Conductive heat transfer across a bolted automotive … heat transfer across a bolted automotive...

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International Journal of Heat and Mass Transfer 50 (2007) 4833–4844

Conductive heat transfer across a bolted automotive jointand the influence of interface conditioning

G.P. Voller a,*, M. Tirovic b,1

a Artemis Intelligent Power Ltd., Sanderson Building, Mayfield Road, Edinburgh EH9 3JL, Scotland, United Kingdomb Cranfield University, Cranfield MK43 0AL, England, United Kingdom

Received 14 June 2006; received in revised form 15 February 2007Available online 10 May 2007

Abstract

As part of commercial vehicle disc brake heat dissipation research, thermal contact resistance (TCR) across a bolted joint is analysed.Studies include new and slightly corroded interface surfaces. Measurements show that corrosion approximately doubles TCR, decreasingconductive heat dissipation, leading to higher brake temperatures. To reduce TCR, two methods of interface conditioning are investi-gated. The application of thermal conductance paste and the use of a thin aluminium gasket at the interface have similar effects, reducingTCR by over 80%. The paper deals with the methodology of measuring TCR and defining its relationship with the change of interfacepressure, temperature and interface conditioning. This approach ensures results of a generic nature applicable to a variety of boltedjoints.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Interface pressure; Bolted joints; Thermal contact resistance; Thermal conductance; Brake disc

1. Introduction

Accurate values of thermal contact resistance (TCR) arevital for heat transfer calculations across bolted joints.Reliable heat transfer predictions are important for manybolted joints, for a variety of reasons. In this study, a com-mercial vehicle (CV) disc brake assembly, shown in Fig. 1,was analysed. The heat generated at disc friction surfaces,and conducted through disc ‘top hat’ section, has two pos-sible conductive paths. One, through the disc/wheel carrierinterface and, the other, through the disc/bearing (outerring) interface. The wheel carrier has substantial massand is exposed to external fast flowing cold air, providingconsiderable potential for heat dissipation, making the heatconduction through disc/wheel carrier interface very desir-able. These boundary conditions lower disc surface temper-

0017-9310/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijheatmasstransfer.2007.03.001

* Corresponding author. Tel.: +44 (0) 131 650 7481.E-mail addresses: [email protected] (G.P. Voller), m.tirovic@

cranfield.ac.uk (M. Tirovic).1 Tel.: +44 (0) 1234 754 648.

ature and increase brake performance and life, asresearched by Voller [1] and Tirovic and Voller [2]. How-ever, at the disc/wheel bearing contact surface (Fig. 1),the heat transfer must be minimised, in order to preventbearing overheating and possible failure.

It is apparent that the disc/wheel carrier interface condi-tion will vary throughout the life of the vehicle. In thisresearch, experimental and theoretical studies have beenperformed to determine bolt clamp force, interface pressuredistribution and TCR variations with pressure, tempera-ture and interface condition. The ultimate aim was toestablish a procedure for reliably predicting TCR, whichwill enable heat flow calculations through bolted joints ina variety of engineering applications, not restricting thefindings to the considered automotive assembly.

2. Published TCR studies

Greenwood and Williamson [3] proposed one of the firstmodels to predict contact area known as the GW model,which can be summarised as an elastic micro-contact

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Nomenclature

Aint interface area (m2)dT/dx thermal gradient (K/m)F total bolt clamp force (MN)hcond thermal contact conductance (W/m2 K)kD thermal conductivity of disc material (W/m K)P contact pressure (MPa)Qcond conductive heat transfer (W)Ra arithmetical mean roughness of a surface (lm)T temperature (K)DTint temperature difference at the interface (K)Rcond thermal contact resistance coefficient (m2 K/W)U uncertaintyx axial coordinate (m)

Symbols

h temperature (�C)

Subscripts

int interfaceavg averagelocal localC carrierD disc

4834 G.P. Voller, M. Tirovic / International Journal of Heat and Mass Transfer 50 (2007) 4833–4844

model. The GW model has been modified by McCool [4]using a random process model of rough surfaces. The mod-ified version of the GW model has been applied byMcWaid and Marschall [5] to predict TCR and resultscompare well with experimental data. The GW model doesnot take into account the presence of surface films, whichmay take the form of natural oxidisation, liquid, powder,foil or plating. Research in the area of oxidisation has beensummarised by Madhusudana and Fletcher [6]. A generallyaccepted conclusion is that oxide films, unless sufficientlythick, do not appreciably increase the TCR. Mian et al.[7] have shown experimentally that TCR increases withthe film thickness and the ratio of the total film thicknessto the mean surface roughness. TCR was found to decreasewith the increased loading and the mean surface roughness.

