from the type of fuel and the design of the internal combustion engine, resulting in the parameters of the working medium:

$$q\_m = f\left(i, d\_{rt}, D\_t, \Delta, r\_{\mathbf{k}\prime} \rho\_{p,t\prime}, \eta\_{p,t\prime}, t, \Delta p, \Delta \varpi\right) \tag{20}$$

The latter relationship will be used to optimize for surface roughness losses Δ, the radius of the spray orifices on the inlet side *rk*, the influence of temperature *t* and pressure Δ*p* and pressure losses Δ*ω*. This means that when Δ*ω* = 1, then there are no friction losses, and when Δ*ω* = 0, there are such large friction losses that the mass flux of dosed fuel is equal to 0. In the simulation, the change in Δ*ω* was assumed in the range from 0.3 to 1.0, in steps of 0.1. Various functions can approximate such a relationship.

From the point of view of friction losses, the value of pressure losses Δ*ω* is interesting, because the quantity is irretrievably lost, depending on the characteristics of the technical state determined at the production stage and changing during operation:

$$
\Delta = f\left(i, d\_{rt}, D\_{t}, l\_{r}\Delta, \rho\_{p,t}, \eta\_{p,t\*}, t, \Delta p\right) \tag{21}
$$

Therefore, there is a need to build a simulation model due to the optimization goal.

Based on the presented dependencies, the influence of selected design features on the pressure loss at the flow through the nozzle holes was presented. Figure 4a shows the effect of the number of flows through the nozzle holes on the pressure curve in the nozzle space. The diagram 4a shows that the influence of the spray hole flow coefficient *μr* on the pressure course in the nozzle space is significant. Therefore, there is a need to build a simulation model due to the optimization goal.

**Figure 4.** Influence of the value of the flow coefficient *μr* on the pressure course in the injector nozzle space *pi* (**a**) and influence of the relative pressure losses Δ*ω* through the spray holes on the fuel mass flow *qm* (**b**).

On the other hand, Figure 4b shows the influence of the relative pressure drop on the mass fluid of the dosage fuel. The figure also shows a significant decrease in the mass fluid of the dosed fuel with an increase in the relative pressure loss caused by friction in the geometric profiles of the spray holes. Figure 4b shows the influence of relative pressure losses through the spray holes on the fuel mass fluid.

Experimental measurements were also made in the injection subsystem outside the engine and on actual selected internal combustion engines in the Marine Power Plants Laboratory of the Maritime University of Szczecin and on sea-going ships.

The total pressure in the injector space *pi* is the sum of the static pressure *ps*, which is the residual pressure *pr*, and the dynamic pressure *pd* depending on the speed of the pumped fuel:

$$p\_i = p\_s + p\_d = p\_r + c\_r^2 \frac{\rho\_{p,t}}{2} \mu\_r \varepsilon\_s + p\_\varepsilon \varepsilon\_s \tag{22}$$

where *pr*—residual pressure, *<sup>ε</sup>s*—controlling indicator, taking the values 1 (when *pi* ≥ *po*) or 0 (when *pi* < *po*) and *po*—nozzle opening pressure. Each throttling in the injection subsystem is associated with a pressure drop, including, among others, the flow factor [23]. The results of the comparison of modeling and measurement on a real engine are shown in Figure 5. The measure of compliance of these two waveforms is the value of the correlation coefficient [5], which was 0.847, which indicates a close relationship. This compatibility is satisfactory, but at the same time, encourages further improvement of the modeling and measurement method of the pressure course with the use of the Kistler sensor.

The course and these are consistent with those presented in the literature [36]. Relative pressure losses may exceed 50%, which leads to significant weight losses of the dosage fuel.

Additionally, carried out experimental measurements of the friction force between the body and the needle of the injectors of new and used injectors, which resulted in the patent award PL 234 940 B1. The friction forces varied, and in some cases very high, expressed in tons.

#### *3.2. Optimization of the Selection of the Geometry of Injector Nozzles*

In technical or economic sciences, optimization is one of the essential research tools. The use of more and more computing power of modern computers has allowed for the development of new optimization methods. Genetic algorithms, gradient methods, evolutionary algorithms, neural networks, finite element method are examples of optimization methods widely used in scientific research [37–39]. In recent years, the application of mathematical optimization methods to practical solutions to material engineering problems [2,24,26,40] has been developed. The main task of optimization is the appropriate formulation of the analyzed problem, defining the set of searches and the optimization

objective function. For this purpose, an area of permissible states *Sp* is introduced among the entire space of decision variables (state space *Ss*). If the objective function *f*: *Ss* → *R* and *Sp* ⊂ *Ss*, then the optimization task is to search for an element *<sup>x</sup>op<sup>t</sup>* ∈ *Sp*, that:

> ∀*x* ∈ *Sp f xopt* ≤ *f*(*x*), (23)

in the case of minimizing the objective function and

$$\forall \mathbf{x} \in S\_{\mathcal{P}} f(\mathbf{x}\_{\text{opt}}) \; \ge \; f(\mathbf{x}), \tag{24}$$

in the case of maximizing the objective function.

