Nomenclature

Q

heat generation

$\alpha$

heat transfer coefficient

F _s

surface area of the smooth wall

F _p

surface area of the fin

$\lambda$ _wall

wall thermal conductivity

Π _p

fin circumference

h _p

height of fin

$\delta$ _wall

wall thickness

C

total drag coefficient

$\beta$

coefficient taking into account the effect of radial clearance on power loss

C _p

specific heat capacity of air

l

diameter of the rolling elements

u

circumferential speed of the bearing cage

Re

Reynold number

Pr

Prandtl number

k

thermal conductivity of the oil

Eu

Euler number

F _m

average load on the bearing

F _r

radial load on the bearing

F _c

centrifugal load of the rolling elements in the bearing

d _m

diameter of the centre of gravity of the bearing

h _bearing

value of radial clearance in the bearing

$\eta$

efficiency of gearing system

N

transmitted power

$\varepsilon$ _s

contact ratio

f

coefficient of friction

$\beta$ _g

angle of inclination of teeth to the axis of rotation of the starting cylinder

Z _K

the number of teeth in driver gear

Z _k

the number of teeth in driven gear

2.0 Support seal system design

As part of the secondary air system, the support seal system plays a vital role in modern turbofan engines, as it flows outside the flow path used to cool vanes, blades, shrouds and disk cavities, and provides seal buffering to oil sumps. The distinctive feature of the method proposed in this work is the design of the support seal system, taking into account the mutual influence ofbuffer air seal and oil seal characteristics on the parameters of the engine oil feed systems. The development of computational fluid dynamics, which enabled study not only of a single seal but of the complete support seal system of the bearing, allowed the examination of the existing parameter connections.

The most significant of the characteristics of the support seal system to be taken into account in the design are leakage and thermo-fluid characteristics. Leakage characteristics directly influence aero engine performance (thrust and specific fuel consumption) and are determined by the geometric parametres of the labyrinth seal. Thermo-fluid characteristics also significantly influence the support seal system design on the one hand, and on the other hand are quite sensitive to the change of the seal geometric parameters (radius location and radial clearance).

The main difficulty in the joint consideration of leakage and thermo-fluid characteristics of the support seal system is that in the early stages of design, the support scheme of the aero engine, which is a determining factor for the radius location of the support seal system structure, is unknown. In addition, the calculation of the leakage without knowledge of the thermal deformations of the rotor/stator can only give approximate results. Therefore, traditionally, the formation of the geometry of the labyrinth seal occurs with a certain priority requirement of leakage, and the requirements of thermo-fluid characteristics (heat generation entering the oil chamber) are taken into account indirectly, either qualitatively or quantitatively, but on the basis of statistics and simplified mathematical models.

This is also reflected in the established order of preliminary design. The initial stage is the cold clearance design, during which the initial geometry of the sealing system is selected, and then follows the stage of synthesis of the structural-heat scheme (SHS) within a given radius location.

Taking into account the previous remarks, designing a support seal system that takes into consideration the features of an oil system is a sophisticated procedure that necessitates the creation of a particular approach, as illustrated in Fig. 2. The CFD-FEA loop is a conventional loosely coupled conjugate heat transfer/fluid-structure analysis, which comprises a numerical computational fluid model for estimating temperature distribution and convective heat transfer coefficients in all internal chambers of the system and a finite-element support seal system model. The devised approach begins with calculating the total quantity of heat entering the system from all sources and selecting the support seal system design. If the overall quantity of heat exceeds the permitted value, the system should be redesigned.

Figure 2. Design method for support seals that takes into consideration the specifications of the aero engine’s oil system.

This approach is justified since, in the great majority of situations, the compressor or turbine support is sealed by multiple seals that comprise the system. The employment of a single seal for this purpose is frequently ineffective. A single seal cannot provide the required disk cooling and cavity purging, turbine shroud cooling and purging or buffering to air/oil interfaces at the bearing chambers. In other words, the seal is designed at a higher degree of design decomposition using this methodology.

The running clearances of all the seals in the system are determined based on the structural calculations and experimental investigation, and the total amount of heat entering the oil chamber and the quantity of oil pumping necessary are completed. If the needed value is not met, the seal design should be modified. The design of a support seal system is an iterative process. It enables the selection of geometric parameters for each seal and their tolerance, while taking into account oil pumping through the support of an aero engine based on a calculation of the total quantity of heat, the heat quantity from each of the sources and an analysis of their percentage ratio, which justifies the use of air cooling of support seals.