Fig. 1. Cross section and whee

To decrease the TCR, the air within interstitial areas atthe contact interface can be replaced with a medium ofhigher conductance. Interstitial fillers may take the formof grease, metal foil, wire screens or powders. Indium foiland silicon grease appear to be the best materials in thiscategory [6]. It has been shown that thermal conductance(the reciprocal of TCR) can be increased up to a factorof seven, by inserting a metallic foil at the interface [8].Reduction of TCR by metallic coatings has been analysedby Antonetti and Yovanovich [9]. It was found that a silverlayer can reduce the TCR of nominally flat, rough,contacting nickel specimens by as much as an order ofmagnitude; and that for a given layer thickness, thesmoother the bare contacting surface the greater theenhancement will be.

l assembly characteristics.

Table 1Brake assembly interface surface finish

Component Surface condition Surface finish,Ra, [lm]

Front CV disc As supplied new machined 2.0Standard CV wheel

carrierAs supplied, Fig. 2a 1.0–3.3Machined, Fig. 2b 2.1

G.P. Voller, M. Tirovic / International Journal of Heat and Mass Transfer 50 (2007) 4833–4844 4835

Published brake thermal analyses deal with conductiveheat dissipation inadequately. Since the contribution ofthis mode to the brake cooling is, in most service condi-tions, lower than convective and radiative heat dissipation,the contribution of conductive cooling is either neglected oroversimplified. Morgan and Dennis [10] found that con-duction coefficients are extremely variable for the theoreti-cal prediction of brake temperatures. Cetinkale andFishenden [11] suggested that the contact coefficient islikely to change dramatically with time or with subsequentdismantling and reassembly during servicing. Sheridanet al. [12] modelled the conductive heat transfer that existsbetween the disc flange and the hub and wheel by doublingthe convective heat transfer coefficients used on the discfriction surfaces (a value of 200 W/m2 K). An even moresimplified approach was taken by D’Cruz [13] with theuse of a ‘blanket’ heat transfer coefficient of 100 W/m2 Kto all disc free surfaces. In order to determine conductiveheat transfer from a disc brake, Fukano and Matsui [14]changed the apparent rate of heat transfer from the discto the hub until calculated temperatures became ‘identical’with experimental values. The best match was achieved forthe thermal contact conductance of 712 W/m2 K. Thisvalue is certainly adequate for the particular vehicle andtest conditions but does not represent a ‘universal’ value.

3. Analysis of interface conditions

The value of TCR at the interface between two matingsurfaces depends on the surface roughness, material prop-erties, temperature and pressure at the interface, and thetype of interstitial medium. In general, TCR will increasewith increased surface roughness and material hardnessand reduce with interface pressure.

In the case of the considered assembly, the interstitialmedium is air, with possible wetting (water, including roadfilm, salt, snow, etc.) causing moisture penetration. Thebrake component materials and their mechanical and ther-mal properties are defined by component function; greycast iron for the disc and spheroidal graphite (SG) cast ironfor the wheel carrier. Material properties of these compo-nents cannot be altered in order to influence TCR. Further-more, surface finish is determined by the manufacturingmethods (face turning), having typical Ra values between

Fig. 2. Interface conditions: (a) corroded, (b) ne

1 and 3.3 lm, see Table 1. The achievement of lower sur-face roughness is not feasible due to high associated costs.

