1 Introduction

Cutting fluids are commonly used in machining processes. They serve a dual purpose: firstly, to provide lubrication, cooling, and facilitate chip removal, thereby extending tool life and enhancing machining quality. Secondly, their use can have detrimental effects on the natural environment, pose health risks to machine operators [1,2,3], and results in high cost and consumption [4,5,6]. In recent years, minimum quantity lubrication (MQL) cutting technology has achieved good machining results [7, 8]. Çamlı et al. [9] investigated the impact of MQL on machining steel parts for rail wheels and found that it reduced surface roughness and tool wear. Ahmad et al. [10] conducted research on tool life and wear mechanisms during AISI 4340 milling, demonstrating increased tool life and reduced wear when employing MQL. Jiang et al. [11] researched the effect of cutting parameters on average surface roughness in MQL during wet and dry cutting and found that the surface roughness under MQL cutting conditions was the best. Huang et al. [12] observed that MQL significantly improved tool life, reduced workpiece surface roughness, and effectively prevented chip deformation. Pervaiz et al. [13] studied the surface roughness, cutting force, and tool life investigation when machining Ti6Al4V with MQL and found that the MQL had the encouraging potential to replace the conventional cutting fluid method. And the MQL is an effective alternative widely used in the machining of aluminum alloy, stainless steel, and so on [14,15,16]. Among them, the performance of the nozzle is an important factor in affecting the performance of MQL, which holds significant research importance. Kumar et al. [17] conducted a study on the influence of nozzle distance on turning temperature when turning EN-31 steel, and they found that the temperature increased with the increase in distance. Zughbi and Rakib [18] pointed out that the angle of the nozzle relative to the horizontal direction was the most important factor determining the mixing time. Zaman and Dhar [19] considered the nozzle size, nozzle angle, air pressure, and oil flow rate to design the double jet nozzle and found that the most influential factor for the nozzle’s performance was the nozzle diameter. Krishnan et al. [20] researched the influence of the positions of the outer two jets in a multi-jet MQL system and showed that the positions of the nozzle can affect the lubricant coverage and cutting force in the cutting area. Maruda et al. [21] studied the effects of nozzle outlet velocity, nozzle distance, angle of the nozzle, and diameters of oil mist on machining performance and found that a smaller droplet of mist can improve the machining performance. Rana et al. [22] analyzed the effects of the nozzle parameters, elevation angle, and spray distance on the processing quality and temperature and obtained the optimal parameters. However, most of the studies mainly focused on the nozzle inlet conditions and relative positions, and there were few studies on the internal parameters of the nozzle, thus indicating the research value in exploring this aspect.

In this paper, the effect of nozzle parameters on MQL is investigated based on computational fluid dynamics and numerical simulation. Nozzle jetting experiments were conducted on existing MQL nozzles, and the internal structure of one commonly used nozzle was examined, leading to the optimization of the nozzle design for MQL. Firstly, we established a nozzle model by observing the physical nozzle. Then, we designed a six-factor, five-level orthogonal experimental scheme to investigate and analyze the influence of different internal structural parameters and pressures on nozzle performance. The three-dimensional model of the nozzle was created, and mesh division was carried out using ANSYS software for numerical simulation. Furthermore, SPSS software was used for post-processing and data normalization of the orthogonal experiments to analyze the order of influence and the change rule of nozzle parameters under different evaluation criteria. The simulation model was validated by comparing it with the results of the nozzle jet experiments. Finally, the optimized design of the micro-lubrication nozzle was achieved, and the comprehensive optimal geometric parameters were obtained.

2 Establishment of nozzle model and CFD numerical simulation

CFD is widely used in various types of fluid simulations [23]. CFD is widely used in various types of fluid simulations [23]. The choice of the CFD method for this study is attributed to its applicability to typical fluid-related problems. In this investigation, we employed an internal-mixing nozzle, where high-pressure gas entered the nozzle through the gas-phase inlet while the cutting fluid was introduced through the central pipe. Subsequently, the cutting fluid and gas were blended in the nozzle’s output pipe and sprayed onto the cutting area. It is noteworthy that research on the internal structural parameters of such nozzles is relatively scarce, rendering it a valuable area of exploration. Therefore, we incorporated the internal structural parameters of the nozzle into our simulation, as illustrated in Fig. 1.