As previously stated, the growing characteristics of modern aero engines need the usage of multiple support seals rather than a single seal to provide pressurisation or cooling. As a result, this section discusses the challenges that occur in the design of support seals as part of these systems. The sealing efficiency of a seal not only impacts the amount of heat entering the bearing, but it also influences the performance of the other seals in the system in issue. As illustrated in Fig. 3, the researched support seal system contains three labyrinth seals (oil seal, buffer air seal, and triple seal), which divide the fluid domain between the rotor and stator into four chambers (oil chamber, pre-oil chamber, buffer chamber and intermediate chamber). As indicated in Table 1, labyrinth seals have an initial cold clearance.

Figure 3. The investigated support seal system scheme.

3.0 Numerical methodology

A 15-degree sector CFD model was proposed to account for 3D flow characteristics in the support seal system. Because the Rinlet-1 has 24 holes and the Rinlet-2 has 32 holes, the decision was regarded as the greatest viable alternative for modeling the true, discrete geometrical characteristics of the cavity at a reasonable cost. The other structures in the cavity are virtually or completely axisymmetric. The inlet flows were considered to have assigned total pressures and uniform total temperatures.

Therefore, a study of the effect of buffer air seal location on oil seal outlet temperature was carried out in three cases. Three support seals arrangements of buffer air seal were chosen for comparison: the bottom one based on minimum cooling air flow rate and minimum heating by providing minimum knife gap area and rotor circumferential speed (D = 314mm), the middle one (D = 334mm) and the top one (D = 354mm). The all meshes of the CFD model were about 9M cells. The dimensionless y+ achieved is between 30 and 100, which is within the range suggested for using the conventional wall function. Figure 4 depicts the whole unstructured mesh utilised for the investigated object. Table 2 contains the boundary conditions for the numerical analysis.

As the working fluid, an ideal gas’s viscosity was computed as a function of temperature using the Sutherland formula, and the heat capacity was determined using a polynomial. The turbulence model adopted was by SST, which produced findings that were most similar to experimental observations. The impact of the following parameters on the needed tightness of the studied labyrinth seal was investigated: radius of the seal location R; value of the sealing radial clearance s. A labyrinth seal, in general, can have many more geometrical parametres to regulate the leakage rate. Knife pitch, knife height, knife front and rear inclined angle and other parameters can be included. These criteria are not fundamentally important within the context of the work, hence their effect was not investigated.

The main goal of the CFD simulations is to measure the heat transfer coefficient on the support seal system’s wall. The heat flux and, thus, the heat transfer coefficient should vary along the wall when flow characteristics like temperature and velocity change. The heat transfer coefficient is determined by ANSYS CFX as

(1) \begin{align} h = Q/\left( {{T_w} - {T_{ref}}} \right)\end{align}

where T _w denotes the static temperature of the cell closest to the wall and T _ref is the reference static temperature, which is set to be the second element from the wall. In this case, this formulation is not very useful, as the near wall temperature differs over the wall. The reference temperature is also used as an input in the modeling on system level, and it is more convenient to have a constant reference temperature. Because of this, the mass flow averaged temperature at the Rinlet-2 has been used as reference temperature, to determine the heat transfer coefficient rather than the near-wall temperature. This is achieved by manually specify T _ref in the simulation set up using an expert parameter. Then, the heat transfer coefficient (HTC) is exported as a matrix comprising the coordinates and HTC for each cell along the wall, to be used in conjunction with the data for the internal cooling system. The heat transfer coefficient is radially averaged and weighted by element length to simplify the 3D heat transfer problem. The post-process calculated result of the heat transfer coefficient will be used in the following deformation analysis.

Table 1. Cold clearance of the support seal system

Figure 4. Unstructured mesh of the CFD model for the support seal system.

The support seal system assembly consists of rotor and stator, which are interconnected through cylindrical roller bearing, for smooth transfer of radial force. The main low-pressure shafts also act as the stiffness element to the engine rotor support system. Therefore, a 2D axisymmetric finite element model was generated in the ANSYS MECHANICAL software program to calculate the consequent deformations, as illustrated in Fig. 5. For the boundary condition, the low-pressure shaft of the rotor part was constrained by the ball bearing at the radial and axial direction, and the cylindrical roller bearing at the radial direction. The entire support seal system model is assumed axisymmetric and element type used is Plane-55. The deformation analysis is carried out in two phases. For the first phase, steady state thermal analysis has been carried out by using heat transfer coefficient (shown in Fig. 6) and bulk temperatures as boundary conditions. The inputs from this analysis are used for getting thermal deformations. For the second phase, the support seal system was analysed under normal speeds (N1 = 2,800rpm), with corresponding thermal expansion, which was calculated from the steady state thermal analysis.

