1 Introduction

The current pursuit to achieve sustainable manufacturing is one of the most important challenges in the manufacturing industry [1,2,3]. The possibility of sustainable manufacturing using abrasive machining processes depends mainly on neutralizing the risks posed by the use of cooling and lubricating fluids (coolants) [4, 5]. Currently, they are still commonly used in the form of oils or aqueous oil emulsions and are most often supplied to the machining zone in high volume mode using the flooding (WET) method with high flow rates, up to 15 l/min (900,000 ml/h) [6, 7]. Despite their known advantages [8], they are nevertheless considered undesirable due to the risk to the environment [9], the health of machine tool operators [10], and the high cost of use [11, 12].

In view of the above, research work on ensuring sustainable manufacturing in various areas of the abrasive machining process is being carried out in global research centers. An example is a novel approach to machining presented in [13], whose authors described a new chemical–mechanical polishing slurry and its polishing mechanisms on a nickel alloy developed for production in the semiconductor, microelectronics, and aerospace industries. Another example in this area is research on the development of a type of new chemical–mechanical polishing slurry for copper including environmentally friendly compositions [14]. Based on the work in [15, 16], it should be stated that theories and models are important for grinding and abrasive machining to understand machining processes and play an important role in the development of new approaches to machining. Their authors have developed a novel model for maximum undeformed chip thickness that is in good agreement with those in grinding experiments. Another example of a novel approach is the work in [17,18,19]. A novel approach to single-grain scratching at 40.2 m/s and nanoscale depth of cut has been carried out, in which the speeds used are four to seven orders of magnitude higher than those used in nano-grinding. Thanks to groundbreaking theories, new methods of mechanical chemical grinding have been developed [20, 21] and high-performance surfaces that are extremely difficult to produce by traditional means have been produced.

Among other ways of neutralizing the deleterious effects of coolants, the method of minimized coolant output, denoted by the acronym MQL (minimum quantity lubrication), has gained great popularity [22,23,24]. In this method, an air-oil aerosol stream is delivered with a coolant output of only 10–500 ml/h [25,26,27,28,29,30,31]. The literature shows [32, 33] that under specific processing conditions, the use of the MQL method gives results comparable to the flood method [34,35,36,37,38,39].

Studies by the authors [40] of this article related to the application of the MQL method in the grinding process of hob cutter contact surfaces, however, pointed out the insufficient cleaning capacity of the MQL method, understood as the limited ability to remove machining products from the grinding zone. A solution to this problem can be the use of an additional cleaning nozzle using a stream of compressed air [41,42,43] or cooled compressed air. A vortex tube refrigeration method using cold air guns (CAG) nozzles [44] is used to cool the compressed air stream. The available literature indicates that the simultaneous application of the MQL and CCA methods creates the possibility of effective cooling and lubrication in the machining zone [45,46,47,48]. This is also confirmed by the authors’ experimental studies [49, 50] describing the grinding of the hob cutter [51]. This method, designated by the acronym MQL-CCA, made it possible to achieve a technological state of the surface layer at the level of values obtained after machining with coolants supplied by the flood method (WET). However, the literature indicates the possibility of further improvement of grinding results, specifying that the effectiveness of the effect of any coolants significantly depends on the angular position of the nozzle relative to the active surface of the grinding wheel (GWAS), and this should be considered individually for each variation of the grinding process [52,53,54,55]. Unfortunately, in this context, experimental studies alone do not make it possible to perform the necessary analysis of the CCA flow in the grinding zone during the grinding of the hob cutter face. For this reason, it is necessary to perform numerical simulations of such flow for different nozzle settings, while due to the different geometry and kinematics of this variety of grinding, it is not possible to use the results of studies available in the literature [56,57,58,59,60,61,62]. These will allow a precise analysis of the phenomena accompanying the flows and the identification of the most favorable angular setting of the grinding wheel from the viewpoint of GWAS cleaning.

In view of the above, the purpose of the research described in this article is to evaluate the ability of CCA to clean GWAS during the sharpening of hob cutters. The evaluation is based on numerical simulations of CCA flow and verification experimental studies. “Section 2” describes the steps related to the preparation of the geometric model and mesh, the establishment of boundary and initial conditions, and the simulation of the flow based on the “SST k-ω” model available in Ansys CFX software using the Kato-Launder modification. Based on the results obtained (“Sect. 3”), a parameter was determined that describes the ratio of the amount of air delivered by the nozzle to the grinding zone to the amount of air entering directly into the zone of contact between the grinding wheel and the hob cutter – the efficiency ηws of the air delivery system. The value of the tangential shear stress τw occurring on the conical surface of the grinding wheel in the area of the wheel’s active surface and the area of the surface Pτ affected by these stresses were also determined. “Section 4” describes experimental studies that verify the results of numerical simulations. The amount of cloggings occurring on the GWAS, defined as the percentage rate Z% of the wheel’s clogging, was used as a parameter for verifying the cleaning efficiency of the grinding wheel. This index was determined by analyzing microscopic images of the GWAS. The methodology, in a similar form, was also used by other researchers [63, 64].