Fig. 1
figure 1

Nozzle and its geometrical parameters. Factor 1, output aperture; factor 2, taper; factor 3, length of gas-phase narrow flow region; factor 4, width of gas-phase narrow flow region; factor 5, output tube length

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In MQL, there is a significant difference in flow velocity between the gas phase and the liquid phase of the micro-lubricating oil. Due to the much higher volume fraction of the gas phase compared to the volume fraction of the micro-lubricating oil, the interaction between micro-lubricating oil particles and the impact of discrete phase particles on the continuous phase can be ignored when conducting CFD fluid simulation. Therefore, the gas phase velocity can be used to approximate the relative velocity, and the main coverage of the cutting fluid in the cutting area can be obtained from the dispersion degree of the gas phase. This work considered the various influencing factors of the nozzle and mainly changed the five model parameters of the nozzle geometry (i.e., width of gas-phase narrow flow region, taper, output tube length, length of gas-phase narrow flow region, output aperture), as well as the pressure of gas phase, so as to control the gas phase axial velocity and gas dispersion degree of the nozzle after ejection. At the same time, a smaller liquid phase output aperture was designed to make the output flow of lubricating oil smaller and the gas–liquid mixing ratio larger, which made the mixing of air and lubricating oil more uniform. The orthogonal experimental method is a multi-factor experimental method based on the orthogonal array [24], and the representative experiments that can represent the overall situation are selected from the full factor tests [25]. In addition, the orthogonal experiment can reduce the test workload and cost [26], and it is very simple and effective for the multi-factor test with optimal parameter combinations [27]. Thus, we chose the above model parameters for experimental study and designed L25 (56) orthogonal tests with six factors and five levels as shown in Tables 1 and 2, where the six factors in Table 2 corresponded to the output aperture, taper, length of gas-phase narrow flow region, width of gas-phase narrow flow region, output tube length, and pressure, respectively. In addition, we plotted the nozzle model as shown in Fig. 2.

Table 1 Parameters of the orthogonal experiment of the nozzle
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Table 2 Orthogonal experimental design for numerical simulation
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Fig. 2
figure 2

Model of the nozzle fluid domain

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Fluent software from ANSYS, Inc., based in the USA, is a general-purpose computational fluid dynamics software used for simulating various complex fluid motions. In this study, the gas phase inside the nozzle was simulated using the 2022 R1 version of Fluent, and the numerical simulation solution steps are illustrated in Fig. 3. The same version of the ICEM meshing software was employed for pre-processing the numerical simulation. A tetrahedral unstructured grid, capable of adapting to complex geometries, was utilized, with local densification near the gas nozzle.

Fig. 3
figure 3

Solution steps of Fluent software

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The discrete phase model (DPM) is a particle-based model for simulating discrete phase flow and particle motion. In DPM, the motion of the discrete phase (particles) is modeled by taking into account collisions between particles, interaction forces between air and particles, etc. Fluent provides a variety of features and options on this model that enable users to better simulate and analyze problems. The model is only suitable for environments where the volume fraction of the particulate phase is less than 10%, while the particle volume is not considered and the interaction force between particles is not considered. In this study, since the volume fraction of the liquid phase is much smaller than that of the gas phase, the DPM model is used, so the continuous phase needs to be solved first, and then the discrete phase is added to the solution domain for coupled solution. The hybrid initialization method is chosen to initialize the fluid domain. For models with complex nozzle internal structures, the hybrid initialization method can improve the initial conditions of the fluid domain and improve the convergence speed. The material of the gas phase inlet is set as air, and “ideal gas” is chosen as the density condition of air, which is affected by pressure and temperature. The inlet temperature is set to 300 K, and the viscosity and specific heat capacity of air are kept as default. The discrete phase model is added by selecting the air-assisted atomizer model, and the specific parameters are set as follows: select the air-assist atomizer model, set the discrete phase as inert particles, set the spray azimuth angle to 0°–360°, set the number of particle streams to 360 so that they are uniformly distributed in all angles, and set the spray time to 0–50 s. According to the structure of the nozzle, the inner diameter of the oil outlet is set to 0 m, and the outer diameter is set to 0.0006 m. According to the structure of the nozzle, the inner diameter of the oil outlet is set to 0 m, and the outer diameter is set to 0.0006 m; the density of the discrete phase is set to 850 kg/m3, the viscosity is set to 0.038 kg/(m·s), the surface tension is set to 0.029 N/m, and the specific heat capacity is set to 1880 J/(kg·K); other parameters and settings are kept as default. For the solution of the continuous gas phase, it is necessary to set the initial inlet condition of the gas phase. The inlet boundary condition of the nozzle model is set as pressure inlet, and the pressure value is set according to the orthogonal test table; the outlet is set as pressure outlet; the operating pressure is set as standard atmospheric pressure; and the temperature setting and gas parameters are kept as default.