The necessity to analyse the combined work of the entire support seal system complicated the investigation. Because of changes in the radial clearance in the seals produced by thermal expansion of the structural sections, a temperature change in one chamber induces a temperature change in all chambers. As a result, the calculation should be run several times to determine the operating radial clearance value for each design alternative, which is illustrated in Fig. 2. At each iteration, the boundary conditions, including the heat transfer coefficient and the displacement of the seal model, were updated. According to the investigated support seal system, it has reached wall temperature convergence after four iterations as usual. Figure 7 shows the effect of the buffer air seal’s radius of position on the temperature distribution in the support seal system.

Table 2. Boundary conditions

Figure 5. Finite element model for structural analysis of the support seal system.

Figure 6. Convective heat transfer coefficient distribution over turbine support seal system surfaces.

Figure 7. Change of temperature distribution depending on the radius of the buffer air seal position: (a) ΔR = −10mm; (b) ΔR = 0mm; (c) ΔR = 10mm.

Figure 7(a) depicts the conventional design. Figure 7(b) depicts the temperature distribution when the radius is lowered by 10mm, whereas Fig. 7(c) depicts the temperature distribution when the radius is raised by 10mm. The radius changes, causing the area of the sealing gap to fluctuate as well. As a result, a 10mm radius decrease reduces the temperature in the pre-oil cavity by 5°C. By increasing the radius, the gas temperature in the pre-oil cavity rises by up to 13°C, causing heating conduction of the oil seal and an increase in the operating seal radial clearance in this seal. It is evident that having as small a radius as feasible is preferable to reducing the magnitude of heat flow into the bearing with leaking working fluid. However, in this scenario, it is vital to consider, first and foremost, the change in clearance between the rotor and stator caused by temperature spreading and centrifugal forces.

It is clear from the temperature field analysis that having as small a buffer air seal radius as feasible is preferable to reducing the magnitude of heat flow into the bearing with leaking working fluid; but, it still needs to consider the pressure distribution difference between a higher radius and a lower radius, as shown in Fig. 8, to help understand how to choose a reasonable buffer air seal radius that is preferable for reducing leakage into the bearing. For the given radius location of the buffer air seal, due to the reduction in flow area and a smaller pressure difference, a lower radius leads to less leakage of the buffer air seal. In this work, it is important to investigate the interaction between the buffer air seal and the oil seal. As shown in Fig. 8, with the reduction of buffer air seal radius, the pressure difference between the oil seal increase from 1.26 to 1.45. For the leakage analysis of the oil seal, it could be concluded that a smaller radius of buffer air seal causes less thermal deformation of the oil seal but increases the pressure difference. Therefore, in this scenario, it is vital to consider conducting the experimental tests, to verify the preliminary analytical results of the CFD-FEA method and investigate the change in clearance between the rotor and stator of the oil seal caused by temperature spreading and centrifugal forces.

Figure 8. Change of pressure distribution depending on the radius of the buffer air seal position: (a) ΔR = −10mm; (b) ΔR= 0mm; (c) ΔR= 10mm.

Figure 9 depicts the integrated deformation results of the support seal system with thermal expansion and centrifugal effect. Figure 10 depicts the change in clearance caused by the above mentioned loads in the oil seal (a), buffer air seal (b), andt seal (c). The figure shows that the change in clearance caused by stator deformations is substantially greater than that caused by rotor deformations, which is explained by the stator’s greater thermal inertia. It is 1.25 times greater for the oil seal, 2.2 times higher for the buffer air seal, and 1.8 times higher for the triple seal. Despite the fact that rotor and stator parts deform in the same direction, the application of temperature and centrifugal loads causes a decrease in the nominal value of the operating radial clearance. In this situation, the oil seal gap was lowered by 0.06mm, the buffer air seal gap was reduced by 0.08mm and the triple seal gap was reduced by 0.15mm. This indicates that a further reduction in radial clearance of the oil seal, together with a matching drop in operating clearance, might result in rotor-stator contact.

Figure 9. Deformation of turbine support seal system.

Figure 10. Change of running clearance in the (a) oil seal, (b) buffer air seal, and (c) triple seal from thermal expansion and centrifugal load.

