Estimation of railway wheel running temperatures using a hybrid approach

Wear 328-329 (2015) 537–551

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Estimation of railway wheel running temperatures using a hybrid approach M.R.K. Vakkalagadda, K.P. Vineesh, V. Racherla n Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India

art ic l e i nf o

a b s t r a c t

Article history: Received 12 December 2014 Received in revised form 23 March 2015 Accepted 28 March 2015 Available online 2 April 2015

Accurate prediction of railway wheel temperatures, for a given running and braking history, is crucial for estimation of service induced thermal stresses which can affect wheel gauge and fatigue crack growth, particularly in tread region. In this work, a three step approach is adopted for estimating wheel temperatures. Firstly, a train running model employing brake block friction, wheel–rail traction-slip, and train running resistance characteristics are used to estimate heat generation rates at brake block–wheel and wheel–rail interfaces. Next, a two dimensional boundary element method, proposed in this work, is used to estimate heat partitioning at the said interfaces as a function of brake block type, geometry, and thermal properties. Lastly, finite element method is used to estimate wheel temperatures taking inputs from the train running model and the boundary element method. The adopted methodology is validated using wheel temperatures, on rim and rim–disc interface, from field trials of a locomotive fitted with cast-iron brake blocks. A good match between simulation and field trial results was obtained despite highly complex speed and braking patterns encountered in the field trial. For periodic braking with no stopovers, heat lost to the rail, cast-iron brake blocks, via radiation, and convection are seen to be comparable to one another. Enhanced convective cooling of railway wheels and use of higher diffusivity brake blocks are seen to be effective ways of reducing the wheel temperatures and resulting thermal stresses. & 2015 Elsevier B.V. All rights reserved.

Keywords: Finite element modelling Rail–wheel tribology Brakes Cast iron Polymer-matrix composite

1. Introduction Indian Railways is one of the world's largest rail networks with over 100,000 track kilometres. It carries over 8 billion passengers and 1 billion tons of freight annually and forms the country's economic backbone. For smooth operation of increasing traffic, there is a need to reduce component failures and maintenance issues. Most of the trains operated by Indian Railways employ tread braking which results in hot running locomotive and wagon/coach wheels. Ekberg and Kabo [1] showed that tread braking induced thermal stresses affect fatigue crack growth, particularly in tread region. In addition, Teimourimanesh et al. [2] showed that braking induced thermal stresses in wheels can cause wheel warping, which can in turn affect wheel gauge and result in derailment. Thus, accurate prediction of running temperatures for tread braked railway wheels is crucial for taking corrective measures to improve wheel life and prevent derailments. Accurate estimation of wheel temperatures for any given train running and braking conditions, using an approach of the kind n

Corresponding author. Tel.: þ 91 3222 282900; fax: þ91 3222 282278. E-mail address: [email protected] (V. Racherla).

http://dx.doi.org/10.1016/j.wear.2015.03.026 0043-1648/& 2015 Elsevier B.V. All rights reserved.

presented in this work, is crucial for the following: (i) Understanding wear mechanism in railway wheels subjected to tread braking. More specifically, correct estimation of wheel temperatures and resulting thermal stresses in tread region is crucial for identification of conditions that can cause/accelerate rolling contact fatigue in wheels [1], (ii) Identifying running and braking conditions that accelerate wear and damage of composite brake blocks. Under certain braking conditions, e.g. application of brakes only on locomotive wheels in a passenger/goods train, brake block temperatures can get quite high (600 °C) because of which several constituents in composite brake blocks vapourize and result in severe cracking and wear of brake blocks [3], (iii) Choosing lubricants to be used on curved segments of rail for reducing wear from wheel–rail flange contact [4] and for avoiding occurrence of rail corrugations [5]. Since lubricant properties are temperature dependent, they are expected to perform differently for different rail–wheel interface temperatures which are in turn affected by prior running and braking history, (iv) Studying formation of wheel flats and microstructural changes in wheel and rail. For example, for partial/full wheel locking, wheel–rail interface temperatures can be quite high and result in formation of wheel flats. White-etching layers are normally found around the wheel flats which signify formation of martensite from heating to above austenizing temperature followed by rapid self-quenching [6], and

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Nomenclature BCP bt c h hwr hax

k Q̇ wr Q̇ wb

Qr (τ) r

brake cylinder pressure (bar) brake block thickness (mm) specific heat (J/kg K) convective heat transfer coefficient (W/m2 K) thermal conductance at wheel–rail interface (W/m2 K) effective convective heat transfer coefficient used to model rail-chill effect in axi-symmetric analysis (W/m2 K) thermal conductivity (W/m K) heat generation rate at wheel–rail interface (W) heat generation rate per brake block at wheel–brake block interface (kW) time varying surface power input (W/m2) radial distance (m)

(v) Investigating wear rates for rail, wheel and brake blocks that are affected by thermal stresses and operating temperatures [7]. Prediction of wheel running temperatures for a given train running and braking history is challenging for several reasons. First, brake block friction characteristics – which depend on sliding speed, sliding distance, and brake load [8,9] – along with wheel– rail traction-slip and running resistance characteristics need to be modelled to estimate heat generation rate at wheel–brake block and wheel–rail interfaces for the given train running and braking history. Second, the fraction of heat generated entering the brake blocks, rail, and wheel needs to be estimated, by solving a moving heat source problem. Lastly, based on the knowledge of net heat entering the wheel, wheel temperatures need to be obtained by using field data calibrated emissivity and convective heat transfer coefficients for locomotive and wagon or coach wheels. Several earlier works use 1D heat transfer analyses to estimate heat partitioning at wheel–brake block and wheel–rail interfaces. However, the efficacy of using 1D analyses has not been shown. Further, attempt to estimate wheel running temperatures based on braking and speed history, and brake block type has not been undertaken. This work has two main novelties. First, 2D boundary element method is used to estimate heat partitioning at wheel– brake block and wheel–rail interfaces and the obtained results are compared with that from 1D analyses to evaluate the accuracy of the latter. Second, effectiveness of the three step approach adopted here is demonstrated by comparing predicted wheel temperatures for a diesel locomotive, fitted with cast-iron brake blocks, with the observed temperatures in field trials. This methodology has for the first time given a clear estimate of heat lost to rail, brake blocks, and ambient air. Several earlier works have used 1D boundary element method to study temperature distributions in sliding and rolling contact problems. Chang et al. [10] showed that Green's function can be used efficiently for solving steady-state and transient heat conduction problems in isotropic and anisotropic media. Beck [11] derived Green's functions for transient heat conduction and applied it to solve a simple 2D problem on a rectangular domain. Jaeger [12] obtained temperature distributions for plane, sliding bodies. Block [13] obtained the flash temperatures by considering lubricant films between two sliding bodies. Block [14] derived the expressions for flash temperatures at contact region of rubbing surfaces with stationary and moving heat sources. Cameron [15] obtained surface temperatures of sliding bodies after accounting for convection. Tanvir [16] estimated the temperature rise at wheel–rail interface due to slip. Expressions for temperature rise were derived using Laplace transforms, by assuming an elliptical