The research work in the area described in this article is new and has not been undertaken before, and the results have application potential for both scientific and industrial aspects.

2 Numerical simulation of the flow of compressed cold air in the grinding zone during the sharpening of hob cutters’ face

2.1 Machining conditions and 3D CAD models

Before carrying out the numerical simulations, 3D CAD models of the actual components involved in the grinding process of the hob cutter face were made. Figure 1 shows the machining space of a specialized tool grinding machine for sharpening hob cutters. During grinding, the hob cutter 1 is mounted on a tool arbor 2 and moves with the grinding table at a speed vw relative to the grinding wheel 3 rotating at a speed ns. The machining allowance a is removed as a result of the mutual movement of the grinding wheel and the hob cutter in successive cycles. Within each cycle, which consists of a grinding and return stroke (Fig. 1), the allowance ae is removed. The coolant is delivered to the grinding zone through a single nozzle. In the case under consideration, this constitutes nozzle 4, spraying compressed cold air (CCA).

Fig. 1
figure 1

View of the machining zone during grinding the face of a hob cutter

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Three 3D CAD models were made for the numerical simulations, namely models for (1) the hob cutter, (2) the grinding wheel, and (3) the CAG (Cold Air Guns) nozzle used to feed compressed cold air (CCA) into the grinding zone. In the case of the CAG nozzle model, it was decided to create only a model of the nozzle tip itself for the purpose of the research being conducted. It was assumed that this model, hereinafter referred to as the CCA nozzle, would be a solid with a circular cross-section and a diameter of d = 5 mm, corresponding to the nozzle outlet diameter used in the experimental tests. The complete set of assumptions made to create the 3D CAD models is summarized in Table 1.

Table 1 Parameters of 3D CAD models
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It should be noted that the value of module m of the hob cutter was selected as the upper value from the range of modules of hob cutters most commonly used in the automotive industry in the production of passenger car parts. In practice, this makes it possible to obtain the most technologically difficult working conditions resulting from the longest maximum contact distance between the abrasive grains of the grinding wheel active surface and the machined surface.

According to the manufacturer’s data [66], the WNT 6910.15.3–7 nozzle can operate at a supply pressure in the range of 0.3–0.7 MPa, allowing it to produce a compressed air stream with a temperature even up to 55 ℃ lower than the temperature value of the air supplying the nozzle. For the purpose of the research, experimental measurements of the TCCA temperature of compressed cold air obtained at the outlet of the CAG nozzle were made, depending on the value of the supply pressure pCCA. An M890G digital multimeter equipped with a K-type thermocouple measuring probe was used to measure the temperature. The reading resolution of the multimeter is 1 ℃. The lowest temperature (− 20 ℃) was obtained at the nozzle outlet when the supply pressure pCCA = 0.6 MPa was set. Due to the fact that the CAG nozzle outlet is located at a distance of about 10 mm from the grinding wheel active surface, the temperature of the compressed cold air was measured at various distances from the nozzle outlet. The measurement results are presented in Table 2, and the value TCCA = − 20 ℃ obtained at the distance l = 15 mm was used in the further part of the research.

Table 2 TCCA temperature of compressed cold air stream (CCA)
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2.2 3D CAD model of the system and supply of compressed cold air by CCA

Before performing the numerical simulations, the created 3D CAD models were mutually arranged in a way that corresponds to the actual positions of the mapped components during the sharpening process. A 3D CAD model of the system was thus created (Figs. 2 and 3).

Fig. 2
figure 2

Location of area on grinding wheel active surface on which flow of cooled compressed cold air was directed (CCA area); 1 – grinding wheel, 2 – hob cutter, X – contact line of the hob cutter cutting edges with the grinding wheel, Y – conventional line parallel to X

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Fig. 3
figure 3

Angular arrangement of the compressed cold air supply (CCA) nozzle relative to the grinding wheel active surface: a general view, b positioning of the CCA nozzle relative to line C, c positioning of the CCA nozzle relative to line D; 1 – grinding wheel, 2 – hob cutter, 3 – CCA nozzle, C – line drawn through the cone formation of the grinding wheel, D – line tangential to the grinding wheel active surface

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As shown in Fig. 2, a stream of compressed cold air is delivered to the grinding zone by means of a nozzle, on the left side of the hob cutter 2, on the CCA fragment of the active surface of the grinding wheel 1. The CCA area is located between line X, which is the line of contact between the apical cutting edges of the hob cutter and the grinding wheel, and the conventional line Y located parallel to line X at a distance of 25 mm from it. Such a location of the CCA area results from the information contained in sources in the literature, a review of which showed that a stream of compressed cold air should be applied to the grinding wheel active surface (GWAS) as close as possible to the contact zone between the grinding wheel and the workpiece [53]. Locating this area, as shown in Fig. 2, eliminates the occurrence of collisions resulting from the respective movement of the grinding wheel and the cutter during machining.