4.0 Experimental test

4.2 Test condition

In general, engine state parameters such as pressure, temperature and rotational speed are high, and direct experimental simulation is expensive. Based on the similarity principle, the pressure, temperature and rotational speed of the experiment should be suitably reduced based on the experimental state being similar to each physical field of the engine condition. Furthermore, the geometrical structural characteristics of the engine model and the experimental model are kept same during the analytical procedure. The state parametres of the experimental test in this example, analysed by the approximate modelling principle, are mass flow discharge coefficient and windage heating coefficient. These two dimensionless parameters are equal in the experimental and engine models, making it easier to apply the results of this work to the engine states. Therefore, according to the working conditions of an aero engine and experiments, the test conditions were investigated, as given in Table 3.

A thermal growth impact is predicted when the stator temperature changes with the increasing radius position of the top buffering seal in Fig. 11. During the experimental test, it was difficult to cause the change of stator deformation by the temperature of R-inlet2 airflow. Therefore, changes in geometric parameters of the stator due to excessive local heat generation in the engine oil feed system was simulated by an electromagnetic rapid heating ring. Heating temperature is obtained from CFD calculations, as shown in Fig. 7. The oil seal radial clearance was dynamically measured during the test using a fiber optic displacement sensor for the dynamic test.

Therefore, the test is divided into two types: those with a heating ring and those without. As a result of these tests, the oil seal outlet temperature and running clearance were determined. The value of oil seal power loss will be compared between experiment and the CFD-FEA method using the experiment-by-hand calculation shown in Section 5, which helps to validate the effectiveness of the method proposed in Section 2.

Table 3. Test conditions

Figure 11. Thermal expansion of the stator/rotor.

5.0 Quantify heat generation

There are large heat fluxes in the support seal system during operation. It is critical to identify and measure heat production since it has a significant impact on engine performance. It raises the working temperatures of the components, reducing their durability. Calculating heat generation necessitates the use of geometry data for the components, material properties and heat transfer coefficients (HTC). These parameters are difficult to determine, thus empirical factors are frequently employed to assess them.

As shown in Fig. 12, the most important heat fluxes in the support seal system are located in the flow channel, support walls, shaft, leakage through the seals, and friction in the bearings, gearings and splines. The total quantity of heat entering the support seal system is then calculated using the following formula:

(2) \begin{align} {Q_\Sigma } = {Q_{wall}} + {Q_{leakage}} + {Q_{shaft}} + {Q_{bearing}} + {Q_{gearings,splines}}\end{align}

In the following section, we will look at a theoretical calculation for calculating the quantity of heat generated by various sources in the investigated support seal system.

Figure 12. Source of heat generation in the support seal system.

5.1 Heat conduction through the walls

It is vital to note that the heat transfer coefficient (HTC) may only be computed using a design value. The HTC values along the seal and the support walls in the investigated support seal system are difficult to determine through the experimental test; thus, the calculated results from CFD-FEA analysis were employed to assess them. The heat flux conducted from the wall to the oil chamber is then calculated using Newton-Richman’s law.

(3) \begin{align} {Q_{wall}} = \sum\limits_{i = 1}^n {\left[ {{\alpha _i}{F_i}\!\left( {{T_{wall}} - {T_{oil}}} \right)} \right]} \end{align}

Where ${\alpha _i}$ denotes the heat transfer coefficient, F _i denotes area of the outer surface of the wall, ${T_{wall}}$ denotes the temperature of the outer side of the wall, ${T_{oil}}$ denotes the temperature of the oil near the wall of the bearing in the oil chamber.

The heat flux entering the outer wall from the air can be represented as:

(4) \begin{align} {Q_{wall}} = \alpha \left[ {\left( {{F_s} - {F_p}} \right) + {\eta _P}{F_P}} \right] \!\left( {{{\bar T}_{in}} - {{\bar T}_{wall}}} \right)\end{align}

(5) \begin{align} {Q_{wall}} = {\alpha _{cyl}}{F_s}\!\left( {{{\bar T}_{in}} - {{\bar T}_{wall}}} \right)\end{align}

(6) \begin{align} {\alpha _{cyl}} = \alpha \left[ {\left( {{\eta _p} - 1} \right)\frac{{{F_p}}}{{{F_s}}} + 1} \right]\end{align}

Where $\alpha $ denotes the heat transfer coefficient of the air, ${F_s}$ denotes the surface area of the smooth wall, ${F_p}$ denotes the surface area of the fin, ${\bar T_{in}}$ denotes the average temperature of air at the inlet to the support seal system, ${\bar T_{wall}}$ denotes the average wall temperature over the cylindrical surface.