R ttrain T0 Tw ΔTw v vt w

α

Δβ

ϵ

ηw

ϕ ω ρ θ

wheel radius (m) train running time (h) initial wheel temperature (°C) current wheel temperature (°C) temperature rise of wheel (°C) tread line speed (km/h) train speed (km/h) width of heat source on translating body (mm) thermal diffusivity (m2/s) angular segment (rad) surface emissivity (–) overall heat fraction entering into wheel (–) angular coordinate (rad) angular speed (rad/s) density (kg/m3) angular displacement (rad)

contact patch at wheel–rail interface. Kennedy [17] studied the effect of sliding velocity and thermal properties of sliding bodies on surface temperatures at contact. Significant changes in surface temperatures were observed with change in velocity and thermal properties. Yuen [18] investigated variation of position at which maximum temperature occurs in the contact zone of sliding bodies as a function of Peclet number. Maximum temperature was shown to occur at the end of the contact zone for higher values of Peclet number. Tian and Kennedy [19] calculated the temperatures of sliding bodies with different contact patches (square, circular and elliptic). Knothe and Liebelt [20] studied the effect of fluctuations in contact pressure, surface roughness and surface damage on maximum contact temperature of sliding bodies. An increase in maximum temperatures was observed with above said conditions. Gupta et al. [21] obtained wheel–rail contact temperatures and heat loss to rail due to frictional heat at wheel–rail interface. Laraqi [22] investigated the effect of contact size and velocity on thermal contact resistance of sliding bodies. A decrease in resistance was observed with an increase in speed and contact size. Ahlstrom and Karlsson [23] derived a 1D analytical model for calculating wheel temperatures at different depths of contact region during wheel– rail skid. A maximum wheel temperature around 900 °C was found during skidding. Hou and Komanduri [24] studied the effect of shape and pressure distribution in contact zone of stationary and moving heat sources. Vick and Furey [25] studied the effect of multiple contacts between sliding bodies instead of a single contact. A decrease in temperature rise was observed with an increase in number of contacts and distance between contacts. Ahlstrom and Karlsson [26] obtained temperatures in wheel–rail skidding using finite element analysis. These temperatures were further used to study the phase transformations in the wheel. High cooling rates were observed in short sliding times which lead to martensite formation in wheels. Kennedy and Traiviratana [27] obtained temperature profiles and heat partition at wheel–rail interface for different sliding times. Spiryagin et al. [4] formulated a mathematical model to understand temperature rises at wheel flange–rail interface. Contact between flange and rail leads to higher temperatures than regular wheel tread–rail contact. Newcomb [29] obtained an analytical solution for estimating heat partition between wheel and brake block for disc braking. Petereson [30] obtained heat partition between wheel and brake block using 2D finite element model with different types of wheel–brake block contact pressure distributions. Analysis was conducted for drag braking at 80 km/h with composite brake block for a single braking event. Vernersson [31–33] studied heat

M.R.K. Vakkalagadda et al. / Wear 328-329 (2015) 537–551

partition at wheel–brake block interfaces with and without railchill from laboratory experiments and analytical model. Analyses were conducted with cast-iron, composite and sinter brake blocks under different brake block configurations. Analyses showed significant amount of heat loss to rail from wheel. Teimourimanesh et al. [34] studied the effect of rail-chill during tread braking. It was found that braking events with intermediate cooling times result in significant heat loss to rail. Abbasi et al. [35] experimentally studied heat partition between cast-iron pin and disc extracted from wheel rim. Experiments were conducted for different initial disc temperatures, it was found that heat fraction entering into disc was varied between 0.77 and 0.895. The paper is organized as follows. Section 2 gives details of the 2D boundary element method proposed and employed in this work. Section 3 provides details of the field tests and numerical analyses employed for estimating wheel temperatures. Section 4 presents the results obtained, while conclusions are presented in Section 5.

2. Mathematical modelling A three step approach is adopted here to estimate the railway wheel running temperatures. Mathematical formulation for the three steps involved, namely (i) estimation of heat generated at wheel–rail and wheel–brake block(s) interfaces, (ii) estimation of heat entering the rail, wheel, and brake block(s), and (iii) estimation of wheel running temperatures based on information of net heat entering the wheel as a function of time, is given in the following three sub-sections.

The train running model requires three main inputs: (i) apparent friction coefficient for the brake blocks as a function of sliding speed, sliding distance, and brake load, (ii) traction coefficient as a function of percentage slip and sliding speed at wheel–rail interface, and (iii) running resistance as a function of train speed. More specifically, the apparent friction coefficient μb at wheel–brake block interface is taken to be of the form 2

2

2

∑ ∑ ∑ Aijk x1i x2j x 3k

with

i=0 j=0 k=0

v , v0 ⎛ ⎞ d x2 = erf ⎜⎜ ⎟⎟, ⎝ d0 ⎠ x1 =

x3 =

⎧ A1x6 + A2 ∀ vt < 10 km/h μ rr = ⎨ ⎩ B1x6 + B2 otherwise, with x6 = vt /v0. ⎪

⎪

(3)

Coefficients appearing in each of the characteristic equations are calibrated from relevant experiments. Next, angular momentum balance equations for the wheel sets and linear momentum balance for the total train are used along with information on train speed, brake load, tonnage per axle, ground inclination, and brake block type to estimate percentage slip at wheel–rail interface and heat generated at wheel–rail and wheel–brake block interfaces. A detailed discussion of the train running model along with coefficients characterizing brake blocks used by Indian Railways and other information needed to apply the model particularly to Indian Railways context is given in [36]. 2.2. Heat partition analysis Fig. 1 shows a schematic of the two dimensional idealization of wheel–rail–brake block system considered in this work. As seen from Fig. 1(a), the aim here is to estimate the temperature and heat flux distributions in the wheel, brake blocks, and rail as a function of time, for a given set of heat inputs at different interfaces, geometric parameters (i.e. wheel diameter, wheel–brake block contact angle, brake block thickness and orientation) and tread line speed ωR ≈ vt . The resulting temperatures and heat fluxes in each of the components should satisfy the governing PDEs (partial differential equations) along with the following interface and boundary conditions:

TRs1+ = TRr1− + (qRr1− − ηwb1Q1)/hwb1, TRs2+ = TRr2− + (qRr2− − ηwb2 Q2)/hwb2 , TRr3− = T yt =0− + ⎡⎣q yt =0− − (1 − ηwr ) Q 3⎤⎦/hwr ,

2.1. Train running analysis

μb =

539

b . b0

(1)

where v, d and b are the tread line speed, the sliding distance and the braking load, respectively, and v0, d0 and b0 are their reference values. Wheel–rail traction coefficient μrw relating slip at wheel– rail interface to traction generated (in terms of normal load) is modelled as

μ rw = A1x 4 + A2 x 4 x12 + A 3 x 4 x14 ,

⎛c ⎞ x 4 = tanh ⎜ ⎟. ⎝ c0 ⎠

(2)

where c = (v − vt )/vt with vt being the train speed is the creep ratio (slip) and c0 is its reference value. Lastly, running resistance coefficient μrr providing an estimate of resistance to motion from aerodynamic drag and rolling resistance (in terms of normal load) is taken as

qRr1−

+

qRs1+

q(s1 R + bt ) −

=

= 0,

Q1,

qRr2−

+

q(s2 R + bt ) −

qRs2+

= 0,

=

Q2,

qRr3−

+

(4) q yt =0−

=

Q 3, (5)

where superscripts “r1, r2, and r3” refer to mid-points of the three interfaces on the wheel side, “s1, s2, and t” refer to the corresponding mid-points on the stationary brake blocks and the translating rail side, T denotes the temperature, and q the normal heat flux (see Fig. 1). The subscripts “R  , R þ , (R + bt )−, and y = 0  ” represent the locations in components where the temperature and heat flux are evaluated for the purpose of enforcing interface and boundary conditions. Further, while h wb1, h wb2 and h wr denote the thermal contact conductances at the wheel–brake block and wheel–rail interfaces, ηwb1, ηwb2 and ηwr denote the corresponding heat apportionment coefficients representing the fraction of heat generated at the interfaces entering the wheel. Note that the solution to the problem of interest here (shown in Fig. 1a) can be obtained by determining heat sources in problems shown in Fig. 1(b–e) such that the resultant temperatures and heat fluxes in the rotating cylinder, stationary brake blocks, and the translating rail in Fig. 1(b–e) satisfy the interface and boundary conditions stated in Eqs. (4) and (5). The challenge now is to estimate temperatures and heat fluxes in problems shown in Fig. 1(b–e) as a function of given heat inputs. In this work, this is done using the appropriate Green's functions for rotating and translating bodies as discussed in the following two subsections. Lastly, it may be noted that for simplicity, temperature continuity and energy conservation (imposed through Eqs. (4) and (5)) are enforced only at mid-points of interfaces. Nonetheless, these conditions can be enforced at any number of points on interfaces. However, the corresponding number of heat sources must then be considered in sub-problems shown in Fig. 1(b–e).