In addition, the steady-state pressure-based solver was used as the solution method in this simulation. The realizable k-epsilon model was selected as the turbulence model, and the influence of gravity was ignored. The coupled algorithm was implemented with the second-order upwind scheme, and the pseudo-transient method was activated. The hybrid initialization method was utilized to initialize the fluid zone. The relevant governing equations of the continuous phase are as shown in Eqs. (1)–(6).

The mass conservation equation is shown in Eq. (1).

$$\frac{\partial \left(\rho {u}_{x}\right)}{\partial x}+\frac{\partial \left(\rho {u}_{y}\right)}{\partial y}+\frac{\partial \left(\rho {u}_{z}\right)}{\partial z}=0$$
(1)

where ρ is density, and the ux, uy, uz are the velocities in x, y, z directions.

The momentum conservation equation is shown in Eq. (2).

$$\frac{\partial }{\partial t}\left(\rho {u}_{i}\right)+\frac{\partial }{\partial {x}_{j}}\left(\rho {u}_{i}{u}_{j}\right)=-\frac{\partial p}{\partial {x}_{i}}+\frac{\partial {\tau }_{ij}}{\partial {x}_{j}}+\rho {g}_{i}+{F}_{i}$$
(2)

where the i, j are the denote tensors; ui is the velocity vector; gi is the gravitational acceleration vector; τij is the stress tensor as shown in Eq. (3); Fi is the external volume force in the i direction, which contains other model-dependent source terms; ρ and P are density and pressure.

$${\tau }_{ij}=\left[\mu \left(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}}\right)\right]-\frac{2}{3}\mu \left(\frac{\partial {u}_{t}}{\partial {x}_{t}}\right){\delta }_{ij}$$
(3)

where t and µ are time and molecular viscosity; δij is the unit tensor.

As mentioned, the realizable k-epsilon model was selected as the turbulence model, which has improvements compared with the standard k-epsilon model [28]. And the turbulent kinetic energy k and turbulent dissipation rate ε are shown in Eqs. (4) and (5).

$$\frac{\partial }{\partial t}\left(\rho k\right)+\frac{\partial }{\partial {x}_{j}}\left(\rho k{u}_{j}\right)=\frac{\partial }{\partial {x}_{j}}\left[\left(\mu +\frac{{\mu }_{t}}{{\sigma }_{k}}\right)\frac{\partial k}{\partial {x}_{j}}\right]+{G}_{k}+{G}_{b}-\rho \varepsilon -{Y}_{M}+{S}_{k}$$
(4)
$$\frac{\partial }{\partial t}\left(\rho \varepsilon \right)+\frac{\partial }{\partial {x}_{j}}\left(\rho \varepsilon {u}_{j}\right)=\frac{\partial }{\partial {x}_{j}}\left[\left(\mu +\frac{{\mu }_{t}}{{\sigma }_{\varepsilon }}\right)\right]+\rho {C}_{1}S\varepsilon -\rho {C}_{2}\frac{{\varepsilon }^{2}}{k+\sqrt{v\varepsilon }}+{C}_{1}\frac{\varepsilon }{k}{C}_{3\varepsilon }{G}_{b}+{S}_{\varepsilon }$$
(5)

where

$${C}_{1}={\text{max}}\left[0.43,\frac{\eta }{\eta +5}\right],\eta =S\frac{k}{\varepsilon },S=\sqrt{2{S}_{ij}{S}_{ij}},{S}_{ij}=\frac{1}{2}\left(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}}\right)$$