The fin efficiency coefficient ( ${\eta _p}$ ) with respect to the fin area ( ${F_p}$ ) is determined by the following equation:

(7) \begin{align} {\eta _p} = \sqrt {\frac{{{\lambda _{wall}}{\Pi _p}}}{{\alpha {F_p}}}}th\left( {\sqrt {\frac{{\alpha {\Pi _p}}}{{{\lambda _{cm}}{F_p}}}}{h_p}} \right)\end{align}

Where ${\lambda _{wall}}$ denotes wall thermal conductivity, ${\Pi _p}$ denotes the fin circumference, ${F_p}$ denotes the surface area of fin, h _p denotes the height of fin.

The overall heat transfer coefficient through the cylindrical wall of the support seal system will be calculated as follows:

(8) \begin{align} K = \frac{1}{{\frac{1}{{{\alpha _{air}}}} + \frac{{{\delta _{wall}}}}{{{\lambda _{wall}}}} + \frac{1}{{{\alpha _{oil}}}}}}\end{align}

Where ${\delta _{wall}}$ denotes the wall thickness. To determine the value of the overall heat transfer coefficient, in addition to the geometric parameters, it is necessary to know the heat transfer coefficients ${\alpha _{air}}$ and ${\alpha _{oil}}$ , which are found from the known criterial equations.

As shown in Fig. 13, there are three major walls to consider in the researched support seal system. Therefore, the amount of heat transferred through both sides of the wall is determined by the following formula:

(9) \begin{align} {Q_{wall}} = \sum\limits_{i = 1}^n {{K_\Sigma }{F_i}\left( {{{\bar T}_{air,i}} - {{\bar T}_{oil,i}}} \right)} \end{align}

Figure 13. Heat generation through the walls in the support seal system.

Where, ${\bar T_{air}}$ denotes the average temperature of the air near the outer walls, ${\bar T_{oil}}$ denotes the temperature of the oil near the inner walls, which can be determined by the following relationship, taking into account the heating of oil in the friction units of the sealing system:

(10) \begin{align} {\bar T_{oil}} = {T_{in,oil}} + \Delta {T_{friction}}\end{align}

For conical walls represented as a set of cylindrical walls, the average heat transfer coefficient is determined as follows:

(11) \begin{align} {K_\Sigma } = \sum\limits_{i = 1}^n {\frac{1}{{\frac{1}{{{\alpha _{air,i}}}} + \frac{{{\delta _{wall,i}}}}{{{\lambda _{wall,i}}}} + \frac{1}{{{\alpha _{oil}}}}}}} \end{align}

Finally, the following dependency was developed to calculate heat generation through the walls:

(12) \begin{align} {K_\Sigma } = \sum\limits_{i = 1}^n {Q_{wall}} = \sum\limits_{i = 1}^n {\frac{{{F_i}\left( {{{\bar T}_{air,i}} - {{\bar T}_{oil}}} \right)}}{{\frac{1}{{{\alpha _{air,i}}}} + \frac{{{\delta _{wall,i}}}}{{{\lambda _{wall,i}}}} + \frac{1}{{{\alpha _{oil}}}}}}} \end{align}

5.2 Heat conduction through the leakage

The second component (the amount of heat produced with leakage through the seal) in Equation (2) is calculated by the formula:

(13) \begin{align} {Q_{leakage}} = \sum\limits_{i = 1}^n {{{\dot m}_i}{C_p}\!\left( {{T_{out}} - {T_{in}}} \right)} \end{align}

Where ${\dot m_i}$ denotes air mass flow rate through seals, ${C_p}$ denotes specific heat capacity of air, ${T_{in}}$ denotes the air temperature at inlet of the seal, ${T_{out}}$ denotes the air temperature at outlet of the seal. Heat transfer and rotational work transfer (windage heating) cause the change in total air temperature.

5.3 Heat conduction through the cylindrical roller bearing

Most rolling bearings operate at relatively low speeds and loads. The heat dissipation from the bearing through the housing and shaft and its lubrication in an oil chamber is sufficient to ensure the satisfactory thermal condition of such bearings. The greatest amount of heat enters the bearing through the oil, and much of the heat could be transferred through the housing and shaft during engine startup and shutdown.

