# Computational fluid dynamics (CFD) analysis and numerical aerodynamic investigations of automotive disc brake rotor.

1. Introduction

Recently, Belhocine and Wan Omar (2017) investigated the structural and mechanical behaviour of a three-dimensional disc-pad model during the braking process under dry contacts slipping conditions. After that, Belhocine (2017) investigated the structural and contact behaviours of the brake disc and pads during the braking phase with and without thermal effects. When a vehicle brake, a part of the friction temperature escapes into the air through convection and radiation. Therefore, the determination of heat transfer coefficients (HTC) is very important. It is, however, very difficult to calculate them precisely because they depend on the shape of the braking system, the speed of movement of the vehicle, and consequently on the air circulation. Here, the modelling of convection proves to be the main problem because it is related to the aerodynamic conditions of the disc.

In this work, we are interested in this part of the calculation of the heat exchange coefficient (h). This parameter must be evaluated to visualise the three-dimensional distribution of the temperature of the disc. The strategy of calculation is based on the software ANSYS CFX 11 which the numerical simulation of the transient thermal field is carried out using ANSYS (Zhang et al. 2009). A comparison was made between the temperature of full and ventilated brake disc showing

2. Principle of disc brakes

The principle of a disc brake is very simple: a metal disc, the rotor, is firmly mounted to the rotating wheel. The pads grip the rotating disc when the brakes are applied. The disc brake system consists of the following parts (see Figure 1):

* The pads -- two flat elements made of friction material. They run perpendicular to the disc and grip it when a brake force is applied.

The rubbing on the moving disc surface causes it to slow down.

* The calliper -- which contains a piston that presses the pads against the disc surface. The calliper is mounted on the chassis.

* The rotor -- which is mounted on the hub and is therefore rotating with the wheel.

The vehicle in motion can be considered as a permanent energy conversion system: chemical energy is transformed through combustion into mechanical energy. This energy is then transformed into kinetic or potential energy of a car. When braking, the vehicle's kinetic energy is entirely converted into thermal energy by the frictional interaction between brake pads and the rotor.

The generated heat is distributed between the rotor and the pads. This partition between pads and disc depends highly on the physical-chemical properties of the two materials. Over 95% of this thermal energy flows into the rotor and is stored, conducted to the hub and the wheel, and for longer brake applications convected to the surrounding air.

3. Brake disc types

There are two types of discs: full discs and ventilated discs. The full discs, of simple geometry and therefore of simple manufacture, are generally placed on the rear axle of the car. They consist simply of a solid crown connected to a 'bowl' which is fixed to the hub of the car (Figure 2(a)). Ventilated discs of more complex geometry appeared later. They are mostly on the front axle. However, they are increasingly in the rear and front of the high-end cars. They are composed of two crowns--called flanges--separated by fins (Figure 2(b)), they cool better than full discs thanks to ventilation between the fins, which in addition promote convective heat transfer by increasing the exchange surfaces. The ventilated disc has more material than the full disc; its heat absorption capacity is therefore better. The number, size and shape (radial fins, curves, circular pins, etc.) of the fins are variable.

The gradients in the throat of the bowl are explained in the same way. At the beginning of braking, the temperature of the bowl is at 20 [degrees]C while that of the tracks is a few hundred degrees. In addition, in order to prevent the hub temperature from being too high (which would cause tyre temperature increases, which is very critical of its behaviour), the throat is machined so as not to transmit too much heat to the hub Bowl (Figure 3). With this machining, the temperature of the bowl actually decreases, but the thermal gradients increase consequently in this zone. These give rise to thermal stresses which explain the rupture of bowl observed during severe experimental tests.

The rotation of the disc causes an air circulation in the channels, resulting in improved cooling (Figure 4).