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Fig. 1. (a) Schematic showing the wheel–brake block–rail system considered in this work for estimating heat partitioning at wheel–brake block and wheel–rail interfaces, (b) schematic of infinite rotating domain with three distributed heat sources in corresponding angular segments, (c, d) infinite stationary domain with two circumferential heat sources and (e) infinite translating domain with heat source on plane characterized by y¼0. Solutions to problems in (b–e) are used to obtain solution to problem in (a). Table 1 Thermo physical properties of wheel, rail and brake block materials used in present analysis. Properties of wheel, cast-iron brake blocks are taken from Vernersson [37]. Component

Wheel/Rail

Cast-iron

time t due to a unit heat, instantaneously released at point (x0, y0 , z0 ) at time t0 > 0 is given by

Composite

G (x, y, z , t) = Thermal conductivity, k (W/mK) Density, ρ (kg/m3) Specific heat capacity, c (J/kgK) Thermal diffusivity, α  106 (m2/s)

49 7850 460 13.57

48 7100 520 13

0.5 2034 941 0.26

2.2.1. Thermal analysis of a rotating body with stationary heat source Consider an infinite body rotating about z-axis in counter clockwise direction at angular speed ω (t). Green's function G (x, y, z, t) denoting temperature rise at point (x, y, z) in space at

1 ρc [4πα (t − t0 )]3/2

⎤ ⎡ 2 2 2 ⎢ (x¯ − x 0 ) + (y¯ − y0 ) + (z¯ − z 0 ) ⎥ exp ⎢ − ⎥ 4α (t − t0 ) ⎥⎦ ⎢⎣

(6)

∀ t > t 0,

where

x¯ = r cos(ϕ − θ), ϕ = tan−1(y /x)

y¯ = r sin(ϕ − θ),

and

θ=

∫t

t 0

ω (τ ) dτ .

z¯ = z,

r=

x 2 + y2 ,

M.R.K. Vakkalagadda et al. / Wear 328-329 (2015) 537–551

Here ρ, c and α denote the density, the specific heat and the thermal diffusivity of the rotating material, respectively. Note that Green's function is valid for an isotropic solid with uniform, temperature independent thermo-physical properties obeying Fourier law of heat conduction. Using Green's function given in Eq. (6), temperature at a point (r , ϕ, z) at time t, due to a distributed heat source Qr (β , t) acting on the cylindrical surface characterized by r = R (see Fig. 1), is given by T (r , ϕ, z , t) = T0 +

t

2π

(7)

Note that the heat source is assumed to be uniform along the z-direction and the infinite body is initially at temperature T0, i.e. T = T0 at time t¼ 0. Substituting Eq. (6) in Eq. (7) and noting that ∞

∫z′=−∞

Next, defining π /2

H (τ ) = − a0 =

∫δ =−π/2 2Qr(ϕ − θ − 2δ, τ)exp[ − (a 0sinδ)2]a 0 dδ, rR , α (t − τ )

⎤ ⎡ (z − z′)2 ⎥ exp ⎢− = α τ 4 ( t ) − ⎥⎦ ⎢ 4α (t − τ ) ⎣ 1

T (r , ϕ, t) = T0 +

T (r , ϕ, t) = T0 + ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

R 4πρcα

2π

∫β=0

t

∫τ=0

⎡ ⎛ϕ − θ − β ⎞⎤ ⎫ ⎟ ⎪ ⎢ rR sin2 ⎝⎜ ⎠⎥ ⎪ 2 ⎥ dβ ⎬ dτ . Q r (β, τ)exp ⎢− α (t − τ ) ⎥ ⎪ ⎢ ⎥⎦ ⎪ ⎢⎣ ⎭

t

∫τ=0

⎡ ⎤ (r − R)2 ⎥ exp ⎢− H (τ ) ⎢⎣ 4α (t − τ) ⎥⎦ rRα (t − τ) 1

(11)

Lastly, expression for inward radial heat flux arising from temperature distribution in Eq. (11), for moderately large times (a0 > > 1) when H (τ) is effectively independent of r, is given by

qr (r , ϕ, t) =

kR 4πρc

expression for temperature at point (r , ϕ, z) at time t can be rewritten as

⎤ ⎡ ⎢ 2 ⎥ r R 1 ( ) − ⎥ exp ⎢− ⎢ 4α (t − τ ) ⎥ (t − τ ) ⎥⎦ ⎢⎣

R 4πρc

π and substituting β = ϕ − θ − 2δ in

dτ .

π, (8)

(10)

with an integrand of period Eq. (9) we get that

∞

∫τ=0 ∫β=0 ∫z′=−∞ G (r , ϕ, z′) Qr (β, τ) R dz′ dβ dτ.

541

t

⎡

∫τ=0 ⎢⎢− 21r ⎣

+

⎤ R−r ⎥ 2α (t − τ) ⎥⎦

⎡ ⎤ (r − R)2 ⎥ exp ⎢− H (τ ) dτ . ⎢⎣ 4α (t − τ) ⎥⎦ rRα (t − τ) 1

(12)

where k is the coefficient of thermal conductivity for the rotating medium. It may be noted that for a0 > > 1, H (τ) can be evaluated analytically. For the scenario shown in Fig. 1, expressions for H (τ) when a0 > > 1 are given in the Appendix.

(9)

Table 2 Dimensions of wheel, brake blocks and other parameters used in boundary element analyses. Brake block type

Cast-iron

Composite

Brake block length (mm) Brake block radius (mm) Brake block thickness (mm) Wheel–brake block contact angle Brake block width (mm) Wheel diameter (mm) wheel–rail contact angle (21 tons axle load) β1 β2 − β1, β4 − β3

425 545 50 46° 85 1092 0.17° 10° 30–60°

320 550 55 34°

β 3 − β2

140–170°

β6 − β5

0.17°

2.2.2. Thermal analysis of a translating body with stationary heat source Consider an infinite body translating along x direction at a speed vt . Green's function G (x, y, z, t) denoting temperature rise at point (x, y, z) at time t due to a unit heat instantaneously released at point (x0, y0 , z0 ) at time t0 > 0 is given by

G (x, y , z, t) =

1 ρc [4πα (t − t0

)]3/2

⎡ ⎤ 2 −r^ ⎥ exp ⎢ ⎢⎣ 4α (t − t0 ) ⎥⎦

∀ t > t0, (13)

where

⎡ ⎛ 2 r^ = ⎢x − ⎜x 0 + ⎝ ⎣

t

∫τ=t

0

⎞ ⎤2 vt (τ) dτ ⎟ ⎥ + (y − y0 )2 + (z − z 0 )2 . ⎠⎦

(14)

Using Green's function given in Eq. (13), temperature at a point (x, y, z) at time t due to a distributed heat source Q t (x, t) acting on the planar surface characterized by y¼0 in the range −w /2 < x0 < w /2 (see Fig. 1) is given by

idling condition

running condition

h = 6 / 12 W/m 2 K, ε =0.95

hax = 1287 W/m2 K qnet

h = 0,

ε =0

Fig. 2. Mesh and boundary conditions of a straight edge locomotive wheel of diameter 1074 mm used in finite element analyses (FEA).