As shown in Fig. 3, the axis of nozzle 3 is inclined at an angle of 90° to line C, defined by the form of the cone describing the grinding wheel body (Fig. 3b), and then at an angle ε to line D (Fig. 3c), which is tangential to the active surface of the grinding wheel and simultaneously intersects line C at an angle of 90° (Fig. 3a). As shown in Fig. 3c, four settings of the angle ε were used in the tests, which were respectively: 30°, 45°, 60°, and 90°. At the same time, it should be mentioned that the angular inclination of nozzle 3 with respect to line D must not be less than 30°, due to technical limitations in nozzle positioning. Moreover, the outlet of nozzle 3 was placed at a distance of 15 mm from the grinding wheel active surface. The grinding wheel, rotating clockwise, after withdrawing from contact with the workpiece surface, is cleaned by the compressed cold air jet (CCA). In the case of simultaneous application of the MQL method and administration of compressed cold air (CCA), as it is the case with the MQL-CCA method, the wheel rotates and lifts the oil mist supplied by the MQL nozzle into the grinding zone and is then cleaned of the grinding products by the compressed cold air jet supplied by the CAG nozzle.

2.3 Numerical simulation of air flow in CCA method

2.3.1 Simulation model and computational grid

The aim of the conducted simulation studies was to determine the capacity of the air supplied by the CAG nozzle during the sharpening of hob cutters for GWAS cleaning. The most important specific objectives of the numerical simulations carried out were as follows.

  1. 1.

    Determination of the parameter defining the ratio of the amount of air delivered by the CAG nozzle to the grinding zone to the amount of air entering directly into the contact zone of the grinding wheel with the hob cutter. It was assumed that this parameter would be defined as the efficiency ηws of the air supply system when using a CCA jet.

  2. 2.

    Determination of the value of wall shear stress τw occurring on the conical surface of the grinding wheel in the area of active surface of the wheel as a result of air supply using the CCA method.

  3. 3.

    Determination of surface area Pτ affected by wall shear stress τw.

In the course of the research, a flow of compressed cold air was simulated, which after being discharged from a nozzle reaches the grinding wheel active surface rotating with the rotational speed ns. The area in which the air flow was simulated was defined by the fluid domain, located between the elements of the 3D model of the system and matching the shape of the grinding wheel, hob cutter, and nozzle tip (Fig. 4).

Fig. 4
figure 4

Location of fluid domain within 3D model of CCA air supply system

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Prior to simulation, the fluid domain was optimized by creating hexagonal grid elements whose regular shapes allow for correct results in numerical simulation. Particular emphasis was placed on the consistency of the grid and the appropriate thickening of the inflation layer at the point of contact between the air and the surface of the rotating grinding wheel. The appearance of the fluid domain grids optimized for the simulation is shown in Fig. 5. Depending on the nozzle angle, the number of nodes in the model ranged from 2,119,321 to 5,128,077, while the number of elements ranged from 2,110,772 to 5,449,291. Detailed values of the number of nodes and elements for each discrete model are presented in Table 3 at the end of this chapter.

Fig. 5
figure 5

Discrete model of the fluid domain: a general view and b cross-section

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Table 3 Characteristics of conditions of simulations
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2.3.2 Boundary and initial conditions

Numerical simulations to determine the efficiency ηws, the wall shear stress τw, and the area Pτ of the τw incidence region were carried out for four angular positions of the nozzle. These angles ε were respectively 30°, 45°, 60°, and 90° (Fig. 3c).

It was assumed that the surface to be used in order to determine what part of the nominal air flow passes through the exit surface from the fluid domain was located under the hob cutter tooth shown in Fig. 6. It should be mentioned that, for the purpose of this simulation, a simplification was made by assuming that the hob gear and the grinding wheel are stationary with respect to each other except for rotation of the grinding wheel with a rotational speed ns = 2950 rpm. This simplification results from the fact that during machining, the length of the contact line between the grinding wheel and the contact surface of the hob cutter, which has the shape of a cog, changes continuously and, consequently, the size of the exit surface from the fluid domain would also change. According to the authors, failure to adopt this simplification would lead to an unnecessary increase in numerical simulation time.

Fig. 6
figure 6

Inflow and outflow surface of the fluid domain (red)

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Finally, the efficiency ηws was defined as the ratio of the mass flow ṁin of air at the inlet surface into the fluid domain to the mass flow ṁout of air at the outlet surface from the fluid domain:

$${\eta }_{ws}=\frac{{\dot{m}}_{\mathrm{in}}}{{\dot{m}}_{\mathrm{out}}}$$
(1)

The numerical simulations were carried out using Ansys CFX 2019 software. The initial and boundary conditions were defined as.