In these equations, Gk is the generation of turbulence kinetic energy because of the mean velocity gradients; Gb is the generation of turbulence kinetic energy due to buoyancy; YM means the fluctuation caused by diffusion transiting in compressible turbulence; C1, C2, C3ε are constants; σk and σε are the turbulent Prandtl numbers for k and ε. Sk and Sε are user-defined data. And the µt is computed from Eq. (6).

$${\mu }_{t}=\rho {C}_{\mu }\frac{{k}^{2}}{\varepsilon }$$
(6)

where Cµ is a variable computed from the field variables and is not a constant in the standard k-epsilon model [29, 30].

The relevant governing equations of the discrete phase are as shown in Eqs. (7)–(10); the resistance and inertia were considered.

The equation of particle force balance is shown in Eq. (7).

$$\frac{d{u}_{p}}{dt}={F}_{D}\left(u-{u}_{p}\right)+\frac{{g}_{x}\left({\rho }_{p}-\rho \right)}{{\rho }_{p}}+{F}_{x}$$
(7)

where FD(u − up) is the drag force of the per-unit mass of particles, and FD is calculated by Eq. (8).

$${F}_{D}=\frac{18\mu }{{\rho }_{p}{d}_{p}^{2}}\frac{{C}_{D}{R}_{e}}{24}$$
(8)

Here, u is the fluid velocity, up is the particle velocity, ρ is the fluid density, ρp is the particle density, µ is the molecular viscosity, and dp is the particle diameter, Fx is the other forces. And Re is the relative Reynolds number as shown in Eq. (9), and CD is the drag coefficient which uses the spherical drag coefficient model [31], as shown in Eq. (10).

$${\text{Re}}=\frac{\rho {d}_{p}\left|{u}_{p}-u\right|}{\mu }$$
(9)
$${C}_{D}={a}_{1}+\frac{{a}_{2}}{{\text{Re}}}+\frac{{a}_{3}}{{{\text{Re}}}^{2}}$$
(10)

Here a1, a2, and a3 are constants.

As mentioned earlier, the volume fraction of atomized cutting oil as discrete phase medium is less than 10%, so the collision between particles is ignored [32].

The momentum exchange of coupling between discrete phase and the continuous phase is shown in Eq. (11).

$$F=\sum \left(\frac{18\beta \mu {C}_{D}{\text{Re}}}{{\rho }_{p}{d}_{p}^{2}24}\left({u}_{p}-u\right)+{F}_{other}\right){m}_{p}\Delta t$$
(11)

Here F is the change in momentum, mp is the mass flow rate of particles, β is the volume fraction of discrete phase, Δt is the time step, and Fother is the force between other phases.

After the simulation, the results were imported into post-processing software such as CFD-Post for data processing. The processing steps are as follows: First, the “.dat” file of simulation results was imported into the software; second, an XY plane was established and the velocity cloud map was generated on this plane (e.g., Fig. 4); third, a line parallel to the fluid axis was established, the broken line diagram was displayed, and the data of the speed change following the x-axis of the model on this line was displayed (e.g., Fig. 5); fourth, surfaces at 0.1 m, 0.2 m, and 0.3 m away from the nozzle outlet that were perpendicular to the direction of the fluid axis were created, and a straight line across the center of the surface was made to get the velocity change on the line, which reflects the wind coverage (e.g., Fig. 6).