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M.R.K. Vakkalagadda et al. / Wear 328-329 (2015) 537–551

sensor 3

sensor 2

gear box

LOCOMOTIVE FRONT END

REAR END

wheel axle

non-contact infrared sensors

cast iron brake block

sensors 1 and 4

Fig. 3. (a) Schematic of wheel and axle sets assembly with locations of non-contact infrared temperature sensors. (b) Photograph showing sensors 1 and 4, mounted on front axle of a WDM3A locomotive.

0.79

No rail, Q = 1 W/mm 2 1

160 km/h

0.88

1 2 2 No rail, Q = Q = 0.5 W/mm

160 km/h

80 km/h

0.785 0.78

0.86 20 km/h

ηw

80 km/h

0.84

0.775

10 km/h 5 km/h

ηw

40 km/h

20 km/h 10 km/h

5 km/h

0.77 40 km/h

0.765 0.76

0.82

0.755 single cast-iron brake block

0.8

0

20

40

60

80

t (s)

100

0.75

120

two cast-iron brake blocks 0

20

40

60

80

100

120

t (s)

υt 65

52

υ t = 80 km/h, t = 30 s 0 mm

50

60

0.5 mm

φ

48

ΔTw ( 0 C )

ΔTw ( 0 C )

0.5 mm 1 mm

55

2 mm

50

1 mm

46 2 mm

44 42

4 mm

45

4 mm

40 single cast-iron brake block

40

υ t = 80 km/h, t = 30 s

0 mm

0

60

120

180

240

300

360

38

two cast-iron brake blocks 0

60

120

180

240

300

360

φ(0 )

φ( ) 0

Fig. 4. Net heat fraction entering the wheel for different drag braking speeds using (a) single and (b) double cast-iron brake blocks per wheel. (c, d) Resultant temperature rises at different depths from outer radius of wheel during drag braking at 80 km/h using single and double cast-iron brake blocks per wheel, respectively.

T (x, y, t) = T0 +

t

w /2

∫τ=0 ∫x′=−w/2

dx′ dτ.

⎡ (x − x^)2 + (y − y )2 ⎤ Q t (x′, τ) 0 ⎥ exp ⎢ − ⎥ ⎢ 4πρcα (t − τ) 4α (t − τ) ⎦ ⎣

(15)

Note that in Eq. (15) the heat source is assumed to be uniform along the z-direction and the infinite sliding body is initially at temperature T0, i.e. T = T0 at time t¼ 0. Further, note that for a stationary body, i.e. vt = 0, with constant heat input, expression for temperature given in Eq. (15) becomes

where

x^ = x′ +

t

T (x, y , t) = T0 +

⎛ ⎞

∫ξ=τ vt ⎜⎝ξ⎟⎠ dξ.

(16)

w/2

∫x′=−w/2

⎡ ⎤ Q t (x′) ⎢ (x − x′)2 + (y − y0 )2 ⎥ dx′. Γ 0, ⎥ 4πρcα ⎢⎣ 4α t ⎦

(17)

M.R.K. Vakkalagadda et al. / Wear 328-329 (2015) 537–551

2.2.3. Specific cases considered The boundary element formulation presented above is used to conduct heat partition analyses for drag and stop braking for two types of brake block materials: cast-iron and composite. The effect of size and number of brake blocks on heat partitioning is also studied in addition to the effect of brake block material. Table 1 gives the temperature independent thermo-physical properties of wheel and brake block materials used. Rail and wheel steel are taken to have identical thermo-physical properties. Geometric parameters used are provided in Table 2. Note that while thermal conductance of zero at wheel–rail interface corresponds to the situation where the rail-chill (heat loss to rail) is neglected, thermal conductance of infinity corresponds to the case of perfect thermal contact between the wheel and the rail in 2D. Heat generated at wheel–brake block interface per unit contact area is kept identical between the actual scenario and the two-dimensional idealization of the problem. Contact length corresponding to wheel–rail contact is chosen such that its product with the width of the brake blocks (over which heat generated is uniformly distributed) is kept identical to the contact area between actual wheel and rail as estimated from Hertzian contact theory. Further, the wheel–rail contact conductance h wr is chosen based on 1D heat conduction analyses with actual wheel–rail profiles and tonnage information as [28]

hwr =

14ϵβr vt (2π)3/2 a

,

(18)

where a is the semi-axis length of the contact ellipse along the rail, b is the semi-axis length of contact ellipse along the transverse direction, D is the tread line diameter of the wheel, and ϵ, βr are defined as

ϵ=

βw Rω βw Rω + βr vt

,

βr =

k r ρr cr ,

βw =

k w ρw cw .

(19)

Here, as rail and wheel have nearly identical thermo-physical properties and as Rω ≈ vt , it can be seen that ϵ = 0.5 and for 21 tons axle load, wheel diameter D= 1074 mm, wheel half cone angle 0.05 radians, rail transverse radius of curvature 300 mm, it follows from Hertzian contact theory that a = 7.82 mm , b = 5.42 mm and hwr = 3 × 105 W/m2 K .

0.95

2.3. Finite element analysis Two dimensional boundary element analyses show that temperature asymmetry in the rotating cylindrical wheel subjected to stationary heat sources is limited to a small depth (∼2 mm) from the tread surface (r¼ R) and that temperatures and resulting heat fluxes in the wheel (except close to the tread surface) can be accurately estimated using an axi-symmetric analysis with net heat entering the wheel uniformly distributed along the circumference. In this work, finite element method is used to obtain the resultant, axi-symmetric, wheel temperatures once net heat entering the wheel is known. Fig. 2 shows the mesh, along with the boundary conditions, for a straight edge locomotive wheel of diameter 1074 mm used in finite element analysis (FEA). The chosen diameter corresponds to the locomotive wheel diameter in the field trials. The locomotive used in the field trials employs cast-iron brake blocks. Thus, heat generation rates at wheel–brake block and wheel–rail interfaces, for cast-iron brake blocks, obtained using the train running model for a given velocity and brake cylinder pressure variation along with heat partition coefficient for the wheel estimated from boundary element method provide information on the heat entering the wheel from wheel–brake block interfaces. The mesh employed in the work consists of 1796, 4-noded linear axi-symmetric heat transfer quadrilateral elements. Heat loss to rail and atmosphere are modelled through appropriate convective heat transfer coefficients and emissivities. Fig. 2 shows the boundaries where convection and radiation boundary conditions are imposed. Uniform convective heat transfer coefficients of 12 and 6 W/m2 K are used during running and idling periods. Rail-chill is taken to be nil during stationary periods. Surface emissivity of 0.95 was used for all surfaces except the surface which comes into contact with the axle. Heat loss from this portion of the boundary is taken to be zero. Since peak wheel running temperatures are at best moderate (∼200 °C) temperature independent thermophysical properties are used. Thermo-physical properties of steel given in Table 1 are also used in FEA. Young's modulus and Poisson's ratio of wheel as well as rail were taken to be 207 GPa and 0.29, respectively.

3. Field trials Field trials were conducted using a WDM3A diesel locomotive fitted with cast-iron brake blocks. WDM3A locomotive is a diesel locomotive that carries passenger as well as freight stock. It has 6 axles, weighs 112.8 tons (18.8 tons per axle), and has a rated power output of 3100 hp. Each of the wheels, on the six axles of the locomotive, was fitted with two cast-iron brake blocks. Temperatures of three of the

0.995

1D, single brake

1D, single brake 2D, single brake

2D, single brake

0.9

543

0.99 1D, two brakes

1D, two brakes

0.85

ηw

0.985

ηw

2D, two brakes

2D, two brakes

0.8

0.98

0.75

0.975

0.7

cast-iron, t = 30 s

1D Analysis [Eq. (20)] 30

35

40

45

50

55 o

Wheel-brake block contact angle ( )

1D Analysis [Eq. (20)]

60

0.97 30

35

40

45

composite, t = 30 s 50

55

60

o

Wheel-brake block contact angle ( )

Fig. 5. Heat fraction entering the wheel as a function of wheel–brake block contact angle during drag braking for single and two (a) cast-iron and (b) composite brake blocks used for drag braking at 60 km/h.