  • working medium: air (air model—ideal gas—was used in the CFX program);

  • velocities at the nozzle inlet: 50 m/s (value for WNT nozzle determined by anemometer measurement);

  • rotational speed of grinding wheel conical surface ns = 2950 rpm, clockwise.

For the purpose of the simulation, it was assumed that the absolute atmospheric pressure was 1013 hPa. Relative to this, the value of the pressure at the exit surfaces from the fluid domain was defined as 0 Pa.

2.3.3 Turbulence model

A review of the literature [67,68,69,70,71] indicates that a properly chosen RANS turbulence model using Reynolds-averaged Navier–Stokes equations should correctly predict the effect of small vortex structures on the overall simulated flow and calculate reliable averaged values of the simulated flow parameters.

In the present study, the “SST k-ω” model, available in Ansys CFX software [72], was used. This model is a hybrid combining the best properties of the “k-ω” and “k-ε” models and allows the introduction of a segment limiting the overproduction of turbulence kinetic energy in areas of large pressure gradients. The equations of the “k-ω” model are used to model turbulent flow in the boundary layer. Since the “k-ω” model is very sensitive to the values of turbulent quantities in free flow, it is replaced by the “k-ε” model in layers further from the wall. This model simulates turbulence in free flow well and is less sensitive to inlet conditions for the quantities describing turbulence. The desirable features of both models are combined into a single model using the fact that the standard “k-ε” model can be transformed into equations for k and ω due to the fact that ω is the inherent kinetic energy dispersion of turbulence; thus, ω = ε/k. The equations of this model are then multiplied by a function which has a value of 1 in free flow and 0 at the wall, while the equations of the standard “k-ω” model are multiplied by the F1 function. The model, through the equation for turbulent viscosity, limits the values of the flow principal stresses. The model equations used are as follows:

  • turbulent kinetic energy:

$$\frac{\partial \left(\rho k\right)}{\partial t}+\frac{\partial \left(\rho {\mathrm{u}}_{j}k\right)}{{\partial x}_{j}}=-P-{\beta }^{*}\rho \omega k +\frac{\partial }{{\partial x}_{j}}\left[\frac{\left(\mu +{\upsigma }_{k}{\mu }_{t}\right)\partial k}{{\partial x}_{j}}\right]$$
(2)

  • dissipation of turbulent kinetic energy:

$$\begin{aligned}\frac{\partial \left(\rho \omega \right)}{\partial t}+\frac{\partial \left(\rho {\mathrm{u}}_{j}\omega \right)}{{\partial x}_{j}}&=-\frac{\gamma }{{\upsilon }_{t}}P-{\beta }^{*}\rho {\omega }^{2}+\frac{\partial }{{\partial x}_{j}}\left[\left(\mu +{\upsigma }_{\omega }{\mu }_{t}\right)\frac{\partial \omega }{{\partial x}_{j}}\right]\\&+2\left(1-{F}_{1}\right)\frac{\rho {\upsigma }_{\omega 2}}{\partial \omega }\frac{\partial k}{\partial {x}_{j}}\frac{\partial \omega }{\partial {x}_{j}}\end{aligned}$$
(3)

where

  • turbulent viscosity

$${\upsilon }_{t}= \frac{{a}_{1}k}{\mathrm{max}({a}_{1}\omega ,S{F}_{2})}$$
(4)

  • production term

$$P={\mu }_{t}SS$$
(5)

whereby

$$S=\sqrt{\frac{1}{2}{\left(\frac{\partial {\mathrm{u}}_{i}}{{\partial x}_{j}}+\frac{\partial {\mathrm{u}}_{j}}{{\partial x}_{i}}\right)}^{2}}$$
(6)

  • functions F1 and F2:

$${\mathrm{F}}_{1} =\mathrm{ tanh}\left({\mathrm{arg}}_{1}^{4} \right), {\mathrm{arg}}_{1} =\mathrm{ min }\left(\mathrm{max}\left(\frac{\surd \mathrm{ k}}{0.09\mathrm{\omega y}} , \frac{500\upnu }{\upomega {\mathrm{y}}^{2}}\right) , \frac{4\mathrm{\rho \sigma }{\upomega }^{2}\mathrm{k}}{\mathrm{CDk\omega }{\mathrm{y}}^{2}}\right)$$
(7)
$${\mathrm{F}}_{2} =\mathrm{ tanh}({\mathrm{arg}}_{2}^{2} ),{\mathrm{arg}}_{2} =\mathrm{ max}\left(2 \frac{\surd \mathrm{ k}}{0.09\mathrm{\omega y }} , \frac{500\upnu }{{\mathrm{y}}^{2}\upomega }\right)$$
(8)