4. Governing equation of the transient heat conduction

Consider an isotropic material in the three-dimensional system in a domain [OMEGA]. If the heat flow in the directions x, y and z axes per unit area and in a unit time is denoted by [q.sub.x], [q.sub.y] and [q.sub.z], respectively, the difference between the outflow and the inflow for a volume element dxdydz is given as,

dydz([q.sub.x] + [[partial derivative][q.sub.x]/[[partial derivative]x]] - [q.sub.x]) + dxdz)([q.sub.y] + [[partial derivative][q.sub.y]/[[partial derivative]y]] - [q.sub.y]) + dxdy([q.sub.z] + [[partial derivative][q.sub.z]/[[partial derivative]z]] - [q.sub.z]) (1)

For conservation of heat, this quantity must be equal to the sum of the heat generated in the element in unit time Qdxdydz and the heat gained in a unit time due to the temperature change, namely -[rho]c[[partial derivative]T]/[[partial derivative]t] dxdydz, where c is the specific heat capacity, p is the density and T (x, y, z, t) is the temperature distribution. The requirement of equality leads to the differential relationship:

[[partial derivative][q.sub.x]/[[partial derivative]x]] + [[partial derivative][q.sub.y]/[[partial derivative]y]] + [[partial derivative][q.sub.z]/[[partial derivative]z]] - Q - [rho]c[[[partial derivative]T]/[[partial derivative]t]]dxdydz (2)

The heat fluxes in the x, y and z directions are:

[mathematical expression not reproducible] (3)

Substitution of these heat fluxes into Equation (2) leads to a higher order differential equation in a single independent variable T,

[[partial derivative]/[[partial derivative]x]](k[[partial derivative]T]/[[partial derivative]x]) + [[partial derivative]/[[partial derivative]y]](k[[[partial derivative]T]/[[partial derivative]Y]]) + [[partial derivative]/[[partial derivative]Z]](k[[[partial derivative]T]/[[partial derivative]Z]]) + Q - [rho]c[[[partial derivative]T]/[[partial derivative]t]]dxdydz = 0 (4)

In a similar way, three-dimensional conduction of heat can be obtained. It may also be written in vector notation as,

[nabla].k[nabla]T + Q = [rho]c[[[partial derivative]T]/[[partial derivative]t]]dxdydz (5)

4.1. Convection

Convection is a transfer of heat in a material medium with a movement of matter. This mode of transfer, therefore, concerns only fluids or exchanges between a solid and a fluid. This transfer mechanism is governed by Newton's law (Figure 5):

[phi] = gS([T.sub.P] - [T.sub.[infinity]]) (6)

with:

[phi] Heat flow transmitted by convection (W), h convective HTC ([Wm.sub.-2] [degrees][C.sup.-1]), [T.sub.p] surface temperature ([degrees]C), temperature of the medium surrounding the surface ([degrees]C), S area of contact surface solid/fluid ([m.sup.2]).

5. Calculating heat flux entering the disc

5.1. The forces acting on the wheels during braking

By observing the situation described in Figure 6, If we define all the parameters appearing on it: [F.sub.RA]: aerodynamic force acting on the body of the vehicle, h: height of the centre of gravity from the ground, [F.sub.G]: weight of the vehicle acting at its (CG), [F.sub.Qv] [F.sub.QH]: dynamic weight carried on the front and rear wheels, [F.sub.RRv] [F.sub.RRH]: rolling resistance at the tyre contact patch, [F.sub.D]: inertia force, [F.sub.RP]: resistance force from the slope, [F.sub.FV] [F.sub.FH]: tractive force, L: wheel base length, [L.sub.H]: distance of the centre of gravity aft of the front axle and [L.sub.V]: distance of the centre of gravity fore of the front axle, the longitudinal and transverse equilibrium of the vehicle can be written along the local axes x, y of the car.