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wheels were monitored with non-contact infrared sensors attached to fixtures mounted on corresponding axle casings. While two non-contact sensors monitored the temperatures at rim and rim–disc interface, at 60 and 130 mm distance from outer flange, for one of the wheels, one non-contact sensor per wheel was used to monitor the rim temperatures for two of the wheels. Wheels whose temperatures are monitored are marked in Fig. 3a. Data from the non-contact

1

1

single brake, η wb

single brake, η w

0.9

η

temperature sensors were continuously recorded in a laptop, using a National Instruments data acquisition system, placed in drivers' cabin. Running speed of the locomotive and brake cylinder pressure (BCP) were obtained using the onboard sensors on the locomotive. Brake load b (in kN) per brake block is related to BCP (in bar) through the relation b = 10 × BCP . Good contact between wheel and brake blocks and equal stroke lengths of all brake cylinders were ensured before the

0.9 single brake, η wb

double brakes,η wb single brake, η w

0.8

0.8

η 0.7

double brakes,η wb

double brakes, η w

0.7 double brakes, η w

0.6

0.6 cast-iron, 460 wheel-brake contact angle

0.5 0

10

20

30

50

40

composite, 340 wheel-brake contact angle

0.5 0

60

10

20

30

t (s)

65

75

hwr = 3 × 105 W/m 2 K

50

60

hwr = 3 × 105 W/m 2 K

70

60

0 mm

0 mm

65

0.5 mm

55

1 mm

ΔTw ( 0 C )

ΔTw ( 0 C )

40

t (s)

2 mm 4 mm

50

0.5 mm 1 mm

60

2 mm 4 mm

55 45

50 cast-iron, 460 wheel-brake contact angle

40 0

50

60

120

0 mm

48

240

300

0

360

60

180

240

300

360

hwr = 3 × 105 W/m 2 K

0 mm 0.5 mm

60

1 mm

ΔTw ( 0 C )

44 42 40

120

65

hwr = 3 × 105 W/m 2 K

0.5 mm

46

ΔTw ( 0 C )

180

composite, 340 wheel-brake contact angle

45

4 mm

2 mm

1 mm

55

4 mm

50

2 mm

38 cast-iron, 460 wheel-brake contact angle

36 0

60

120

180

240

300

360

composite, 340 wheel-brake contact angle

45 0

60

120

180

240

300

360

Fig. 6. Fraction of heat generated at wheel–brake block interfaces entering the wheel ηwb along with net heat fraction entering the wheel ηb (which accounts for heat loss to rail) as a function of braking time, during drag braking at 60 km/h with single and double (a) cast-iron and (b) composite brake blocks. Resultant temperature rises at different depths from outer radius of wheel using single and double (c, e) cast-iron and (d, f) composite brake blocks, per wheel.

M.R.K. Vakkalagadda et al. / Wear 328-329 (2015) 537–551

test. Trials were conducted between Kharagpur and Tatanagar, India over a stretch of around 130 km. This distance was covered by the locomotive in around 3.75 h with several full and partial stop braking events. The locomotive had to stop at multiple locations for long times during the journey. It had a peak speed of around 100 km/h. Wheels on the locomotive had straight edge profile and wheel diameter was around 1074 mm. Dimensions of the cast-iron brake blocks used in the trials were as shown in Table 2 with the exception of brake block thickness which varied from 55 to 75 mm.

4. Results and discussions 4.1. Heat partitioning at wheel–brake block and wheel–rail interfaces Fig. 4 shows heat fraction entering the wheel along with resultant temperatures at different depths for drag braking conditions, i.e. for the case where braking occurs at constant tread speed, when castiron brake blocks are used and rail-chill effect (heat loss to rail) is neglected, i.e. thermal conductance at wheel–rail interface h wr = 0 . Dimensions of wheel and brake blocks used in the analyses are shown in Table 2. The rate of heat input per unit area at the wheel– brake block interface is 1 W/mm2 for the single brake block case and 0.5 W/mm2 for the two brake blocks' case. Note that as the problem being studied is linear, heat fraction entering the wheel would remain unchanged for any constant value of heat input at the interfaces, while the temperature rise at any point would scale linearly with heat input. The following can be noted from the figure:

545

(i) Heat fraction entering the wheel is marginally higher at higher tread line speeds, (ii) Without the rail-chill effect, heat fraction entering the wheel asymptotically tends to a constant in relatively short time, e.g. it tends to about 0.87 and 0.78 within about 30 s for drag braking at 40 km/h with single and double brake blocks, respectively, (iii) Even though braking is asymmetric, temperature asymmetry is limited to small depths from tread surface, e.g. temperature profile even at about 2 mm depth from the tread surface is effectively axi-symmetric, (iv) Despite that the thermo-physical properties of wheel and brake block material are similar, only about 12–14% of heat generated at the interface enters the brake block for a single brake and about 22–23% of heat generated enters the brake blocks for two brake blocks, and (v) Peak temperature at the interface reduces by about 17% as we use two instead of one brake block. Fig. 5 shows net heat fraction entering the wheel as a function of wheel–brake block contact angle during drag braking for cast-iron and composite brake blocks in the absence of rail-chill. Heat fractions were estimated from the theory proposed in Section 2 as well as from the analytical expression proposed by [29] based on 1D heat conduction:

ηw =

2π k w ρw cw 2π k w ρw cw + Δβ k b ρ b cb

, (20)

where k , c , and ρ denote the thermal conductivity, the specific heat, and the density, respectively, subscripts ‘w, b’ denote the properties for wheel and brake block material, respectively, and Δβ denotes the net contact angle for the brake block(s). Note that while Δβ = β2 − β1 for the single brake block case, Δβ = (β2 − β1) + (β4 − β3 ) for the two

1

0.9

single composite brake block

single cast-iron brake block 0.95

0.85

60 s

60 s

0.9

0.8

120 s

120 s

0.75

300 s

240 s

180 s

0.85

180 s

240 s

300 s

0.8 0.7

0.75

0.65 0.6 104

0.7

105

106

107

0.8

108

109 5 109

two cast-iron brake blocks

60 s

105

180 s

106

1

107

108

109 5 109

two composite brake blocks

0.95

120 s

0.75

0.65 10 4

60 s

0.9 240 s

120 s

0.85

0.7 300 s

0.65

180 s

240 s

300 s

0.8 0.75 0.7

0.6

0.65 0.55 104

105

106

107

108

109 5 109

0.6 104

105

106

107

108

109 5 109

Fig. 7. Net heat fraction entering the wheel as a function of wheel–rail contact conductance, for drag braking at 60 km/h with single and double (a, c) cast-iron and (b, d) composite brake blocks.

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brake blocks' case. The following can be noted from the figure: (i) Net heat fraction entering the wheel decreases nearly linearly with brake block contact angle, at least in the range of interest, (ii) Eq. (20) provides a reasonably accurate estimate of net heat entering the wheel in the absence of rail, (iii) Nearly all the heat generated during braking enters the wheel when composite brake blocks are used, and (iv) The effect of number of brake blocks is higher at larger brake block contact angles.