  • fixed coefficients

$${\upsigma }_{\mathrm{k}1}=0.85 {\upsigma }_{\upomega 1}=0.5 {\upbeta }_{1}=0.075 {\upbeta }^{*}=0.09\upkappa =0.41 {\upgamma }_{1}= \frac{{\upbeta }_{1}}{{\upbeta }^{*}}-{\upsigma }_{\upomega 1} {\upkappa }^{2} \sqrt{{\upbeta }^{*}}$$
$${\upsigma }_{\mathrm{k}2}=1.0 {\upsigma }_{\upomega 2}=0.856 {{\upbeta }_{2}=0.0828\upbeta }^{*}=0.09,\upkappa =0.41 {\upgamma }_{2}= \frac{{\upbeta }_{2}}{{\upbeta }^{*}}-{\upsigma }_{\upomega 2} {\upkappa }^{2} \sqrt{{\upbeta }^{*}}$$

2.3.4 Modification of the production conditions

In the “SST k-ω” model, Kato-Launder modifications [73] were applied. Normally, the “SST k-ω” turbulence model tends to artificially overproduce turbulence in the pile-up regions due to the large S values that are generated in these regions. Kato-Launder’s proposal is to replace shear stress S in the turbulence production equation by vorticity Ω, then

$$P={\mu }_{t}S\Omega$$
(9)

where

$$\Omega =\sqrt{\frac{1}{2}{\left(\frac{\partial {\mathrm{u}}_{i}}{{\partial x}_{j}}-\frac{\partial {\mathrm{u}}_{j}}{{\partial x}_{i}}\right)}^{2}}$$
(10)

2.3.5 Wall shear stresses on the grinding wheel surface

Due to the shear effect in the abrasive layer [74, 75], the wall shear stress τw occurring on the conical surface of the grinding wheel (wall) was used in the analysis of numerical simulations in order to determine the possibility of removing contaminants from the grinding wheel active surface by use of the CCA jet. These stresses are described by the formula

$${\tau }_{w}=\mu {\left(\frac{\partial u}{\partial y}\right)}_{y=0}$$
(11)

where τw is the shear stress on the conical surface of the grinding wheel (wall), µ is the dynamic viscosity, u is the velocity over the conical grinding wheel surface (wall), and y is the distance from the wall of the conical grinding wheel surface.

2.3.6 Conditions of the numerical simulations

ANSYS CFX 2019 software was used to carry out the numerical simulations of the flow, using a 3D CAD model whose geometric parameters are described in Table 1. Table 3 summarizes the conditions applied during the numerical work carried out. Among other things, the assumptions and simplifications made are listed, as well as the parameters of the finite element mesh. Furthermore, the properties of the fluid used and the conditions of the simulation process are listed.

Mesh convergence analysis was carried out by incrementally increasing the number of elements and verifying the estimations to ensure convergence of the numerical solution. The chart in Fig. 7 shows that for each of the four geometric variants differing in the angular position of the nozzle, the densification of the grid causes the results to converge around a single value. The fact that the efficiency ηws for each of the four variants is lower with lower mesh densification is due to the fact that in the gap between the abrasive tool and the workpiece, the geometry of the closely spaced walls is reproduced incorrectly and the calculations in the wall layer erroneously reduce the amount of fluid that enters this area. By properly densifying this area and the area around the main fluid stream, it is possible to achieve a plateau in the ηws efficiency results.

Fig. 7
figure 7

Mesh independence study

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3 Results of numerical simulations of compressed cold air flow in grinding zone of hob cutter blades using CAG nozzle

3.1 Efficiency ηws of air supply system

Figure 8 shows the distribution of air flow streamlines in the fluid domain obtained by numerical simulation for four nozzle angle settings: 30°, 45°, 60°, and 90°.

Fig. 8
figure 8

Velocity vector distribution on the current line obtained for the angle ε of the nozzle inclination: a 30°, b 45°, c 60°, and d 90°

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As can be observed in Fig. 8, decreasing the angle ε of inclination of the nozzle in relation to the grinding wheel active surface causes the lines representing air streams to be more directed and directed “under the tooth” of the hob cutter (opposite to the direction of rotation of the grinding wheel). This results in a greater volume of air flow reaching the contact zone of the hob cutter with the grinding wheel due to lower losses resulting from multidirectional dispersion of air flow from the point of contact with the rotating grinding wheel surface, which occurs when the nozzle is inclined at an angle of 90°, for instance (Fig. 8d).

A confirmation of the above-described effect of changes in the inclination angle ε on the directionality of the CCA jet is provided by the results of efficiency ηws, which defines the ratio of the amount of air delivered by the nozzle to the grinding zone to the amount of air entering directly into the zone of contact between the grinding wheel and the hob cutter. The results are summarized in Table 4 and shown in Fig. 9.