V.M. Demidovich et al. [Reference Petrov and Lavrentyev12] devised a technique for detecting heat emission in aviation engine rolling bearings in the second part of the 1970s. The author identified friction losses and losses from oil mixing as causes of heat release using similarity criteria. The following dependency was developed to estimate the heat dissipation (W) of friction in these bearings:

(14) \begin{align} {Q_{bearing}} = C\beta z\rho {l^2}{u^3}\end{align}

Where $C$ denotes total drag coefficient, $\beta $ denotes coefficient taking into account the effect of radial clearance on power loss, $z$ denotes number of rolling elements in the bearing, $\rho = 1180.37 - 0.753T$ denotes oil density, $l$ denotes the diameter of the rolling elements, $u$ denotes circumferential speed of the bearing cage.

The total drag coefficient can be found by the formula:

(15) \begin{align} C = 1.26{{\mathop{\rm Re}\nolimits} ^{ - 0.5}}E{u^{-0.5}} + 46.5 \times {10^3}{{\mathop{\rm Re}\nolimits} ^{ - 1}}{\Pr}^{-0.8}\end{align}

The Reynolds criterion is found from the ratio:

(16) \begin{align} {\mathop{\rm Re}\nolimits} = \frac{{ul}}{v}\end{align}

Where $v = 0.0276 \times {2.71828^{((3609.58(\frac{1}{T} - \frac{1}{{311.11}}) + 776568{{(\frac{1}{T} - \frac{1}{{311.11}})}^2}))}}$ denotes the kinematic viscosity of the oil.

Prandtl criterion is equal to:

(17) \begin{align} \Pr = \frac{{{C_p}v\rho }}{k}\end{align}

Where $k = 0.148676 - 0.000222153\left( {T - 366} \right)$ denotes thermal conductivity of the oil.

The Euler criterion can be found by the formulas:

(18) \begin{align} Eu = \frac{{10{F_m}}}{{\rho {{\left( {ul} \right)}^2}}}\end{align}

(19) \begin{align} {F_m} = \frac{{2.92{F_r} + z{F_c}}}{{2z}}\end{align}

(20) \begin{align} {F_c} = 1225\frac{{{l^3}{u^2}}}{{{d_m}}}\end{align}

Where ${F_m}$ denotes the average load on the bearing, ${F_r}$ denotes the radial load on the bearing, ${F_c}$ denotes the centrifugal load of the rolling elements in the bearing.

The circumferential speed of the bearing cage can be found using the formula:

(21) \begin{align} u = \frac{{\pi\!\left( {{d_m} - l} \right)n}}{{120}}\end{align}

Where ${d_m} = \frac{{D + d}}{2}$ denotes the diameter of the centre of gravity of the bearing, equal to the half-sum of the inner and outer diameters.

The coefficient $\beta $ in Equation (14) is found by the formula:

(22) \begin{align} \beta = 1 + 1.7\!\left( {0.1 - {h_{bearing}}} \right)\end{align}

Where ${h_{bearing}}$ denotes the value of radial clearance in the bearing.

5.4 Heat conduction through the shaft

The third component of Equation (2) is analogous to the conduction of heat through walls. The following formula could be used to determine it:

(23) \begin{align} {Q_{shaft}} = \alpha A \!\left( {{T_{air}} - {T_{oil}}} \right)\end{align}

5.5 Heat conduction through the gears, splines

In general, each bearing has a number of gears and splines. Because of the geometry and kinematics of teeth, friction in gear systems differs in certain ways. It comprises rolling and sliding actions. To calculate the heat generated by friction in gear and spline systems, use the following equation:

(24) \begin{align} {Q_{gearings,splines}} = \left( {1 - \eta } \right)N\end{align}

Where $\eta $ denotes the efficiency of gearing system, $N$ denotes the transmitted power (W). For more accurate calculations, the special literature recommends using the experimental dependence obtained by A.I. Petrusevich [Reference Petrov13]:

(25) \begin{align} {Q_{gearings,splines}} = N\frac{{\pi {\varepsilon _s}\,f}}{{K\cos \beta_{g} }}\!\left( {\frac{1}{{{Z_K}}} \pm \frac{1}{{{Z_k}}}} \right)\end{align}

Where ${\varepsilon _s}$ denotes the contact ratio, $f$ denotes coefficient of friction, ${\beta _g}$ denotes the angle of inclination of teeth to the axis of rotation of the starting cylinder (for helical and chevron gears), ${Z_K}$ and ${Z_k}$ denote the number of teeth. The ratio $f/K$ varies from 0.015 to 0.045.

Figure 14. Investigate the radius of the buffer air seal position on the (a) oil seal outlet temperature, (b) oil seal leakage flow rate, (c) oil seal power loss and (d) oil seal power loss.