During initial assembly, the component’s contact sur-faces are corrosion free and it is apparent that the disc/wheel carrier interface condition, and therefore TCR, willvary through out the life of the vehicle. This is mainlydue to moisture penetration and corrosion developmentat the disc/wheel carrier interface. The ‘typical’ wheel car-rier interface condition found in service, for most of the lifeof the vehicle, is shown in Fig. 2a, a newly machined sur-face is shown in Fig. 2b. As the components are boltedtogether, separation is possible for repair and maintenance.However, this is not normally expected for at least 6 yearsof vehicle operation. The main reasons for joint separationwould be disc and/or bearing replacement. Obviously, suchan activity would ‘disturb’ the interface and cause a changein the TCR.

The above two conditions, considering new and slightlycorroded components, represent most bolted jointsexposed to the outside elements. In order to study the pos-sibilities of reducing TCR to increase the conductive heattransfer (and therefore improve brake cooling), two meth-ods have been tested on the slightly corroded ‘typical’ inter-face condition.

The first method of interface conditioning was the inser-tion of a thin aluminium foil ‘gasket’ at the interface, seeFig. 2c. Aluminium has a high thermal conductivity andis very soft, providing good contact at the interface. Thefoil used in the tests was 15 lm thick, standard 99.5% alu-minium foil (see Table 2).

The second method of interface conditioning was theuse of a high thermal conductivity paste, which has theadded benefit of easy application by brush or spray (anadvantage in the vehicle servicing environment). The man-ufacture’s specification is shown below in Table 3, unfortu-nately specific heat and density values were not available.

w machined, and (c) aluminium foil gasket.

Table 2Aluminium gasket properties (at 100 �C) [8]

Foilcomposition

Thickness Density Thermalconductivity

Specificheat

99.5% Al 15 lm 2707kg/m3

240 W/m K 900J/kg K

Table 3Technical specification of heat sink compound

Property Silicone heat sink compound, RS

Dielectric strength 18 kV/mmVolume resistivity 1015 W cmThermal conductivity 0 � 9 W/m KTemperature range �100 to +200 �C


The heat sink compound used is a two metal oxide filledpaste of high thermal conductivity, approximately 35 timeshigher than air at 20 �C. The paste is commonly used toimprove heat transfer between semiconductor devices andheat sinks.

Both methods are considered practically applicable in anindustrial environment, in particular the application of thepaste. To determine the feasibility of the conditioningmethod, the actual gains in heat dissipation must be justi-fied with associated costs.

4. Investigation of interface pressure distribution

The brake assembly is fastened together using ten 12.9grade high tensile alloy steel M16 bolts (see Figs. 1 and2), tightened to the nominal torque of 300 N m. Theexperimental investigations confirmed a linear relationshipbetween the bolt tightening torque and clamp force. Forthe nominal torque, individual bolt clamping forcewas 120 kN, giving an average interface pressure of56.2 MPa.

Fig. 3. High (a) and medium (b) pressure sensitive pape

Interface pressure distributions have been studied theo-retically, using finite element modelling and experimentally,using pressure sensitive paper. For illustration purposes,Fig. 3 shows the Pressurex pressure sensitive paper used(Sensor Products Inc.), after being clamped between the discand the wheel carrier. Due to large variations in pressure,two grades of paper were used, High grade (measures pres-sures from 49.0 to 127.6 MPa) and Medium grade (measurespressures from 9.7 to 49.0 MPa). The change in colour of theinitially white pressure sensitive paper to the shades of red isdirectly proportional to the pressure applied.

It can be clearly seen that the pressure is higher in theproximity of the fixing bolts, and reduces with increaseddistance from the bolts (note two smaller jacking holes dia-metrically opposed). After conducting the experiments, thepressure paper samples were sent to the manufacturer forprocessing. The results were processed in different forms(2D contour plots, line plots, 3D contour plots, and histo-grams) giving detailed information about the pressure dis-tribution on the interface.

It is very important to emphasise that practically identi-cal results were obtained using FE modelling, with ade-quate interface modelling (using gap elements) [15].Obviously, the advantage of FE modelling is the possibilityof designing new joints and predicting pressure distribu-tions. For the considered bolted joint, FE analyses andpressure sensitive paper measurements (see Fig. 3) indi-cated two specific areas, area r in the proximity of thebolts with high pressure (approx. 101 MPa), and area s

between the bolts, with a much lower pressure of approxi-mately 35 MPa.