Table 4 Efficiency ηws of air supply system using CCA method
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Fig. 9
figure 9

Efficiency ηws of CCA air supply system obtained for four angles ε of nozzle inclination, namely, 30°, 45°, 60°, and 90°

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As can be seen from the data in Table 4 and Fig. 9, decreasing the angle ε increases the efficiency of CCA supply to the grinding zone expressed as ηws efficiency. The highest efficiency ηws, i.e., the most favorable from the point of view of cooling and cleaning of the grinding zone, was achieved when the nozzle was positioned at an angle ε = 30° with respect to the grinding wheel active surface. It should be noted that in the case of two successive angular settings of the nozzle (45° and 60°), the efficiency achieved for them reaches almost identical values, but is lower than the highest one by about 30%. Such a large difference in the values obtained is due to the nature of the flow. The numerical simulation indicates that at a nozzle inclination of 30° in relation to the GWAS, the air stream, due to the presence of an air cushion around the rotating grinding wheel, does not reach the GWAS for the most part. Thus, it is not dispersed as a result of contact with the grinding wheel and is directed directly to the point of contact between the grinding wheel and the hob cutter.

In comparing the above-mentioned values of efficiency with the values obtained using the MQL method (described by the authors in [40]), it should be noted that the efficiency of the air supply system obtained for identical angles of nozzle inclination in both cases changes in an analogous way – for ε = 30° it takes on the highest value, while for ε = 90° it is the lowest. Nevertheless, it should be noted that the efficiency values obtained for the MQL method are 90–160% higher than the corresponding values obtained for the CCA method. It must be assumed that this is due to the way the nozzle is positioned with respect to the direction of rotation of the grinding wheel. In the case of the MQL method, the air stream leaving the nozzle encounters an air cushion around the rotating grinding wheel in a clockwise direction. As a result, it is lifted by the air cushion and directed directly to the point of contact between the grinding wheel and the hob cutter. When a CAG nozzle is used, the clockwise rotation of the grinding wheel impedes air flow to the contact area between the grinding wheel and the hob cutter.

3.2 Wall shear stress τw and area Pτ of action of wall shear stress τw

Figure 10 shows the distributions of the wall shear stress τw occurring on the conical surface of the grinding wheel in the area of the active surface of the wheel. The distributions are shown for four nozzle angle settings, namely, 30°, 45°, 60°, and 90°.

Fig. 10
figure 10

Distribution of wall shear stress τw obtained for angle ε of nozzle inclination: a 30°, b 45°, c 60°, and d 90°

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Table 5 and Fig. 11 show the maximum wall shear stress τw-max values obtained for nozzle inclination angles ε of 30°, 45°, 60°, and 90°, respectively.

Table 5 Maximum wall shear stress τw-max
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Fig. 11
figure 11

Maximum wall shear stress τw-max obtained for four angles ε of nozzle inclination: 30°, 45°, 60°, and 90°

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From the data in Table 5 and Fig. 11, it can be concluded that the highest value of wall shear stress occurs when the CAG nozzle is set at an angle ε = 90°, similar to the authors’ study on air utilization using the MQL method [40]. Moreover, as the angle of inclination of the CAG nozzle decreases, the wall shear stress τw also decreases and reaches, for an angle ε = 30°, a τw-max value less than half the value obtained for an angle ε = 90°. This results from the already mentioned occurrence of the air cushion around the rotating grinding wheel, which causes that the smaller the value of the nozzle inclination angle, the less air reaches the GWAS, while more enters the contact zone between the grinding wheel and the hob cutter.

It should be noted that the ability to remove contaminants from the active surface of the grinding wheel is also influenced, apart from the value of wall shear stress τw, by the area Pτ on which these stresses act. The greater the surface area Pτ, the greater the ability to remove contaminants from the grinding wheel surface. Table 6 summarizes the values of the surface area Pτ of the action of wall shear stress τw induced by the CCA jet, determined for four cases of nozzle angular settings (ε = 30°, 45°, 60°, and 90°). For each of the four angles ε, the area is defined by three intervals of wall shear stresses with values of (1) 30–170 Pa, (2) 40–170 Pa, and (3) 50–170 Pa.

Table 6 Surface area Pτ of wall shear stress τw
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To illustrate the shape and size of the Pτ area obtained as a function of the nozzle inclination angle ε, they are summarized in Fig. 12. This figure shows the Pτ area obtained for case 1, when the wall shear stress τw is between 30 and 170 Pa. The surface area is marked in green.

Fig. 12
figure 12

Area Pτ of occurrence of wall shear stress τw in the range 30–170 Pa obtained for angles ε of nozzle inclination: a 30°, b 45°, c 60°, and d 90°

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In order to facilitate the analysis of the results in Table 6, their values were used to make the graph shown in Fig. 13.

Fig. 13
figure 13

Area Pτ of occurrence of wall shear stress τw obtained for three ranges of stress values, namely, 30–170 Pa, 40–170 Pa, and 50–170 Pa

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From Table 6 and Fig. 13, it can be seen that for all three ranges of τw, the largest Pτ surface area was obtained when the nozzle was positioned at an angle ε = 45° to the grinding wheel active surface. Similarly, although smaller on average by about 5.9%, values were obtained when the nozzle was positioned in a direction normal (ε = 90°) to the GWAS. When an angle of ε = 60° was used, the difference averaged 16.2%.