120

120

two cast-iron brake blocks 120 km/h

100 80 60

80 km/h

40

two composite brake blocks 120 km/h

100

υ (km/h)

υ (km/h)

Fig. 6(a,b) shows the fraction of heat generated at wheel–brake block interfaces entering the wheel ηwb along with net heat fraction entering the wheel ηb (which accounts for heat loss to rail) as a function of braking time, during drag braking at 60 km/h with single and double cast-iron and composite brake blocks, respectively. Dimensions of wheel and brake blocks used in the analysis are shown in Table 2. The heat inputs at the wheel–brake block interfaces are chosen such that net heat generated during braking

80 80 km/h

60 40

40 km/h

40 km/h 20 0

20

0

10

20

30

40

50

60

0

70

t (s)

0

5

10

800

25

30

35

two composite brake blocks

two cast-iron brake blocks 700

300

600

250 40 km/h

200

Qwr (W)

Qwr (W)

20 t (s)

350

80 km/h

150

500 400 40 km/h

300 120 km/h

100

80 km/h

120 km/h

200

50 0

15

100 0

10

20

30

40

50

60

0

70

0

5

10

15

20

25

30

35

t (s)

t (s) 200

140

two composite brake blocks

two cast-iron brake blocks 120 150 40 km/h

Qwb (kW)

Qwb (kW)

100 80 80 km/h

60

120 km/h

80 km/h

100

120 km/h

40 km/h

40

50

20 0

0

10

20

30 40 t (s)

50

60

70

0

0

5

10

15

20

25

30

35

t (s)

Fig. 8. Locomotive speed (v), heat generation rates at loco wheel–rail interface (Q̇ wr ) and loco wheel–brake block interface (Q̇ wb ) when braked with (a, c, e) cast-iron, (b, d, f) composites brake blocks, from different initial speeds.

M.R.K. Vakkalagadda et al. / Wear 328-329 (2015) 537–551

is identical for the two brake block types despite the different angles they subtend at the wheel centre. More specifically, the rate of heat input per unit area at the wheel–brake block interface is taken as 1 W/mm2 for single and 0.5 W/mm2 for double cast-iron brake blocks, while it is taken as 1.35 W/mm2 for single and 0.675 W/mm2 for double composite brake blocks. The following observations can be made from the figure: (i) Heat fraction ηwb entering the wheel at wheel–brake block interfaces is nearly constant during braking, while net heat fraction decreases with increasing wheel temperature, (ii) For composite brake blocks, about 98% of heat generated at wheel–brake block interfaces enters the wheel even when two of them are used, (iii) Even with wheel–rail contact, which results in sharp temperature gradients on the tread surface, temperature profiles are asymmetric only at small depths from the tread surface, and (iv) Even though the railchill effect appears to be small during the braking period, it is expected to cause significant cooling as it is operative throughout the train running period while braking occurs only intermittently. Fig. 7 shows net heat fraction entering the wheel as a function of wheel–rail contact conductance, for drag braking at 60 km/h with single and double cast-iron and composite brake blocks. Conditions considered are identical to that for Fig. 6. Note that hwr = 0 corresponds to the case where rail-chill effect is absent, hwr = 3 × 105 W/m2 K (vertical, dashed line in the figure) corresponds to the case where wheel–rail thermal conductance is estimated from 1D heat conduction analysis, and h wr = ∞ indicates Table 3 Total heat generated at loco wheel–brake block and loco wheel–rail interfaces during stop braking from different speeds with cast-iron and composite brake blocks. Cast-iron brake blocks

Composite brake blocks

40 km/h 80 km/h 120 km/h 40 km/h 80 km/h 120 km/h 0.64

2.5

5.4

0.64

2.5

5.4

1.2

3.5

4.1

0.5

3

7.1

0.64

2.5

5.4

0.64

2.5

5.4

100

76.4

Fig. 8 shows results for stop braking for braking from different initial speeds using cast-iron and composite brake blocks. A sixaxle, 126 ton locomotive fitted with two brake blocks, running on a flat track, is considered in the train running analysis. The following are evident from the figure: (i) While cast-iron brake blocks are more effective, resulting in shorter braking times, at lower speeds, composite brake blocks are significantly more effective at braking from higher speeds, e.g. 120 km/h. Lower friction coefficient of cast-iron brake blocks for braking from higher speeds [36] is believed to be responsible for this, (ii) During braking, the rate of heat generated at wheel–rail interface is about 300 times smaller than that generated at wheel–brake block interface. Thus, its contribution to overall heat generated during braking is negligible. This is also evident from Table 3 which shows overall heat generated at wheel–rail and wheel–brake block interfaces, and (iii) Peak power generated during braking for cast-iron brake blocks varies moderately with initial braking speed. However, there is a strong effect of initial braking speed on peak power generated for composite brake blocks. At high speeds, due to higher peak power generated and high heat fraction entering the wheel, peak temperatures are expected to be significantly higher for composite as compared to cast-iron brake blocks. Fig. 9 shows the percentage of overall heat generated during braking entering the wheel, brake blocks, and rail for cast-iron and composite brake blocks for braking from different initial speeds. It is clear that heat fraction entering the wheel is effectively independent of the initial speed and independent of the brake block type used. However, while heat fraction entering the rail increases with increasing initial braking speed, that entering the brake blocks decreases with

120 100

75.2 % Heat dissipated

75

60 40 24

21.6

composite, hwr = 3 × 105 W/m 2 K

96.4

95.8

94.8

80 60 40

21

20

20

80 km/h

120 km/h

40 km/h

80 km/h

Rail

1.5 3.7 Wheel Brake blocks

Rail

1.5 2.7 Wheel Brake blocks

Rail

1.6 2 0

Rail

Wheel Brake blocks

Rail

Wheel Brake blocks 40 km/h

3.8

2

1

0

Rail

% Heat dissipated

80

4.2. Heat generation during braking

cast iron, hwr = 3 × 105 W/m 2 K

Wheel Brake blocks

At wheel– brakes interface (MJ) At wheel–rail interface (kJ) Total (MJ)

the perfect thermal contact (in 2D). It can be seen from the figure that the rail-chill effect increases with increasing wheel temperature and contact thermal conductance. During drag braking, the net heat fraction entering the wheel asymptotes to a constant value with increasing time. Further, as expected, the final heat fraction entering the wheel decreases with increasing thermal conductance (rail-chill effect). Lastly, it may be noted that contact conductance influences net heat fraction entering the wheel predominantly for h wr varying from 105 to 107 W/m2 K.

Wheel Brake blocks

Total heat generated

547

120 km/h

Fig. 9. Percentage of overall heat generated during braking, entering the wheel, brake blocks, and rail for (a) cast-iron and (b) composite brake blocks, for stop braking from different initial speeds as estimated from train running model.