5. Measurement of TCR

5.1. Experimental set-up

Experiments have been conducted on the CV vehiclebrake disc and wheel carrier assembly (see Fig. 1) installed

r after pressure application at 300 N m bolt torque.

Fig. 4. Experimental set-up for measuring thermal contact resistance.

Fig. 5. Angular position of drilled holes for embedded thermocouples (indegrees).


onto the Spin Rig (dedicated equipment for studying brakecooling characteristics). Fig. 4 shows the experimental set-up. The brake disc and wheel carrier assembly had aslightly corroded ‘typical’ interface condition. The discwas heated using electric air heaters, enabling the regula-tion of the heating power between 0 and 4 kW. To ensureuniform heating, a special heater box was designed andused, and the components were insulated as shown inFig. 4.

Pressure distribution investigations indicated two dis-tinctive areas; high pressure around the bolts, and lowpressure between the bolts (see Fig. 3). Therefore, holeshave been drilled into the disc and carrier in these twoareas as shown in Fig. 5 and correspond to positions r

and s. The holes were positioned with a bilateral toler-ance of 0.1 mm and sized to allow secure fitting of thethermocouples. K-type welded tip glass fibre insulatedthermocouples are used. For each area investigated,the thermocouples measured the temperature gradient ateight points across the interface of the two components(disc and carrier, see Fig. 4). Heat sink compound wasapplied to the bottom of the drilled holes to improve thecontact with the thermocouple tip. Measurements weretaken when steady-state conditions were reached, allowingthe thermocouple tip to reach the same temperature as thesurrounding material. The thermocouple hole depth wasas close to the centre of the components as possible, thiswas to ensure that only conductive heat flow was mea-sured and not heat flow to the surface, generated by anypossible (despite insulation) convective or radiative heatloss. Further thermocouples were installed and experi-ments conducted in order to determine heat losses. Inthe worst test conditions, for the interface temperatureof approx. 170 �C, the heat losses are estimated to bebelow 5%.

5.2. Experimental procedure

The standard CV brake disc and wheel carrier are boltedtogether as shown in Fig. 4. A specific procedure of tight-ening bolts was established in order to obtain consistentresults. The bolts were tightened first to 50 N m, opposingbolts being tightened alternately to avoid high pressureconcentration. The assembly was heated until steady-stateconditions were achieved. By controlling the heater power,three temperature levels were achieved at the interface, inthe regions of approximately, 70, 120 and 170 �C. The 8thermocouple temperatures were logged during a typicalheating cycle at each position (r and s). The criteriafor reaching steady-state condition was set as a tempera-ture change of less than 1.0 �C during 400 s.

The assembly was then cooled down to approximatelyroom temperature (within 1 �C). Bolts were tightened to


the higher torque, in 50 N m increments, and the heating(and data logging) process repeated. It is assumed thatthermal stresses and deflection generated across the threecomponents (brake disc, wheel carrier and bolts) haveinsignificant influence on pressure distribution. It must benoted that reference interface pressure at room tempera-ture was used in all analyses and derived formulae. Themaximum measured temperature variation is typically15 �C across the bolted flanges, having total thickness of30 mm. The thermal expansion coefficients for the grey irondisc, SG iron wheel carrier and steel bolts are very similar(but still somewhat different). Steady state conditions wereexamined, allowing quite uniform heating of the compo-nents, all manufactured from metals of relatively highthermal conductivity. There is no doubt that thermalgradient will influence the interface pressure distribution,despite all the effects mentioned before and the high ‘stiff-ness’ of the bolted joint. This effect will be discussed inmore detail later, with other influencing factors and uncer-tainty analysis.

It must be noted here that one measurement for eachtest condition (pressure and temperature) was conducted,which makes the experiments single-sample type. In theprocess of developing the procedures presented in thispaper, a number of measurements were repeated, ensuringgood repeatability and building confidence in developedmethodology. However, the testing is very time consumingand for all measured conditions, under the imposed limita-tions, it was not possible to conduct multiple-sample exper-iments nor to vary some other parameters authors wouldhave liked to experiment with (such as repeat the measure-ments when releasing the bolts in 50 N m increments).