[summation][F.sub.x] = 0 [??] [F.sub.RRV] + [F.sub.V] + [F.sub.RRH] + [F.sub.PH] + [F.sub.RA] - ([F.sub.RP] + [F.sub.D]) = 0 (7)

[F.sub.FV] + [F.sub.FH] = [F.sub.RP] + [F.sub.D] - [F.sub.RRV] - [F.sub.RA] - [F.sub.RRH] (8)

[F.sub.F] = [F.sub.RP] + [F.sub.RF] - [F.sub.RR] - [F.sub.RA] (9)

with

[F.sub.F] = [F.sub.FV] + [F.sub.FH] [F.sub.RR] = [F.sub.RRV] + [F.sub.RRH]

[summation][F.sub.Y] = 0 [??] [F.sub.G] COS [alpha] - ([F.sub. QV] + [F.sub. QH]) = 0. (10)

[F.sub. QH] = [F.sub. G] COS [alpha] - [F.sub.QV]. (11)

[summation][M.sub.B] = 0 [??] [F.sub.QV] L + ([F.sub. RA]h - h[F.sub. RF] + [F.sub. RP] ) - h[F.sub.G] sin [alpha] (12)

[F.sub.QV] = [([F.sub.RF] + [F.sub.RP])h + [F.sub.G][L.sub.H] - [F.sub.RA]h]/L (13)

For a road vehicle, the rolling force [F.sub.RR] = [F.sub.G] [f.sub.r] cos [alpha] is due to the flat formed by a tyre on the road, [f.sub.r] is the rolling resistance coefficient. For a high pressure tyre ([f.sub.r] = 0.015).

[F.sub.RP] = [F.sub.G] sin [alpha] (14)

The aerodynamic force is given by:

[F.sub.RA] = [C.sub.X][A.sub.F][[[rho].sub.a]/2][v.sup.2] (15)

With [C.sub.X] coefficient of form, equal to: 0.3 to 0.4 on car [A.sub.F] ([m.sup.2]) frontal surface; in the approach, for a road passenger vehicle, we can take:

[A.sub.F] = 0.8 x height x width S

[[rho].sub.a] air density.

5.2. Total braking power

[P.sub.tot] = [P.sub.R] + [P.sub.F] (16)

[P.sub.F] = [summation] [F.sub.F]v = ([F.sub.FV] + [F.sub.FH]) v (17)

[P.sub.R] = [summation] [F.sub.R]v = ([F.sub.RR] + [F.sub.RP] + [F.sub.RA]) v (18)

where v is the velocity of the vehicle,

In the case of flat braking (Figure 7), the resistances due to rolling and to the slope are neglected [F.sub.RR] = 0 ([F.sub.RR] = 0 and [F.sub.RP] = 0), the penetration into the air is generally negligible, for this reason, ([F.sub.RA] = 0).

[P.sub.R] = [summation][F.sub.R]v = ([F.sub.RR] + [F.sub.RP] + [F.sub.RA])v = 0. (19)

[P.sub.F] = [summation] [F.sub.F]v = ([F.sub.FV] + [F.sub.FH])v. (20)

([F.sub.FV] + [F.sub.FH]) = [F.sub.D] = ma. (21)

[P.sub.tot] = [P.sub.F] = mav. (22)

Let [phi] the coefficient which represents the proportion of the braking force relative to the rear wheels [P.sub.FH] = [phi]mav then, [P.sub.FV] = (1-[phi])mav if a is constant, we have:

v(t) = [v.sub.0] - at. (23)

[P.sub.F] = (1 - [phi])ma([v.sub.0] - at). (24)

The braking power delivered to the brake disc is equal to half the total power:

[P.sub.FVI] = [(1 - [phi])/2]ma([v.sub.0] - at). (25)

At time t = 0, we have

[P.sub.FVI] = [(1 - [phi])/2]ma [[v.sub.0]](26)

The braking efficiency is then defined by the ratio between the deceleration (a) and the acceleration (g):

Z = [[a.sub.d]/g]. (27)

[P.sub.FVI] = [(1 - [phi])/2]. (28)

The purpose of the brake discs is to dissipate mechanical energy into heat. For trains or cars, it is the kinetic energy of the vehicle that is dissipated by the friction of the skates on the discs. The disc pad assembly heats up under this action and cools in the ambient air. As these brakes are repeated, the brake discs are subjected to thermo-mechanical fatigue. In the automotive industry, many studies have shown that braking can generate temperatures in excess of 700 [degrees]C in a matter of seconds.