Fig. 4
figure 4

Velocity distribution

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Fig. 5
figure 5

Velocity in the different distances (this result was obtained from the first set of data in the parameter table of the nozzle orthogonal experiment described above)

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Fig. 6
figure 6

Velocity distribution in the jet away from nozzle section (this result was obtained from the first set of data in the parameter table of the nozzle orthogonal experiment described above)

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3 Optimization of nozzle parameters based on numerical simulation

Considering the various influencing factors of the nozzle and the pressure factor during the actual cutting process, the nozzle parameters, including the output aperture (1.5–3.5 mm), the taper (40–60°), the length of the gas-phase narrow flow region (1.0–3.0 mm), the width of the gas-phase narrow flow region (0.2–0.4 mm), the output tube length (9.5–17.5 mm), and the pressure (3.0–5.0 bar), were studied and optimized within a certain range. This numerical simulation uses the orthogonal design of experiment method and the data normalization method to optimize the parameters, taking into account a variety of factors, levels, and the effects of these factors on each other.

In order to verify the simulation results, we verified them by nozzle tests, and the main design experiments measured the axial velocities at 0.1 m and 0.2 m from the nozzle outlet. The nozzle structure parameters used in these tests were the same as those in the seventh group of orthogonal tests. The experimental equipment used the Armorine minimum quantity lubrication system, which can provide stable gas phase input. The anemometer is used to measure the wind speed at different distances from the nozzle axis. The measurement is repeated three times at the same position, and the average value is taken as the wind speed value at the distance. Measuring equipment selection includes a measuring accuracy of 0.1 m/s with an anemometer and a measurement range of 0–100 m/s. In the nozzle experiment, adjust the input parameters of the micro-lubrication system to maintain a stable inlet pressure of 3 bar. A V-shaped fixture will be fixed in the cylindrical section of the anemometer, and a magnetic suction device at the end of the nozzle will be fixed to the nozzle to keep the nozzle jet direction axis and the anemometer measurement end sensor on the same level. Keep the axis of the nozzle jet and the sensor at the measuring end of the anemometer at the same level. In addition, a steel ruler with a length of 500 mm was used as the reference for the distance from the nozzle outlet; the nozzle outlet was placed at the 0-mm scale position; and the fixture holding the measuring instrument was moved horizontally to 100 mm and 200 mm positions. The wind speed at different positions was measured repeatedly, and the average value was taken. The velocity results and average values of the repeated measurements of the nozzle jet experiments are shown in Table 3.

Table 3 Nozzle experimental results
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In the simulation experiment, the nozzle inlet boundary condition was set to pressure inlet, whose value was 3 bar; the outlet condition was set to pressure outlet; and the operating pressure was set to standard atmospheric pressure. The wind speed at different distances from the nozzle outlet was measured in the experiment and also analyzed in the simulation, as shown in Table 4. It can be seen that the simulation value at a distance of 0.1 m from the nozzle and the experimental value error of 18.6%. At 0.2 m for 19%, the simulation value and the experimental value of the error are controlled within 20%. The source of error here is mainly the measurement of the nozzle jet experiments brought about by the error and the fluid simulation of the existence of its own error. For the numerical simulation of CFD, the simulation value and the experimental value of the error control in the 20% range are acceptable. Therefore, the error analysis verifies the correctness of the nozzle simulation model.

Table 4 Comparison of experimental and simulation data
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The results of each nozzle numerical simulation test were sorted out, and the data obtained from the analysis is shown in Table 5. Above these indexes, the attenuation rate of injection direction represents the velocity attenuation at 0.05 m to 0.06 m along the nozzle axis, and the dispersion of spray section represents the coverage of the wind at a certain distance in the direction perpendicular to the nozzle axis. For example, index 4 (dispersion of spray Sect. (0.1 m from the outlet)) is the coverage of the wind at 0.1 m from the outlet of the nozzle.

Table 5 Orthogonal experimental results for nozzle simulation
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After data normalization, the entropy weight method was used for processing, which was an objective weighting method. The weight of each index was calculated by information entropy, which was calculated according to the variation degree. For simulation data, the excellence of the simulation results needed to be evaluated using the evaluation method. Thus, the comprehensive score was set to evaluate the simulation results as the standard in this study. The entropy weight method was adopted in the calculation method of the comprehensive score, and the calculation formula was expressed by Eqs. (12)–(15).