Fig. 6 shows the average steady-state temperatures atthe eight points (position s) for the interface temperature

Fig. 6. Average temperatures at the mod

of approximately 170 �C. The distance of the thermocouplemeasurement from the interface is plotted on the x-axis ofthe graph. The temperature gradients are proportional tothe conductivity of the material and the steeper gradientof the carrier is a result of its lower conductivity. The pre-dicted temperature drop at the interface is shown in Fig. 6(DTint equal to 2.3 K).

So far, the term ‘thermal contact resistance’ (Rcond

[m2 K/W]) has been mainly used to describe the phenome-non of resistance to conductive heat transfer through theinterface of two solid surfaces in contact. However, despitethis term being most appropriate for describing the phe-nomenon, for actual calculations of conductive heat trans-fer, thermal contact conductance hcond [W/m2 K], is oftenmore suitable. Thermal contact conductance is the recipro-cal value of the TCR coefficient (hcond = 1/Rcond). Theaverage thermal contact conductance (hcond) at the inter-face can be determined from the heat transfer equation:

hcond ¼Qcond

AintDT int

ð1Þ

where Qcond is conductive heat transfer through the inter-face, Aint is the contact area, DTint temperature differenceat the interface, between the disc and carrier surfaces:

DT int ¼ T D � T C ð2Þ

It is assumed that there are no heat losses in the proximityof the interface (again, this will be discussed later) and allthe heat (Qcond) is conducted from the disc to the wheel car-rier. The heat flow is determined by Fourier’s law of heatconduction:

Qcond ¼ �kDAint

dTdx

ð3Þ

ified CV disc/wheel carrier interface.


where kD is disc material thermal conductivity, and thermalgradient dT/dx is already known from Fig. 6 (dT/dx =�278.25 K/m). It should be noted that a correspondingequation can be established for the carrier, by includingthermal conductivity of carrier material (kC) and corre-sponding temperature gradient dT/dx, with the interfacearea (Aint) being identical for both components.

6. The influence of interface pressure and temperature on

TCR

Numerous experiments have been conducted to examinethe change in hcond with contact pressure and temperature.As explained above, the fixing bolts were tightened to sixtorque levels (all bolts to the same torque) graduallyincreased from 50 to 300 N m, in 50 N m increments.Fig. 7 shows hcond values at the two positions (at bolt r

and between bolts s), for 6 average pressures (bolt tor-ques), at the three interface temperatures. The results indi-cate that the intercept at zero average interface pressure isequal for all the measurements. This is a good indication ofthe validity of the approach and measurements, since atzero average pressure (component only ‘gently’ broughtinto contact) there will be no difference in the pressureacross the interface area (for nominally flat contactsurfaces).

Fig. 8 shows hcond values at the two positions (at bolt r

and between bolts s), for the three interface temperatures,at the nominal interface pressure (at a 300 N m bolt tor-que). The data is quite limited and assuming the interfacepressure distribution remains unaffected with the tempera-ture increase (the assumption already mentioned, which

Fig. 7. Thermal contact conductance at t

will be further discussed later), the relationship indicatesslight linear increase of hcond with temperature.

It can be seen from Figs. 7 and 8 that hcond ranges from2800 to 11,400 W/m2 K, increasing with the increase incontact pressure and temperature. There is some scatterbut the trend is a linear increase with both average pressureand temperature. Figs. 7 and 8 also indicate that hcond ishigher in the proximity of the bolts (position r) thanbetween the bolts (position s). This is seen throughoutthe pressure range and corresponds to the increase in inter-face pressure in that region. Increase of thermal conduc-tance hcond with temperature, as indicated in Fig. 8, isrelatively small but consistent within the temperature rangeinvestigated. The increase of hcond is most probably due tohigher radiative heat transfer within the interstitial volumes(filled with air) between the two surfaces (components) incontact. The surfaces are very close and even smallincreases in temperature may well influence radiative heattransfer. However, it is unlikely that such a small differencein the temperature range would significantly influence con-ductive heat transfer between the asperities in contact.Obviously, further work both on the micro and macro scaleis required to investigate the influence of the temperature inmore detail. Some further comments will be given indiscussions.