Considering that the brake disc can totally absorb the amount of heat produced.

[Q.sub.v] = [(1 - [phi])]/2] [m.sub.tot]gv. (29)

In the brake applications, there are usually two brake pads per disc, with each rotor, there are two contact surfaces (2[A.sub.d]) swept by an inner and outer brake pad. The expression of the transformed friction power per unit area is thus:

[Q'.sub.v] = [[(1 - [phi])]/2][[m.sub.tot]gv]/2[A.sub.d][[Nm]/[[sm.sup.2]]] = [[W/[m2.sup.m2]]. (30)

The quantity [Q'.sub.v] characterises the heat flux injected into the disc, it must therefore be located only on the actual contact surface. Where [A.sub.d], the disc surface swept by a brake pad.

If we introduce the factor of exploitation of the friction surface

[[epsilon].sub.p] = [[Q'.sub.v]/[Q.sub.vmax]. (31)

Thus, the equation of the initial thermal flow of friction entering the disc, which is calculated as follows:

[Q'.sub.vmax] = [[(1 - [phi])]/2][[[m.sub.tot]gv]/2[A.sub.d][[epsilon].sub.p]][[Nm]/[sm.sup.2]] = [[W/[m.sup.2]]. (32)

5.3. Heat flux entering the disc

In the course of braking, the kinetic energy and potential energy of the automobile are converted into thermal energy through the frictional heating behaviour between the interfaces of the friction pair: the frictional heat is generated on the interfaces of the brake disc and brake pads. The initial heat flux [q.sub.0] into the rotor face is directly calculated using the following formula (Reimpel 1998):

[q.sub.0] = [[1-[empty set]]/2]mg[v.sub.0]z/2[A.sup.d][[epsilon].sub.p]]. (33)

where a is the deceleration of the vehicle [[ms.sup.-2]] and g is the acceleration of gravity (9.81) [[ms.sup.-2]].

Our work consists in studying the thermal behaviour of a three-dimensional brake disc, which includes the heat flux generated inside the brake disc, the maximum and minimum temperatures, and so on. The scenario analysed is a stop brake. In practice, the braking system bathes in an air flow, more or less forced according to the system, which participates in the cooling of the disc and of the plates. This air flow is governed by the laws of aerodynamics. The values of the heat exchange coefficient h as a function of time are calculated using the ANSYS CFX code. These values will be used to determine the thermal behaviour of the disc in transient mode. The brake disc consumes the major part of the heat, usually greater than 90% (Cruceanu 2007), by means of the effective contact surface of the friction coupling. Considering the complexity of the problem and the limitation in the average data processing, one identifies the pads by their effect, represented by an entering heat flux (Figure 8).

The loading corresponds to the heat flux on the disc surface. The study is based on a ventilated brake disc of high carbon grey cast iron FG; it is the disc (262 x 29 mm) that is fitted to certain versions of the vehicles (Figure 9).

To facilitate comparison of simulation results, the geometric dimensions of the two disc variants, full disc and ventilated disc are the same

The dimensions and the parameters used in the thermal calculation are described in Table 1.

The disc material is grey cast iron (FG 15) with high carbon content (Gotowicki, Nigrelli, and Gabriele Virzi 2005), good thermophysical characteristics, whose thermo-elastic characteristics adopted in this simulation in the transient analysis of the disc are listed in Table 2. It is very difficult to exactly model the brake disc, in which there are still researches are going on to find out transient thermal behaviour of disc brake during braking applications. There is always a need of some assumptions to model any complex geometry. These assumptions are made, keeping in mind the difficulties involved in the theoretical calculation and the importance of the parameters that are taken and those which are ignored. In modelling, we always ignore the things that are of less importance and have little impact on the analysis. The assumptions are always made depending upon the details and accuracy required in modelling. To simplify the analysis, several assumptions have also been made as follows (Khalid et al. 2011):

* All kinetic energy at disc brake rotor surface is converted into frictional heat or heat flux.