Of note is the large difference (about 215 mm2 on average) in the Pτ values obtained for the cases when the nozzle is positioned at 30° and 45°. Such a large difference results from the nature of the flow and is caused by the occurrence of an air cushion around the rotating grinding wheel, which blocks most of the air stream from reaching the GWAS, a fact that was mentioned when discussing the efficiency diagram ηws in Fig. 9 and the diagram of maximum wall shear stress τw-max (Fig. 11).

4 Experimental verification of numerical simulation results

4.1 Grinding of face of hob cutters

The aim of the experimental study was to verify the results of numerical simulations of compressed cold air (CCA) flow in the grinding zone. The amount of clogging occurring on the grinding wheel active surface was selected as the parameter verifying the efficiency of the air supply. This is due to the fact that one of the functions of the cooling lubricant is to remove the swarf from the grinding zone and to clean the wheel.

In the course of experimental investigations, the contact surface of a hob cutter was ground, and then the surface proportion of deposits on the active surface of the wheel was measured. It was on this basis that a percentage index Z% of the grinding wheel cloggings was determined. The obtained results allowed an indirect evaluation of the air supply conditions in the CCA method determined by the angle ε of inclination of the CAG nozzle.

During grinding, the face of solid hob cutters NMFc-3/20°/B, made of HS6-5–2 high-speed steel without antiwear coatings, was sharpened. These cutters are designed for the production of cylindrical gears according to ISO 53 and ISO 54. Table 7 summarizes the grinding conditions used during the described tests.

Table 7 Grinding conditions
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The cutters were sharpened on a special conventional grinding machine for sharpening hob cutters using a Norton 38A60KVBE disc wheel [65]. This is a grinding wheel made of white fused alumina grains with a vitrified bond. The wheel was conditioned before each test using a single-grain diamond dresser. The grinding parameters for the hob cutter blades were selected based on literature data [76, 77] and workshop practice.

During the first stage of the research, compressed cold air (CCA) was applied using a single CAG nozzle with the symbol WNT 6910.15.3–7. The nozzle was inclined with respect to the active surface of the grinding wheel using four angles ε amounting to 30°, 45°, 60°, and 90°. As a reference, machining was also carried out with the minimum quantity of machining fluid using the MQL method. In this method, coolant in the form of an air-oil aerosol was supplied using a MicroJet MKS-G100 MQL system from MicroJet (Germany) with a flow rate of QMQL = 50 ml/h. Biocut 3000 synthetic ester-based oil from Molyduval (Germany) was used as the liquid [78]. The nozzle was an inclined relative of the grinding wheel active surface using four ε angles of 30°, 45°, 60°, and 90°. The air-oil aerosol supply conditions were determined based on previous studies described in detail in [40].

In the second stage of the experimental study, the grinding of the hob cutter face was carried out using the MQL-CCA method and the flood (WET) method. During machining with the MQL-CCA method, the machining conditions adopted in the first stage of the research were applied during individual trials for each method. The exception was the variant in which the nozzles were positioned at an angle ε that allowed the lowest Z% to be obtained. In the case of the nozzle used in the MQL method, this was an angle ε = 90°, while in the case of the CCA method, the angle ε was 45°. The flood method (WET) was carried out using an aqueous oil emulsion with Emulgol ES-12 (5%) from Orlen Oil (Poland) [79]. The emulsion supply conditions to ensure the lowest index of wheel flooding were established based on previous studies [55]. A single nozzle inclined with respect to the active surface of the wheel by an angle ε = 30° was used, which supplied the emulsion to the grinding zone at a nominal flow rate QWET = 7 l/min.

4.2 Conditions for grinding wheel clogging measurements

Measurements of wheel bonding were commenced by recording images of a fragment of the grinding wheel active surface (GWAS) using a digital microscope DO Smart 5MP PRO from Delta Optical (Poland) [80]. The images were recorded using Delta Optical Smart Analysis 1.0.5 software using 50 × magnification (Fig. 14a). Images recorded in grayscale were analyzed to detect the occlusions and then to determine the degree of filling of the inter-grain spaces of the grinding wheel by the chips. For this purpose, the Met-Ilo v.5.1 image analysis program developed at the Silesian University of Technology [81] was used. This program enables the detection of chips accumulated between the abrasive grains on the basis of the intensity of the light reflected from the chips in relation to the intensity of the light reflected from the surface devoid of them. This difference in reflected light intensity is used to segment the image area by setting the color of the pixels corresponding to the chips as white and giving the remaining pixels a red color (Fig. 14b). As a result of dividing the number of black pixels by the total number of pixels, the grinding wheel clogging index Z% was determined. The final value of the Z% index of grinding wheel clogging representative for a single grinding trial was determined as the arithmetic mean of the index values determined from five GWAS images taken at different locations on the circumference of the grinding wheel and in total covering approximately 6% of the GWAS area [55].