Using the data shown in Figs. 7 and 8 and the detailedinformation about local interface pressure from Section 4(see also Fig. 3), it is possible to calculate hcond as a func-tion of local pressure and temperature:

hcond ¼ 0:2hP þ 56P þ 2300 ð4Þ

where P is local interface pressure in MPa and h is ‘nomi-nal’ interface temperature in �C. Since, at the interface, the

he standard CV disc/carrier interface.

Fig. 8. Thermal contact conductance at the standard CV disc/carrier interface, change with temperature.


difference between the temperatures of the two components(disc and carrier) is relatively small (of the order of onlyseveral �C), the nominal interface temperature can be as-sumed to be equal to the mean value of the interface tem-peratures of the two components in contact:

h ¼ ðhD þ hCÞ=2 ð5Þ

The linear relationship in Eq. (4) is justified assuming nochange of interface pressure with temperature (no influenceof thermal stresses and deflections, which will be discussedlater), and can be used to calculate average thermal con-

Fig. 9. Comparison of measured and p

ductance (hcond(avg)) for the entire bolted joint, based onaverage interface pressure (Pavg):

hcondðavgÞ ¼ 0:2hP avg þ 56P avg þ 2300 ð6Þ

The average interface pressure can be calculated from thetotal bolt clamp force (F) and interface area:

P avg ¼ F =Aint ð7Þ

The above procedure is fast and simple, therefore suitablefor a variety of reliable engineering calculations. Fig. 9shows the comparison between measured and predicted lo-cal thermal conductance (hcond) using Eq. (4). The results,

redicted local thermal conductance.


assumptions and uncertainties will be discussed in more de-tail later.

7. The influence of corrosion and interface conditioning on

TCR

As explained earlier, in order to reduce TCR, twomethods have been investigated, the use of aluminium foilgasket (see Fig. 2c) and the application of a silicon heatsink compound at the interface. This was done for theslightly corroded surfaces, as expected for the majorityof the life of the components in contact. As in previouscases, the measurements were taken at three tempera-tures, in the range of 70–170 �C, for a nominal bolt tor-que of 300 N m. Average hcond values have been used tocompare the modification techniques with the two ‘initial’interface conditions (new and corroded), as shown inFig. 10.

The average hcond values range from 7 to 67 kW/m2 K.The new joint has an average hcond value of 12 kW/m2 K.This value is in good general agreement with the limiteddata published. Due to corrosion, thermal conductance isreduced to only 7 kW/m2 K. However, the use of heat sinkcompound paste increases conductance to 59 kW/m2 K.The use of aluminium foil provides the highest increasein hcond (approximately 10 times) with an average valueof 67 kW/m2 K. These two practical and inexpensivemethods increase the thermal conductance very dramati-cally. At the same time, it is very likely that such highconductance will remain throughout the lives of compo-nents. In the experiments conducted, the interfaces werecorroded, and both methods provide a seal preventingany deterioration of the contact conditions. Applications

0

10

20

30

40

50

60

70

80

New Corroded

h con

d/k

W/m

2 K

Fig. 10. Average hcond at the interf

on new components are expected to achieve even higherconductance.

8. Discussion and uncertainty analysis

In the presentation so far some of the factors influencingthe measurement results and ultimately thermal conduc-tance (hcond) have been mentioned but not discussed indetail. In order to address these and other factors influenc-ing the uncertainties related to the measurements, thesource of errors are summarised in Table 4. This may alsohelp in deciding the application of the derived formulae toother joints. The list is not complete but all importantsources of error have been included, they have been classi-fied as five broad groups for reasons of clarity.

It should be noted that some sources of error vary rela-tively little, some have a minor influence on the results(such as hcond) whilst the variation and uncertainties relatedto others are much higher. Obviously, the important factoris how much these variations influence the actual resultsand/or predictions. The sources of error should be alsolooked at in two different ways: firstly, the effect and influ-ences in the conducted measurements and formulae deriva-tions (presented in this paper), and secondly the effects andinfluences when applying these formulae to other (howeversimilar) designs.

Including all parameters in the uncertainty analysiswould be very complex task, in particular since identicalcomponents were used, all components were dimensionallyand geometrically accurately measured, the assembly pro-cedure (tightening and positioning of bolts) carefully con-trolled and temperature increase was relatively low. Thesources of error considered particularly important were

Paste Aluminium

ace at the nominal bolt torque.

