Maximum Temperature and Relaxation Time in Wet Surface Grinding ...

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 5387612, 14 pages http://dx.doi.org/10.1155/2016/5387612

Research Article Maximum Temperature and Relaxation Time in Wet Surface Grinding for a General Heat Flux Profile J. L. González-Santander Universidad Católica de Valencia “San Vicente mártir”, C/ Guillem de Castro 94, 46001 Valencia, Spain Correspondence should be addressed to J. L. González-Santander; [email protected] Received 26 November 2015; Accepted 24 February 2016 Academic Editor: John D. Clayton Copyright © 2016 J. L. González-Santander. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We solve the boundary-value problem of the heat transfer modeling in wet surface grinding, considering a constant heat transfer coefficient over the workpiece surface and a general heat flux profile within the friction zone between wheel and workpiece. We particularize this general solution to the most common heat flux profiles reported in the literature, that is, constant, linear, parabolic, and triangular. For these cases, we propose a fast method for the numerical computation of maximum temperature, in order to avoid the thermal damage of the workpiece. Also, we provide a very efficient method for the numerical evaluation of the transient regime duration (relaxation time).

1. Introduction Grinding is a machining industrial process of metallic plates that consist in an abrasive wheel rotating at a high speed over the surface of a workpiece. The abrasion that produces the grinding wheel removes the surface material of the metallic plate being ground. During the grinding process, most of the energy is converted into heat, which accumulates within the contact zone between the wheel and the workpiece [1]. The high temperatures reached during this machining process may produce an unacceptable decrease in the quality of workpieces, such as burning or residual stresses [2]. In order to avoid this thermal damage of the workpiece, liquid coolant is usually injected into it, so power generation by friction is reduced and cooling by convection occurs. Therefore, a theoretical model that can predict the maximum temperature in wet grinding for the online monitoring process is demanded. In surface grinding, the heat transfer inside the workpiece usually is modeled by a strip heat source infinitely long and of 2ℓ width (m in SI units), which moves at a speed V⃗𝑓 = V𝑓 𝑖 ⃗ (m s−1 ) over a semi-infinite solid surface (see Figure 1). Setting the Cartesian coordinate system 𝑋𝑌𝑍 fixed to wheel, as it is shown in Figure 1, the temperature field of the

workpiece 𝑇(𝑡, 𝑥, 𝑧) with respect to the room temperature 𝑇0 (K) must satisfy the convective heat equation [3, §1.7(2)]: 𝜕𝑇 𝜕𝑇 𝜕 2 𝑇 𝜕2 𝑇 = 𝑘 ( 2 + 2 ) − V𝑓 , 𝜕𝑡 𝜕𝑥 𝜕𝑧 𝜕𝑥

(1)

where 𝑘 is the thermal diffusivity (m2 s−1 ). Since initially the workpiece is at room temperature, (1) is subjected to an homogeneous initial condition: 𝑇 (0, 𝑥, 𝑧) = 0.

(2)

Wet grinding can be modeled assuming a constant heat transfer coefficient ℎ (W m−2 K−1 ) on the workpiece surface [4]. Assuming as well a dimensionless heat flux profile 𝑓(𝑥) within the contact area between wheel and workpiece, we have the following boundary condition: 𝑘0

𝜕𝑇 (𝑡, 𝑥, 0) = ℎ𝑇 (𝑡, 𝑥, 0) 𝜕𝑧

(3)

− 𝑞𝑓 (𝑥) 𝜃 (ℓ − 𝑥) 𝜃 (ℓ + 𝑥) , where 𝜃(𝑥) denotes the Heaviside function, 𝑘0 (W m−1 K−1 ) is the thermal conductivity, and 𝑞 (W m−2 ) is the average heat

2

Mathematical Problems in Engineering

Grinding wheel

Leading edge

󰪓

−󰪓

x

Workpiece

→

f z

y

Heat flux profile

Figure 1: Setup in surface down-grinding.

→

f

P −󰪓

x󳰀

󰪓

x y

y󳰀

z

z󳰀

Figure 2: Relationship between the coordinates systems, one fixed to the wheel and the other one to the workpiece.

flux entering into the workpiece along the contact width 2ℓ; thus, 1 ℓ ∫ 𝑓 (𝑥) 𝑑𝑥 = 1. 2ℓ −ℓ

(4)

In the case of dry grinding, that is, ℎ = 0, boundary-value problem (1)–(3) can be solved in integral form for the case of a constant heat flux profile [5, Eqn.22]. Also, considering a general heat flux profile, a closed form expression for the surface temperature in the stationary regime, that is, lim𝑡→∞ 𝑇(𝑡, 𝑥, 0), can be found [6]. For the wet case, that is, ℎ > 0, (1)–(3) has been solved for constant [4] and linear [7] heat flux profiles, both in the