**Figure 5.** Comparison of the pressure course of the calculated *pic* in the injector nozzle space and measured before the injector valve *p*im.

Gradient optimization methods use, in addition to the known values of the objective function, also its gradient values. They can be used only in those problems where the objective function has the form *f*: *f*: *Rn* → *R* and is differentiable in the *Rn* space.

The strategy of gradient methods comes down to two stages:


In the case of the conjugate gradient method used in this paper, the search direction is determined by designating conjugate directions. This method allows for the minimization of the objective function in *n* steps, where *n* is the number of variables present in the objective function [40].

#### *3.3. Optimization of the Parameters of Multi-Hole Nozzles*

The research subject is multi-hole nozzles, both conventional and with electronically controlled injection, and the aim of the research is the impact of changes in the design of nozzles on fuel flow losses. The research object is a four-stroke medium-speed marine engine. The dependencies of the pressure course in the atomizer well and the fuel stream flowing out through the spraying hole for a given fuel setting can be approximated by various functions. Based on the formulas (8), (14) and (15), the pressure loss Δ*ω* can be represented as a function of six variables:

$$\Delta\omega = f(d\_{lt}, D\_{lt}, \Delta, l\_{lt}, p\_i, p\_c) = \frac{\sqrt{1.5 \left(D\_t^4 - d\_{rt}^4\right) + \frac{\left(D\_t^4 - d\_{rt}^4\right) l\_{lt}}{d\_{rt} \left(1.14 + 2\log\left(\frac{d\_{rt}}{\Lambda}\right)\right)^2}} - d\_{rt}^2}{\sqrt{1.5 D\_t^3 (D\_t - d\_{rt}) + \frac{D\_t^3 (D\_t - d\_{rt}) l\_{rt}}{d\_{rt} \left(1.14 + 2\log\left(\frac{d\_{rt}}{\Lambda}\right)\right)^2} + d\_{rt}^2}} \cdot (p\_i - p\_c) \tag{25}$$

four of which relate to the geometric parameters of the atomizer: hole diameter *drt*, the diameter of the inflow channel to the hole *Dt*, surface roughness Δ and channel length *lrt*. These four variables were analyzed in terms of minimizing pressure losses Δ*ω*. As the form of the function *f* is too complex, the analytical solution turned out to be impossible to implement. Therefore, in order to optimize the function *f*, the numerical conjugate gradient method was used. The following restrictions were imposed on the values of the variables:

0.00020 m < *drt* < 0.00032 m; 0.0004 m < *Dt* < 0.0010 m; 0.001 m < *l* < 0.002 m; 2.5×10−<sup>8</sup> <Δ<2.8×10−<sup>7</sup> m.

The pressure increase was assumed at the average level: Δ*p* = 2 × 108 Pa.

For the assumed limits of variability, the function *f* reached the minimum for the extreme values of the variability ranges of the atomizer geometric parameters: Δ*ωmin* = 5.22858 × 10<sup>7</sup> Pa for *dr*t = 0.00032 m, *Dt* = 0.0004 m, *lr*t = 0.001 m and Δ = 2.5 × 10−<sup>8</sup> m.

The course of the function variability depends on the individual variables, with the values of the remaining variables fixed, witch makes it possible to determine the influence of these variables on the rate of changes in the value of the function Δ*ω*.

In the case of Δ, the change in the value of pressure loss Δ*ω* is the smallest and amounts to 1.03312 × 10<sup>6</sup> Pa, which is about 2% of the minimum value of Δ*ω* (Figure 6).

**Figure 6.** The course of absolute pressure loss Δ*ω*, as a function of surface roughness Δ.

In the case of *lrt*, the change in the value of pressure loss Δ*ω* amounts to 1.55469 × 10<sup>6</sup> Pa, which is about 3% of the minimum value of Δ*ω* (Figure 7).

In the case of *Dt*, the change in the value of pressure loss Δ*ω* is the biggest and amounts to 1.48785 × 10<sup>8</sup> Pa, which is about 285% of the minimum value of Δ*ω* (Figure 8).

In the case of *drt*, the change in the value of pressure loss Δ*ω* amounts to –1.17087 × 10<sup>8</sup> Pa, which is about 240% of the minimum value of Δ*ω* (Figure 9).

In the case of the variables Δ and *lrt*, *Dt*, as the variable grows, the values of Δ*ω* also increase. Only for hole diameter *drt*, the values of Δ*ω* decrease. The variables *Dt* and *drt* have the greatest influence on the change of the pressure loss value.

**Figure 7.** The course of absolute pressure loss Δ*ω*, as a function of channel length *lrt*.

**Figure 8.** The course of absolute pressure loss Δ*ω*, as a function of diameter of the inflow channel to the hole *Dt*.

**Figure 9.** The course of absolute pressure loss Δ*ω*, as a function of hole diameter *drt*.