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increasing speed. It is also clear that net heat entering the wheel during braking is about 25% higher for composite brake blocks. 4.3. Field studies

υ t (km/h)

Fig. 10(a, b) shows, respectively, the speed and brake cylinder pressure variation in a WDM3A diesel locomotive during a 3.75 h field trial covering 130 km. Note from Fig. 10(b) that brakes were applied, at speeds of about 100 km/h, about 35 times during this time period. Wheels were at an ambient temperature of about 33 °C at the start of the trial. Evolution of wheel temperatures at the rim and rim–disc interface as the trial progressed is shown by filled circles in Fig. 10(c, d). While rim temperatures were measured on three wheels, rim–disc interface temperature was measured on one wheel. Predictions from the three step approach, employing train running model, boundary element analyses, and finite element method, are shown by solid lines. It is clear that the approach adopted here gives

120 100 80 60 40 20 0

0

0.5

1

1.5

reasonably good predictions for wheel temperatures, even for the complex running and braking conditions encountered in the trial. By matching the cooling rates during idling conditions (vt = 0) and during running conditions it is found that the effective convective heat transfer coefficient for locomotive wheels during idling and running conditions is around h¼6 and 12 W/m2 K, respectively. Further, from the good match it may be deduced that heat loss to rail could be effectively modelled using effective heat convective transfer coefficient hax on wheel–rail contact width (see Fig. 2), given by

hax =

ahwr , 2D

(21)

where h wr is the thermal conductance between wheel and rail (see Eq. (18)). Here, as rail and wheel have nearly identical thermo-physical properties and as Rω ≈ vt , it can be seen that ϵ = 0.5 and for 21 tons axle load, wheel diameter D¼ 1074 mm, wheel half cone angle 0.05 radians, rail transverse radius of curvature 300 mm encoun-

2

2.5

3

3.5

4

BCP (bar)

ttrain (h) 3.5 3 2.5 2 1.5 1 0.5 0

Partial braking: 1.25 to 2 bar, Full braking: 2.5 bar, Emergency braking: 3.8 bar

0

0.5

1

1.5

2

2.5

3

3.5

4

ttrain (h) 80 70

Tw ( 0C )

monitored 60 location

simulation 50 40 30

field measurements 0

0.5

1

time instances at which FEA contour plots are shown 1.5

2

2.5

3

3.5

4

ttrain (h) 80

Tw ( 0C )

70 60 monitored

location simulation

50 40

field measurements

30 0

0.5

1

1.5

2

2.5

3

3.5

4

t train (h) Fig. 10. (a) Speed and (b) brake cylinder pressure variation in a WDM3A diesel locomotive during a 3.75 h field trial covering 130 km between Kharagpur and Tatanagar, India. Comparison of wheel running temperatures measured from field studies and the approach proposed in this work at (c) rim and (d) rim disc interfaces.

M.R.K. Vakkalagadda et al. / Wear 328-329 (2015) 537–551

tered in field trials, it follows from Hertzian contact theory that a = 7.82 mm , b = 5.42 mm and hax = 1287 W/m2 K . It may lastly be noted that heat fraction entering the wheels at wheel–brake block interfaces, based on the results shown in Fig. 6, is taken as 0.77. Temperature profiles in the wheel at five different time instances marked in Fig. 10, determined from FEA, are shown in Fig. 11. The following can be noted from the figure: (i) Cooling predominantly NT11 33 33 33 33 33 33 33 33 33 33 33 45 44 43 42 40 39 38 37 36 34 33 92 87 81 76 70 65 59 53 48 42 37 53 52 51 50 49 49 48 47 46 45 44

549

occurs from the disc region during idling condition and from disc and wheel–rail contact region during running condition as seen from sharp thermal gradients in these regions, (ii) Thermal gradients in the rim region where the temperature is being monitored are small. Consequently, vibration of the fixture holding the sensors or exact locations where the sensors are mounted will not affect the rim temperature measurement. Thus, moderately large standard deviations in rim temperature measurements shown in Fig. 10c are believed to be from non-uniformity in performance of brake blocks mounted on different wheels, (iii) Temperature gradients in rim– disc interface are significantly greater than that in rim region. Thus, error in measurement of temperature is expected to be greater at rim–disc interface than in the rim region, and (iv) Due to effective cooling provided by the disc region, peak hub temperature never exceeds even 45 °C during the entire trial. Fig. 12 shows wheel rim temperatures obtained from FEA under different sets of assumptions. Quantum of heat loss to brake blocks, ambient air (through radiation and convection) and rail (through conduction) is evident from the figure. It can be seen that radiation and convection play an important role in cooling the wheels, particularly when running temperatures are high due to frequent breaking from high speeds. Consequently, prevention of dirt and oil layers on the wheels which can significantly hinder radiation and convection is crucial to avoiding high wheel temperatures. Rail-chill effect and heat loss to brake blocks are equally significant in comparison to convection and radiation. Brake blocks with high thermal diffusivity and appropriate choice of rail wheel profiles providing large contact areas can be chosen to increase wheel cooling from these.

5. Conclusions A three step approach employing a train running model, boundary element method, and finite element model is proposed to estimate railway wheel temperatures. The approach is validated using data from a field trial. A detailed study of heat loss to brake blocks, rail and ambient air (through convection and radiation) under different conditions is also conducted. The following conclusions are drawn from the work:

63 61 59 57 55 53 51 49 47 45 43

 Propensity for occurrence of fatigue wear in wheels subjected

Fig. 11. (a–e) Temperature profiles in the wheel at five different time instances marked in Fig. 10, determined from FEA.



to tread braking is expected to be lower for cast iron brake blocks – which result in lower peak temperatures and lower thermal gradients – as compared to high friction coefficient composite brake blocks. The hybrid approach which can accurately predict rail–wheel interface temperatures as a function of train-running history

180

temperatures obtained without any heat losses

160

Tw ( 0 C )

140 120 monitored

location 100

kes bra o t

80 60

+ ad +r

ill l ch rai + v con

heat lost only to brakes

only to brakes + radiation only to brakes + rad + conv

40

field measurements 20

0

0.5

1

1.5

2

2.5

3

3.5

ttrain (h) Fig. 12. Wheel rim temperatures obtained from FEA of the trial under different sets of assumptions.

4

550













M.R.K. Vakkalagadda et al. / Wear 328-329 (2015) 537–551

can be used to better select lubricants used on curved rail segments for minimizing flange wear and avoiding formation of rail corrugations. The approach can also be used for improved specification of thermal stability of composite brake blocks, to ensure uniform wear rate under typically encountered running and braking conditions. Even with rail-chill effect which introduces sharp thermal gradients on tread surface, wheel temperatures are asymmetric only close to wheel tread surface. Temperatures even at about 2 mm depth from tread surface are effectively axi-symmetric. Consequently, axi-symmetric analysis is suffice to estimate wheel running temperatures once net heat entering the wheels is known. Effect of wheel speed on the fraction of heat generated at wheel–brake block interfaces entering the wheel is only moderate. Thermal diffusivity of the brake block material and surface area of the brake blocks coming into contact with the wheel predominantly determine the heat fraction entering the wheel. For brake blocks used by Indian Railways, net heat entering the wheels during braking for composite brake blocks (with low thermal diffusivity) is about 25% higher than that for cast-iron brake blocks. Heat loss to brake blocks, ambient air through convection and radiation, and rail are all equally significant in cooling the railway wheels. Thus, ineffective cooling from any of the factors, e.g. from use of low thermal diffusivity brake blocks or formation of thick oil and dirt layers on the wheel, can significantly raise the wheel temperatures. Effective convective heat transfer coefficient for locomotive wheels during idling and running condition was found to be 6 and 12 W/m2 K, respectively.

To summarize, the three step approach adopted was found to predict wheel temperatures reasonably accurately. The analyses show that rail-chill effect, cooling from convection and radiation, and heat loss to brake blocks are all equally important in keeping the wheel temperatures low. In the context of Indian Railways, castiron brake blocks are found to be more appropriate for operation at speeds lower than 80 km/h (e.g. for pulling goods stock) when their braking effectiveness is on par with composite brake blocks.