* The heat transfer involved for this analysis only conduction and convection process. This heat transfer radiation can be neglected in this analysis because of small amount which is 5-10% (Limpert 1999).

* The disc material is considered as homogeneous and isotropic.

* The domain is considered as axisymmetric.

* Inertia and body force effects are negligible during the analysis.

* The disc is stress free before the application of brake.

* In this analysis, the ambient temperature and initial temperature has been set to 20 [degrees]C

* All other possible disc brake loads are neglected.

* Only certain parts of disc brake rotor will apply with convection heat transfer such as cooling vanes area, outer ring diameter area and disc brake surface

The thermal conductivity and specific heat are a function of temperature, Figures 10 and 11.

6. Modelling in ANSYS CFX

The air flow characteristics around the brake components are highly complex and they can vary significantly with the underbody structure as well as the component shapes. Instead of using empirical equations, which are commonly used in the thermal analysis (Dittrich and Lang 1984; Fukano and Matsui 1986), the average HTCs are calculated from the measured cooling coefficients by an iteration algorithm. Since the cooling coefficients account for all three modes of heat transfer, the estimated HTCs include the equivalent radiation HTC. The solution scheme employees the [kappa]-[epsilon] model with scalable wall function and sequential load steps. For the preparation of the mesh of computational fluid dynamics (CFD) model, one defines initially, various surfaces of the disc in ICEM CFD as shown in Figures 12 and 13; we used a linear tetrahedral element with 30717 nodes and 179798 elements. In order not to weigh down calculation, an irregular mesh is used in which the meshs are broader where the gradients are weaker (nonuniform mesh) (Figure 14). The CFD models were constructed and were solved using ANSYS-CFX software package. The model applies periodic boundary conditions on the section sides; and the radial and axial lengths of the air domain surrounding the disc. The disc is modelled attached to an adiabatic shaft whose axial length spans that of the domain. The air around the disc is considered at [T.sub.8] = 20 [degrees]C and open boundaries with zero relative pressure were used for the upper, lower and radial ends of the domain.

6.1. Preparation of the mesh

This stage consists in preparing the mesh of the fluid field. In our case, one used a linear tetrahedral element with 30717 nodes and 179798 elements (Figure 15).

Considering symmetry in the disc, one took only the quarter of the geometry of the fluid field (Figure 16) using software ANSYS ICEM C[F.sub.D].

In this step, one declares all of the physical characteristics of the fluid and the solid. It was introduced into the library, the physical properties of materials used. In this study, we selected three grey cast iron materials (FG25AL, FG20, FG15) having a thermal conductivity, respectively, (43.7 W/m[degrees]C, 55 W/m[degrees]C and 57 W/m[degrees]C). Since the aim of this study is to determine the temperature field in a disc brake during the braking phase of a vehicle of the average class, we take the following temporal conditions:

* Braking time = 3.5 [s]

* Increment time = 0.01 [s]

* Initial time = 0 [s]

The disc is attached to four adiabatic surfaces and two surfaces of symmetry in the fluid domain with an ambient air temperature of 20 [degrees]C.

Figure 17 shows the elaborated C[F.sub.D] model to be used in ANSYS cfx Pre.

The airflow through and around the brake disc was analysed using the ANSYS CFX software package. The ANSYS-CFX solver automatically calculates HTC at the wall boundary. Afterwards the HTCs considering convection was calculated and organised in such a way, that they could be used as a boundary condition in thermal analysis. Averaged HTC had to be calculated for all disc using ANSYS CFX Post as it is indicated in Figures 18-21.