$${P}_{ij}={\psi }_{ij}/{\sum }_{i=1}^{n}{\psi }_{ij}$$
(12)
$${E}_{i}=-\frac{1}{{\text{ln}}n}{\sum }_{i=1}^{n}{P}_{ij}{\text{ln}}{P}_{ij}$$
(13)

where Ψij is the value of the jth item under the ith index, Pij is the proportion of the index value of the jth item under the ith index, and Ei is the information entropy of the ith index. And if the Pij = 0, \(\underset{{P}_{ij}\to 0}{{\text{lim}}}{P}_{ij}{\text{ln}}{P}_{ij}=0\).

$${W}_{i}=\frac{1-{E}_{i}}{n-\sum {E}_{i}}\left(i=\mathrm{1,2},\cdots ,n\right)$$
(14)
$$Z_i=\sum\nolimits_{i=1}^n\psi_{ij}^{'}W_{ij}$$
(15)

Wi is the weight of the ith index, Zi is the comprehensive score of each group calculated according to the weight of each index, and Ψ'ij is the value of the jth item under the ith index after normalization.

For the indexes selected in this study, the performance of the nozzle was better when the absolute speed was higher and the other indexes were smaller. Therefore, the nozzle with the best parameters for the simulation group was selected. And the standardized table of nozzle experimental data is shown in Table 6.

Table 6 Experimental standardized data of nozzle
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3.1 Influence of different parameters on nozzle spray performance

Relevant performance values of nozzles under different parameters are shown in Table 4. IBM SPSS Statistics 24 software was used to test the inter subject effect, and the results of influence of different parameters on nozzle spray performance were analyzed.

By analyzing the absolute velocity of gas at a distance of 0.05 me from the outlet, it can be determined from the analysis that the factors influencing the absolute velocity follow the sequence: width of gas-phase narrow flow region > pressure > output tube length > taper > output aperture > length of gas-phase narrow flow region. Through the analysis of the estimated marginal average, the absolute velocity increased significantly with the increase in the width of the gas-phase narrow flow region, as shown in Fig. 7(a), and also increased with the rise in pressure as shown in Fig. 7(b), and the influence of other factors was not significant.

Fig. 7
figure 7

Estimated marginal average line chart of absolute velocity (0.05 m away from the outlet): a width of gas-phase narrow flow region; b pressure

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The results of the intersubject effect test for the attenuation rate of injection direction showed that the order of factors affecting the attenuation rate of injection direction followed the following sequence: width of gas-phase narrow flow region > pressure > length of gas-phase narrow flow region > output aperture > output tube length > taper. Figure 8 shows the estimated marginal average line chart of the attenuation rate of the injection direction. It can be found that with the increase in width of the gas-phase narrow flow region, the overall trend of the attenuation rate of injection direction was upward, and the attenuation rate increased evidently when the width of the gas-phase narrow flow region was at 0.25 mm–0.35 mm. In addition, with the increase in pressure, the overall trend of the attenuation rate of the injection direction was upward and decreased slightly at 3.5 bar–4.5 bar. And the influence of other factors on the attenuation rate of the injection direction was not irregular.

Fig. 8
figure 8

Estimated marginal average line chart of attenuation rate of injection direction: a width of gas-phase narrow flow region; b pressure

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The results of the intersubject effect test for the velocity at 0.1 m from the outlet were analyzed, and it can be obtained from the analysis that the influence of factors on the absolute velocity at 0.1 m from the outlet followed the following sequence: width of gas-phase narrow flow region > pressure > taper > output tube length > output aperture > length of gas-phase narrow flow region. As shown in Fig. 9(a), the absolute velocity increased prominently when the width of the gas-phase narrow flow region was 0.20 mm–0.35 mm. And the absolute velocity tended to increase as a whole, as shown in Fig. 9(b), and the law of influence of other factors was not obvious.

Fig. 9
figure 9

Estimated marginal average line chart of absolute velocity (0.1 m away from the outlet): a width of gas-phase narrow flow region; b pressure

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By analyzing the intersubject effect test for the dispersion of the spray section at 0.1 m from the outlet, the order of factors affecting the dispersion followed the following sequence: taper > width of gas-phase narrow flow region > output tube length > pressure > output aperture > length of gas-phase narrow flow region. The estimated marginal average line chart of the dispersion of the spray section at 0.1 m from the outlet is shown in Fig. 10. It can be found that with the increase of taper, the dispersion of the spray section fluctuated, there was a minimum value of the dispersion of the spray section when the width of the gas-phase narrow flow region changed; and there was no obvious law in the influence of other factors.