Double brackets in the above equation represents floor function. Considering the assembly as shown in Fig. 1, with Q1, Q2 and Q 3 as heat generation rates at wheel–brake block and wheel–rail interfaces, with contact angles shown in Fig. 1, H (τ) for different cases can be evaluated as ⁎ ⁎ ⁎ Case 1: If β1 ≤ β1 ≤ β2 and 0 ≤ β2 ≤ β2 or β6 ≤ β2 ≤ 2π ,

H (τ) = Q1 π ⎡⎣erf(a 0 y2 ) − erf( − a 0 y1 ) ⎤⎦,

(A.3)

where

y1 = y ,

⎧ ⎛ β − β2⁎ ⎞ ⁎ ⎪y − ⎜ 1 if 0 ≤ β2 ≤ β1, ⎟ ⎪ ⎝ 2 ⎠ y2 = ⎨ ⎛ β1 − β2⁎ ⎞ ⎪ ⁎ ⎟ if β6 ≤ β2 ≤ 2π . ⎪ y − π − ⎜⎝ 2 ⎠ ⎩

Case 2: If β1 ≤

⁎ β1 ,

⁎ β2

(A.4)

≤ β2 ,

H (τ) = 2Q1 π ⎡⎣erf(a 0 y) ⎤⎦. Case 3: If β2 ≤

⁎ β1

≤ β3 and β1 ≤

(A.5) ⁎ β2

≤ β2 ,

H (τ) = Q1 π ⎡⎣erf(a 0 y2 ) − erf( − a 0 y1 ) ⎤⎦,

(A.6)

where

⎛ β ⁎ − β2 ⎞ y1 = y − ⎜ 1 ⎟, ⎝ 2 ⎠

y2 = y .

⁎

(A.7) ⁎

Case 4: If β3 ≤ β1 ≤ β4 and β2 ≤ β2 ≤ β3,

H (τ) = Q2 π ⎡⎣erf(a 0 y2 ) − erf( − a 0 y1 ) ⎤⎦,

(A.8)

where

y1 = y ,

⎛ β3 − β2⁎ ⎞ y2 = y − ⎜ ⎟. ⎝ 2 ⎠

Case 5: If β3 ≤

⁎ β1 ,

⁎ β2

(A.9)

≤ β4 ,

H (τ) = 2Q2 π ⎡⎣erf(a 0 y) ⎤⎦. ⁎

(A.10) ⁎

Case 6: If β4 ≤ β1 ≤ β5 and β3 ≤ β2 ≤ β4 , Acknowledgements

H (τ) = Q2 π ⎡⎣erf(a 0y2 ) − erf(−a 0y1)⎤⎦,

Authors would like to thank Indian Railways for the financial support for project DGG under Centre for Railways Research via the sanction letter 2012/M(L)/466/2(2701) dated 09-01-2012. We are also grateful to Shri A.K. Tripati, ADME, Diesel Loco Shed who made the field trials possible.

where

⎛ β ⁎ − β4 ⎞ y1 = y − ⎜ 1 ⎟, ⎝ ⎠ 2

(A.11)

y2 = y .

⁎

(A.12) ⁎

Case 7: If β5 ≤ β1 ≤ β6 and β4 ≤ β2 ≤ β5,

H (τ) = Q 3 π ⎡⎣erf(a 0 y2 ) − erf( − a 0 y1 ) ⎤⎦,

Appendix A

(A.13)

where For a0 > > 1, from Eq. (10), for various situations H (τ) can be calculated as follows. Let

⎧ ⎡⎡ ⎤⎤ ⎪ ϕ − θ + 2y + 2π ⎢⎢ ϕ − θ + 2y ⎥⎥ ∀ (ϕ − θ + 2 y ) ⎣ ⎦⎦ ⎣ 2π ⎪ ⎪ β1* = ⎨ < 2π , ⎪ ⎡⎡ ϕ − θ + 2y ⎤⎤ ⎪ otherwise ⎪ ϕ − θ + 2y − 2π ⎢⎣⎣⎢ ⎦⎥⎥⎦ 2π ⎩ and

⎡⎡ ϕ − θ − 2y ⎤⎤ β2* = ϕ − θ − 2y + 2π ⎢⎢ ⎥⎦⎥⎦ . ⎣⎣ 2π

y1 = y ,

⎛ β5 − β2⁎ ⎞ y2 = y − ⎜ ⎟. ⎝ 2 ⎠ ⁎

(A.14)

⁎

Case 8: If β5 ≤ β1 , β2 ≤ β6 ,

H (τ) = 2Q 3 π ⎡⎣erf(a 0 y) ⎤⎦.

(A.1)

⁎

(A.15) ⁎

H (τ) = Q 3 π ⎡⎣erf(a 0 y2 ) − erf( − a 0 y1 ) ⎤⎦,

(A.2)

⁎

Case 9: If β6 ≤ β1 ≤ 2π 0 or 2π ≤ β1 ≤ β1 and β5 ≤ β2 ≤ β6 ,

(A.16)

M.R.K. Vakkalagadda et al. / Wear 328-329 (2015) 537–551

where

⎧ ⎛ β ⁎ − β6 ⎞ ⁎ ⎪y − ⎜ 1 if β6 ≤ β1 ≤ 2π , ⎟ ⎝ 2 ⎠ ⎪ y1 = ⎨ , ⎛ β1⁎ − β6 ⎞ ⎪ ⁎ ⎟ if 2π ≤ β1 ≤ β1. ⎪ y − π − ⎜⎝ 2 ⎠ ⎩ Case 10: If β2 ≤

⁎ β1

≤ β3 and β6 ≤

⁎ β2

y2 = y . (A.17)

≤ 2π or 0 ≤

⁎ β2

≤ β1,

H (τ) = Q1 π ⎡⎣erf(a 0 y2 ) − erf( − a 0 y1 ) ⎤⎦,

(A.18)

where

⎛ ⎞ ⎜ ⁎ ⎟ ⎜ β − β2 ⎟ y1 = y − ⎜ 1 ⎟, ⎜ 2 ⎟ ⎜ ⎟ ⎝ ⎠

y2

⎧ ⎛ β − β2⁎ ⎞ ⁎ ⎪y − ⎜ 1 if 0 ≤ β2 ≤ β1, ⎟ ⎪ ⎝ 2 ⎠ =⎨ ⎛ β1 − β2⁎ ⎞ ⎪ ⁎ ⎟ if β6 ≤ β1 ≤ 2π . ⎪ y − π − ⎜⎝ 2 ⎠ ⎩ Case 11: If β4 ≤

⁎ β1

≤ β5 and β2 ≤

⁎ β2

(A.19)

≤ β3 ,

H (τ) = Q2 π ⎡⎣erf(a 0 y2 ) − erf( − a 0 y1 ) ⎤⎦,

(A.20)

where

⎛ β ⁎ − β4 ⎞ y1 = y − ⎜ 1 ⎟, ⎝ ⎠ 2

⎛ β3 − β2⁎ ⎞ y2 = y − ⎜ ⎟. ⎝ 2 ⎠

(A.21)

⁎

⁎

⁎

Case 12: If β6 ≤ β1 ≤ 2π or 0 ≤ β1 ≤ β1 and β4 ≤ β2 ≤ β5,

H (τ) = Q 3 π ⎡⎣erf(a 0 y2 ) − erf( − a 0 y1 ) ⎤⎦,

(A.22)

where

⎧ ⎛ β ⁎ − β6 ⎞ ⁎ ⎪y − ⎜ 1 if β6 ≤ β1 ≤ 2π , ⎟ ⎪ ⎝ 2 ⎠ y1 = ⎨ , ⎛ β1⁎ − β6 ⎞ ⎪ ⁎ π β β − − ≤ ≤ y if 0 . ⎜ ⎟ 1 ⎪ 1 ⎝ 2 ⎠ ⎩

y2

⎛ ⎞ ⎜ ⁎⎟ ⎜ β5 − β2 ⎟ =y−⎜ ⎟. ⎜ 2 ⎟ ⎜ ⎟ ⎝ ⎠ ⁎

(A.23) ⁎

⁎

⁎

⁎

⁎

Case 13: If β2 < β1 , β2 < β3 or 0 < β1 , β2 < β1 or β6 < β1 , β2 < 2π or ⁎ ⁎ β4 < β1 , β2 < β5,

H (τ) = 0.

(A.24)

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