Figures 19-21 show the distribution fields of the exchange coefficient (h) for the three types of materials. It is found that the behaviour of (h) in the disc does not depend on the material chosen. The distribution of (h) in the disc is not the same as that existing in the literature (see Table 3).

Table 4 shows the mean values of the heat exchange coefficient (h) calculated by the minimum and maximum values of the various surfaces of the ventilated disc. It is found that the type of the material does not have a great influence on the variation of the heat exchange coefficient (h). Contrary to the first case, it is found that the value of the heat exchange coefficient (h) is strongly influenced by the ventilation system for the same material (FG 15).

The modelling of the disc temperature is carried out by simulating a stop braking of a middle class car (braking of type 0).The vehicle speed decreases linearly with time until the value 0.

Braking down from the maximum speed of 100.8 km/h to a standstill as shown in Figure 22.

The variation of the heat flux during the simulation time is represented on Figure 23.

8. Meshing details

The goal of meshing in Workbench is to provide robust, easy to use meshing tools that will simplify the mesh generation process. The model using must be divided into a number of small pieces known as finite elements. Since the model is divided into a number of discrete parts, in carry out a finite element analysis. A finite element mesh model generated is shown in Figure 24.

9. Thermal boundary conditions

The boundary conditions are introduced into module ANSYS Workbench [Multiphysics], by choosing the mode of first simulation of the all (permanent or transitory), and by defining the physical properties of materials. These conditions constitute the initial conditions of our simulation. After having fixed these parameters, one introduces a boundary condition associated with each surface. Total time of simulation = 45 s

* Increment of initial time = 0.25 s

* Increment of minimal initial time = 0.125 s

* Increment of maximal initial time = 0.5 s

* Initial temperature of the disc = 20 [degrees]C

* Material: three types of Grey Cast iron (FG25AL, FG20, FG 15.

* Convection: one introduces the values of the HTC (h) obtained for each surface in the shape of a curve (Figures 25, 26).

* Flux: one introduces the values obtained by flux entering by means of the code CFX.

10. Results and discussions

10.1. Results of the calculation of the heat transfer coefficient (h)

The HTC is a parameter relates with the velocity of air and the shape of brake disc, and many other factors. In different velocity of air, the HTC in different parts of brake disc changes with time (Zhang and Xia 2012). HTC will depend on air flow in the region of brake rotor and vehicle speed, but it does not depend on material.

From the comparison between Figures 25 and 26 concerning the variation of heat transfer coefficients in the non-stationary mode for the two types of design full and ventilated disc, one notes that the introduction of the system of ventilation directly influences the value of this coefficient for same surface, which is logically significant because this mode of ventilation results in the reduction in the differences in wall-fluid temperatures. Figures 27 and 28 show, by way of example, the change in the exchange coefficient (h) for the surface SPV2 and SV1 for each time step.

10.2. Thermal analysis

Figure 29 shows for each type of the selected cast iron (FG 25 Al, FG 20, FG 15), the temperature variation at the moment when the temperature is maximum. More the thermal conductivity of the material is low, more its temperature is high. The FG 15 is differentiated from the two other cast iron by smaller temperatures. According to Figure 29, the cast iron FG 15 has the best thermal behaviour. For the full disc, the temperature reaches its maximum value of 401.55 [degrees]C at the moment t = 1.8839 s, and then it falls rapidly at time t = 4.9293 s until the end of simulation (t = 45 s), the variation in the temperature becomes slow. It is noted that the interval [0-3.5] s represents the phase of forced convection as seen by Figures 31,32. During this phase, one observes the case of the free convection until the end of the simulation. In the case of the ventilated disc, one observes that the temperature of the disc falls approximately by 60 [degrees]C compared with the first case. It is noted that the ventilation in the design of the discs of brake plays an important role in producing a better system of cooling.