Fig. 10
figure 10

Estimated marginal average line chart of dispersion of spray (0.1 m from the outlet): a taper; b width of gas-phase narrow flow region

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It can be seen from the results of the intersubject effect test for the dispersion of spray at 0.2 m from the outlet that the influence of factors on the dispersion of spray at 0.2 m followed the following sequence: width of gas-phase narrow flow region > output aperture > taper > pressure > length of gas-phase narrow flow region > output tube length. Figure 11 shows the estimated marginal average line chart of dispersion of the spray section at 0.2 m from the outlet. It can be seen that the dispersion of the spray section decreased first and then increased with the rise of the width of the gas-phase narrow flow region, with the minimum value at 0.25 mm. And the dispersion of the spray section fluctuated with the increase in output aperture and had the minimum value at 2.5 mm. In addition, the dispersion of the spray section at 0.2 m from the outlet decreased first, then increased, and reached its minimum value with the increase in taper.

Fig. 11
figure 11

Estimated marginal average line chart of dispersion of spray (0.2 m from the outlet): a width of gas-phase narrow flow region; b output aperture

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3.2 Influence of different nozzle parameters on comprehensive index

According to the above numerical simulation analysis results, the information entropy and weight of each index can be obtained by Eqs. (12)–(15) and Table 5, as shown in Table 7 and 8.

Table 7 Information entropy of each index
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Table 8 Weight of each index
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It can be found that the order of priority affecting the composite index was absolute gas velocity at 0.1 m from the outlet > attenuation rate of injection direction > absolute gas velocity at 0.05 m from the outlet > dispersion of spray section at 0.1 m from the outlet > dispersion of spray section at 0.2 m from the outlet.

The comprehensive scores of each group were calculated by Eq. (15) according to Table 4, 7, and 8, as shown in Table 9. And the result of the intersubject effect test was analyzed, and it can be found that the influence of factors on the comprehensive score followed the following sequence: width of gas-phase narrow flow region > pressure > output tube length > length of gas-phase narrow flow region > output aperture = taper. The comprehensive score increased with the increase in the width of gas-phase narrow flow region and pressure, as shown in Fig. 12. Besides, the comprehensive score decreased with the increase in output aperture, and the influence of other factors on the comprehensive score was not clear.

Table 9 Comprehensive score
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Fig. 12
figure 12

Estimated marginal average line chart of comprehensive score: a width of gas-phase narrow flow region; b pressure

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4 Conclusion

This work investigates the effect of nozzle parameters on MQL performance based on computational fluid dynamics. Taking the nozzle as the research object, we mainly analyzed the internal structural parameters, pressure, and other influencing factors; conducted a CFD numerical simulation of the nozzle using ANSYS Fluent software; verified the simulation model by comparing it with the nozzle jet experiments; realized the optimized design of the nozzle for micro-lubrication; and came up with the following conclusions:

  1. 1.

    For the six factors of the width of gas-phase narrow flow region, taper, output tube length, length of the gas-phase narrow flow region, output aperture, and pressure, their order of priority is different under different evaluation criteria. Among them, the absolute gas velocity at 0.1 m from the outlet was the most important parameter affecting the comprehensive index, and the order of factors followed the following sequence: absolute gas velocity at 0.1 m from the outlet > attenuation rate of injection direction > absolute gas velocity at 0.05 m from the outlet > dispersion of spray section at 0.1 m from the outlet > dispersion of spray section at 0.2 m from the outlet.

  2. 2.

    The optimal combination of nozzle parameters under the comprehensive score as the evaluation standard was an output aperture of 2 mm, taper of 40°, a length of the gas-phase narrow flow region of 1.5 mm, width of the gas-phase narrow flow region of 0.3 mm, output tube length of 15.5 mm, and a pressure of 5 bar.