Table 4Sources of error

Geometrical effects Material effects Clamp force effects Procedure effects Measurementeffects

Geometrical tolerances:� Component flatness at interface� Component flatness at the free

flange side (opposite side to theinterface)� Parallelism of flange surfaces

Dimensional tolerances:� Variation of OD & ID� Bolt holes diameter variation.

Variations in surface finish(Ra, Rz) and surface texture

Variations in:� Young’s

modulus� Hardness� Material con-

dition andmicrostructure� Thermal

conductivity

� Specific heatcapacity

Variation in clamp force:� Bolt geometry� Friction effects� Tightening procedure/

order� Torque measurement

Bolt positioning withinthe hole (‘eccentricity’)

Disturbance of heat flow caused by

temperature measurements (drilling the

holes and inserting thermocouples)

Errors in positioning the bolt holes forthermocouple insertion

Thermal contact resistance at the

thermocouple tip

Interface pressure change due tocomponent deflection at increasedtemperatures

Thermocouple

signal errorsData logger

errorsHeat losses


disc thermal conductivity and temperature measurements,related to the thermocouple signal and data logging errors.These three influencing factors have been marked in bold

italic in Table 4.Grey cast iron thermal conductivity can vary quite sub-

stantially (see [16,17]) and its influence is direct and sub-stantial (see Eq. (3)) on measured results. Howeverthroughout the tests, the same components were used andno change in these properties can be expected, since theachieved temperatures were relatively low. Obviously,some variation with temperature is inevitable. The particu-lar problem with this property is that its measurement isvery expensive and different procedures often give some-what different results. This is one of the most importantproperties for brake disc materials but, for the very reasonsexplained, is (usually) not specified by brake manufactur-ers, neither on documentations nor used for quality controlpurposes. This property is controlled indirectly, by specify-ing chemical composition, microstructure and manufactur-ing processes (casting, heat treatment etc.). Such anapproach ensures consistent thermal, mechanical, frictionand wear properties. Numerous other thermal analysesand measurements were conducted with this type of discand the authors are confident that the value used is veryclose to the ‘true value’. The uncertainty analyses were con-ducted for two values of uncertainties of disc thermal con-ductivity (kD), ±2% and ±5%, considered to be realisticvariation of this property.

Temperature measurements are crucial for the con-ducted experiments, directly influencing the results (Eqs.(1)–(3)). For uncertainty analysis, the thermocouple signalerrors will be given a wider ‘meaning’ and will also includethe ‘procedure effects’ related with the disturbance of heatflow by temperature measurements and thermal contactresistance at the thermocouple tip (marked in italic inTable 4). The uncertainties related to the thermocouple sig-nal error are difficult to realistically estimate. Two values,assuming thermocouple signal error of ±0.1 �C and±0.2 �C were used. It is considered, in particular since mea-surements were conducted for steady state conditions, thatthe first quoted value (±0.1 �C) is more realistic. It must be

noted here that actual thermocouple signal errors are muchsmaller but the ‘allowance’ was made for influences of dis-turbance to heat transfer due to thermocouple installationand thermal contact resistance of the thermocouple tip.

Data logger errors are taken from detailed manufac-turer’s specification. The equipment used is the RS Data-scan 2200 data acquisition system, with an onboardprocessor. The uncertainty in temperature measurementdue to data logger errors is ±0.011 �C, which is of the orderof the magnitude lower than the estimated thermocouplesignal error.

Following the previous considerations the effects of theuncertainties described were analyzed using the uncertaintyapproach explained in [18]. Uhcond

(the uncertainty of hcond)can be calculated in Eq. (1), data logging and thermocouplemeasurement uncertainty directly gives estimates for U T D

and U T C(uncertainties in TD and TC at the interface).

The UAint(uncertainty in Aint) is neglected as ultimately

the method of calculating hcond using Eq. (3) zeros Aint out.