11. Conclusion

In this work, it was presented the analysis of the thermo-mechanical behaviour of the dry contact between the brake disc and pads during the braking process; the modelling is based on the ANSYS 11.0. We have shown that the ventilation system plays an important role in cooling discs and provides a good high temperature resistance. The analysis results showed that, temperature field and stress field in the process of braking phase were fully coupled. The temperature, Von Mises stress and the total deformations of the disc and contact pressures of the pads increases as the thermal stresses are additional to mechanical stress which causes the crack propagation and fracture of the bowl and wear of the disc and pads. Regarding the calculation results, we can say that they are satisfactory commonly found in the literature investigations. It would be interesting to solve the problem in thermo-mechanical disc brakes with an experimental study to validate the numerical results, for example on test benches, in order to show a good agreement between the model and reality.

Disclosure statement

No potential conflict of interest was reported by the authors.

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Ali Belhocine (a) and Wan Zaidi Wan Omar (b)

(a) Department of Mechanical Engineering, University of Sciences and the Technology of Oran, Oran, Algeria; (b) Faculty of Mechanical Engineering, universiti teknologi Malaysia, skudai, Malaysia

KEYWORDS

CFD; heat flux; disc brake; heat transfer coefficient; temperature

ARTICLE HISTORY

Accepted 26 April 2017

https://doi.org/10.1080/14484846.2017.1325118
```Table 1. Geometrical dimensions and application parameters of
automotive braking.

Inner disc diameter, mm                                         66
Outer disc diameter, mm                                        262
Disc thickness (TH), mm                                         29
Disc height (H), mm                                             51
Vehicle mass m, kg                                            1385
Initial speed [v.sub.0], m/s                                    28
Deceleration a, m/[s.sup.2]                                      8
Braking time [t.sub.[for all]]s                                  3.5
Effective rotor radius [R.sub.rotor], mm                       100.5
Rate distribution of the braking forces [phi], %                20
Factor of charge distribution of the disc [[epsilon].sub.p]      0.5
Surface disc swept by the pad [A.sub.d], [mm.sup.2]          35993

Table 2. Thermo-elastic properties used in simulation.

Material Properties                            Disc

Thermal conductivity, k (W/m[degrees]C)          57
Density, p (kg/[m.sup.3])                      7250
Specific heat, c (J/kg. [degrees]C)             460
Poisson's ratio, v                                0.28
Thermal expansion, a ([10.sup.-6]/[degrees]C)    10.85
Elastic modulus, E (GPa)                        138

Table 3. Value of the HTC of different surfaces in the stationary case
for a full disc (FG 15).

FG 15
Surface  [h.sub.moy] = [W [m.sup.-2] [k.sup.-1]]

SC1                    25,29168
SC2                     5,18003
SC3                    2,922075
SC4                    11,77396
SF1                   111,20765
SF3                    53,15547
ST2                    23,22845
ST3                     65,6994
ST4                    44,26725
SV1                    81,37535
SV2                    71,75842
SV3                    41,83303
SV4                    65,82545

Table 4. Values of the HTC of different surfaces in the stationary case
for a ventilated disc (FG 25 AL, FG 20 and FG15).

Materials   FG 25 AL         FG 20                   FG15

Surface                   [h.sub.moy] =
[W [m.sup.-2] [k.sup.-1]]
SC1         54,16235         53,926035              53,8749
SC2          84,6842          83,7842               83,6516
SC3          44,4171          44,3485              44,32945
SF1 et 2    135,4039         135,0584             135,00065
SF3         97,17095          95.0479               94,8257
SPV1       170,64715         171.4507             171,56955
SPV2       134,08145         134.3285              134,3615
SPV3        191,2441         191.9436              192,0391
SPV4       175,16665         176,13395             176,2763
ST1         113,6098         114,3962            114,391555
ST2          35,0993          34,47225              34,3473
ST3         68,33155          66,33155              66,0317
ST4         75,09445          72,1235               71,6642
SV1         135,5299         131,11825            131,20745
SV2        119,25715         118,464835           118,20395
SV3         46,70225          44,8195              44,52635
SV4        111,57685         108,5044              108,1817
```
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