Fig. 14
figure 14

Measurements of bonding of grinding wheel: a GWAS photo after grinding, magnification 50 × ; b GWAS photo processed to determine Z%

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4.3 Experimental results and their analysis

Table 8 and Fig. 15 show the results of the measurement of the active surface indentation of the grinding wheel during grinding with a stream of compressed cold air (CCA) and coolant supplied at a minimum flow rate using the MQL method. The percentage of surface clogging on the GWAS is described by the Z% index of grinding wheel clogging.

Table 8 Z% index of clogging for CCA and MQL
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Fig. 15
figure 15

Grinding wheel clogging index Z% for CCA and MQL methods

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As can be seen from Table 8 and Fig. 15, when the CAG nozzle is used, the Z% index assumes the smallest (most favorable) value for the nozzle inclination angle ε = 45° and increases its value for the other angle settings. This character of the occurrence of clogging coincides with the values obtained from numerical simulations described in “Sect. 2.” The smallest percentage of cloggings on the GWAS observed at a CAG nozzle inclination of ε = 45° is mainly due to the largest surface area Pτ, which is affected by the second highest wall shear stress τw induced by the CCA jet. Moreover, the cleaning effect of the grinding wheel is enhanced by the air jet directed into the contact zone of the active abrasive grains with the workpiece material, as confirmed by the second highest value of the efficiency ηws of the air supply system.

In the case of the results obtained for the MQL method, the Z% index takes the smallest (most favorable) value for the nozzle inclination angle ε = 90°. It should be noted here that the values of the Z% index obtained for the other angles ε are similar. The difference between the extreme Z% values obtained for the angle ε = 30° and ε = 90° is only 2.3%. Therefore, it can be concluded that a change in the angle ε does not have a significant effect on the efficiency of chip removal from the inter-grain spaces of the grinding wheel. This property can be explained by analyzing the results of numerical simulations and assuming that the influence of air entering the contact zone of the workpiece/wheel (expressed by the efficiency ηws) on grinding wheel clogging is equivalent to the influence of the wall shear stress τw acting on the GWAS [40]. For an angle ε = 90°, the wall shear stress τw has a decisive influence on the GWAS cleaning. With decreasing the nozzle inclination angle, the influence of τw stresses also decreases, and the cleaning function is taken over by the air entering directly into the contact zone of the grinding wheel with the attack surface of the sharpened hob cutter.

A comparison of the Z% values obtained during grinding using the CCA method and the MQL method indicates a much better cleaning ability of compressed cold air at an angular position of the CCA nozzle in the range from 30 to 60°. The difference between the best Z% values obtained for both methods is 8.2%, with the Z% value for the MQL method being 2.5 times higher than that for the CCA method.

Table 9 and Fig. 16 summarize the results of the most favorable (smallest) values of the Z% index of wheel clogging determined for four methods of supplying machining fluid to the grinding zone: CCA, MQL, MQL-CCA, and WET.

Table 9 Lowest values of Z% index for CCA, MQL, MQL-CCA, and WET methods
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Fig. 16
figure 16

Smallest values of Z% index of grinding wheel clogging for CCA, MQL, MQL-CCA, and WET methods

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As can be seen from the analysis of the results presented above, the smallest value of the Z% index was obtained by supplying the cooling lubricant using the flood method (WET). The application of an additional GWAS cleaning nozzle using compressed cold air (CCA) and supporting the MQL method resulted in the Z% value determined for the hybrid MQL-CCA method being higher by a mere 0.5% in comparison with the indicator for the priming method (WET).

The above observations and the fact that the Z% value for the MQL method is almost seven times higher than the value of the index determined for the flood method (WET) give rise to the conclusion that the function of evacuating chips from the contact zone of the GWAS with the workpiece surface and cleaning the GWAS during grinding with the MQL method can be effectively supported by feeding an additional stream of CCA in the manner described in this work. This makes it possible to achieve a level of wheel clogging similar to that achieved when feeding a water-in-oil emulsion using the flood method (WET) at an 8400-fold reduction in coolant output.

5 Conclusions

By carrying out numerical flow simulations and verification experimental studies, it was established that the ability of the CCA jet to clean GWAS depends primarily on the value of the tangential shear stress τw and the surface area Pτ on which it acts. The greater these two values are, the smaller (more favorable) the value reached by the grinding wheel clogging index Z%. In the presented range of test conditions, the smallest Z% value of 5.3 was obtained with the nozzle positioned at an angle of ε = 45° to the GWAS and is 2.5 times lower than the value determined for the most favorable positioning of the nozzle using the MQL method, for which Z% = 13.5. This observation indicates the validity of using the CCA jet directed at the GWAS at an angle of 45° to assist in cleaning the grinding wheel during machining with the simultaneous application of the MQL method. This is confirmed by the wheel clogging condition obtained during grinding using the hybrid MQL-CCA method, for which the Z% is 2.5 and its value is comparable to the Z% = 2.0 – value obtained for a wheel working under cooling conditions with a water-based oil emulsion supplied by the flood method (WET).