Uhcond¼ ohcond

oQcond

U Qcond

� �2

þ ohcond

oT D

UT D

� �2

þ ohcond

oT C

UT C

� �2( )1=2

ð8ÞThe calculation of U Qcond

(the uncertainty of Qcond), used inEq. (8), involves the uncertainty in the Fourier law equa-tion applied in the direction across the interface, calculatedin Eq. (3). Material thermal conductivity uncertainty di-rectly gives a value for UkD

, (uncertainty in kD). Giving afinal expression for U Qcond

:

UQcond¼ oQcond

okD

U kD

� �2

þ oQcond

odTdx

U dTdx

!28<:

9=;

1=2

ð9Þ

where Udx/dT (the uncertainty of dx/dT), neglecting Udx, isassumed to be:

U dTdx

��max¼ UT D

þ U T Cð10Þ

Uncertainty analysis was carried out for randomly selectedsets of experimental data and the results can be considered

Table 5Uncertainty predictions

Thermalconductivity (kD)uncertainty (±%)

Thermocouplesignaluncertainty(±�C)

Data loggeruncertainty(±�C)

Thermalconductancehcond uncertainty(±%)

2 0.1 0.011 10.65 0.1 0.011 11.55 0.2 0.011 20.22 0.2 0.011 19.7


as an overall value for data shown in Figs. 7–9. The resultsare summarised in Table 5. The authors have confidence inthe first result, indicating around 10% uncertainty in thecalculated thermal conductance (hcond) at the interface. Itmust be noted here that thermal contact resistance is acomplex phenomenon and the research is conducted at‘bulk’ level, aiming at establishing global relationships.

When uncertainty in disc material thermal conductivityis increased from 2% to 5%, the influence on the ultimateresult hcond is very low, under 1%. However, the increasein the thermocouple signal error, from 0.1 �C to 0.2 �Cpractically doubles the uncertainty of the thermal conduc-tance (hcond) to 20%. Reduction of thermal conductivityuncertainty has practically no effect. These calculationswere performed only for illustration purposes, clearly dem-onstrating crucial importance of accurate temperaturemeasurements.

9. Conclusions

Conduction has been the least studied mode of heattransfer in the past. Published work dealing with brakethermal modelling takes a simplified approach to conduc-tive heat dissipation, applying general heat transfer coeffi-cients to brake areas conducting heat.

The work presented here clearly demonstrates that reli-able predictions of thermal conductance can be made. Sim-ple relationships relate interface pressure (local or global)and temperature to thermal conductance at the interface.The influence of interface pressure on thermal conductanceis particularly pronounced. The interface pressure can bereliably theoretically predicted (using FEA) or measured(using pressure sensitive paper). Due to the much lowerinfluence of temperature (compared to interface pressure)on thermal conductance, reasonable temperature estima-tion is sufficient for accurate calculations of heat transferthrough a variety of bolted joints, at either local or globallevel.

It has been shown that corrosion approximately halvesthermal conductance. The use of high thermal conductivitypaste or a thin aluminium gasket at the interface has shownthat very substantial improvements in the conductive heattransfer coefficient can be achieved. The increase is practi-cally of the order of magnitude and likely to remain sothroughout the life of the components.

The phenomenon studied in this paper is not limited tofriction brake heat dissipation analysis; the thermal contactresistance results can be applied to similar multi-solidapplications for accurate theoretical prediction of conduc-tive heat transfer across a variety of bolted joints. Obvi-ously, the joint behaviour with increase in temperatureshould be carefully examined, since high interface thermalcontact resistance (low hcond), ‘flexible joints’ (subject tolarge variations in interface pressure due to thermal stressesand deflections) and/or large thermal gradients may sub-stantially influence interface pressure and therefore thermalconductivity across the bolted joint.

It must be taken into consideration that the thermalcontact resistance of the bolted joint remains a very com-plex phenomenon and derived equations should be onlyapplied to suitable designs and with adequate care. Discus-sion of results and uncertainty analyses gave useful furtherinsight into the associated sources of error and their influ-ence on results obtained.

Acknowledgements

The work presented in this paper was made possible bythe assistance obtained from the EPSRC (Engineering andPhysical Sciences Research Council), ArvinMeritor (UK)and Brunel University, Department of Mechanical Engi-neering. The authors are grateful for this help.

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