stationary regime. The scope of this paper is to generalize these results and calculate, in terms of an integral expression, the time-dependent temperature field 𝑇(𝑡, 𝑥, 𝑧) satisfying (1)– (3) for any constant ℎ ≥ 0, considering any heat flux profile. From this general expression, we will recover the expressions for the particular cases found in the literature (constant [8, 9] and linear [7] heat flux profiles), calculating also solutions for other heat flux profiles found in the literature, such as triangular [10–15] and parabolic [16]. These latter solutions do not seem to be reported in the literature. Further, we will provide a simple and rapid method for the calculation of the maximum temperature reached in the workpiece and some approximated expressions for the duration of the transient regime, for all the heat flux profiles considered (i.e., constant, linear, triangular, and parabolic). These approximate expressions seem to be novel as well. It is worth noting that the computation of the relaxation time is essential for analyzing the transient regime in which the wheel is engaged and disengaged from the workpiece (cut in and cut out) [17]. This paper is organized as follows. Section 2 derives the solution of the value boundary problem stated in (1)–(3) for any analytic heat flux profile 𝑓(𝑥). Section 3 particularizes the general solution of the previous section to the most common heat flux profiles reported in the literature, that is, constant, linear, and parabolic. We also give an expression for a triangular heat flux profile, although in this case 𝑓(𝑥) is not an analytic function. Section 4 provides a very efficient method for searching the maximum temperature in the workpiece for the different heat flux profiles considered. In Section 5, for each heat flux profile considered, we propose the root searching of an equation in order to evaluate numerically the duration of the transient regime. For this purpose, we provide analytical approximations to these roots, to be used as starting iteration point in the root finding. Section 6 presents some numerical simulations of the workpiece surface temperature in the stationary regime as well as the maximum temperature computation by using the expressions derived in previous sections. We also plot the surface temperature evolution during the transient regime. Our conclusions are summarized in Section 7.

2. General Heat Flux Profile Solution In order to solve the problem given in (1)–(3), we can build the solution by using the method of Green’s function. For this purpose, let us denote the Cartesian coordinates fixed to the workpiece as 𝑋󸀠 𝑌󸀠 𝑍󸀠 (see Figure 2), where, at 𝑡 = 0, both coordinates systems are overlapped. Green’s function 𝐺(𝑡, 𝑟󸀠⃗ ; 𝑡0 , 𝑟󸀠0⃗ ) is interpreted in this case as the temperature field rise evaluated at (𝑡, 𝑟󸀠⃗ ) = (𝑡, 𝑥󸀠 , 𝑦󸀠 , 𝑧󸀠 ) in a semi-infinite body (𝑧 > 0) in which, at position 𝑟󸀠0⃗ = (𝑥0󸀠 , 𝑦0󸀠 , 𝑧0󸀠 ), an instantaneous point heat source of energy 𝐸 (J) appears at the instant 𝑡0 < 𝑡 and where the surface (𝑧 = 0) is subject to a constant heat transfer coefficient ℎ [3, §14.9 (4)]: 2

󸀠

𝐺 (𝑡, 𝑟⃗

; 𝑡0 , 𝑟󸀠0⃗ )

2

(𝑥󸀠 − 𝑥0󸀠 ) + (𝑦󸀠 − 𝑦0󸀠 ) 𝐸 = exp (− ) 4𝜋𝜌𝑐𝑘 (𝑡 − 𝑡0 ) 4𝑘 (𝑡 − 𝑡0 )


3 2

2

󸀠 󸀠 󸀠 󸀠 { { exp (− (𝑧 − 𝑧0 ) /4𝑘 (𝑡 − 𝑡0 )) + exp (− (𝑧 + 𝑧0 ) /4𝑘 (𝑡 − 𝑡0 )) ⋅{ { 2√𝜋𝑘 (𝑡 − 𝑡0 ) { 2 − ℎ̃ exp (ℎ̃ (𝑧󸀠 + 𝑧0󸀠 ) + ℎ̃ 𝑘 (𝑡 − 𝑡0 )) erfc (ℎ̃√𝑘 (𝑡 − 𝑡0 ) +

} } )} , 2√𝑘 (𝑡 − 𝑡0 ) } } 𝑧󸀠 + 𝑧0󸀠

(5)

where 𝜌 is the density of the body (kg m−3 ) and 𝑐 its specific heat (J kg−1 K−1 ) and where we have defined the radiation coefficient (m−1 ) as ℎ ℎ̃ = . 𝑘0

(6)

A formal derivation of (5) is given in [18, Appendix]. According to Figure 2, a point 𝑃 of the heat source has (𝑥0 , 𝑦0 , 0) coordinates in 𝑋𝑌𝑍 and (𝑥0󸀠 , 𝑦0󸀠 , 0) in 𝑋󸀠 𝑌󸀠 𝑍󸀠 . The relationship between both coordinates systems is 𝑥󸀠 = 𝑥 + V𝑓 (𝑡 − 𝑡0 ) , (7)

𝑧󸀠 = 𝑧. Therefore, the temperature rise in the workpiece at coordinates (𝑡, 𝑥, 𝑦, 𝑧) due to a point 𝑃 of the heat source with respect to the coordinate system fixed to the wheel 𝑋𝑌𝑍 is given by

2

2

(𝑥 + V𝑓 (𝑡 − 𝑡0 ) − 𝑥0 ) + (𝑦 − 𝑦0 ) 4𝑘 (𝑡 − 𝑡0 )

{ { exp (−𝑧2 /4𝑘 (𝑡 − 𝑡0 )) ̃ ℎ ̃2 𝑘(𝑡−𝑡0 ) ̃ ℎ𝑧+ ⋅{ − ℎ𝑒 { 2√𝜋𝑘 (𝑡 − 𝑡0 ) {

(9)

thus, integrating in 𝑦0 ∈ (−∞, ∞), we obtain the field temperature due to an instantaneous line heat source as 𝑞𝐿 𝑇line (𝑡, 𝑥, 𝑧) = 2𝜋𝜌𝑐𝑘 (𝑡 − 𝑡0 ) ⋅ exp (−

(𝑥 + V𝑓 (𝑡 − 𝑡0 ) − 𝑥0 ) 4𝑘 (𝑡 − 𝑡0 )

)

{ { ̃ ̃2 −𝑧2 ⋅ {exp ( ) − ℎ̃√𝜋𝑘 (𝑡 − 𝑡0 )𝑒ℎ𝑧+ℎ 𝑘(𝑡−𝑡0 ) { 4𝑘 (𝑡 − 𝑡0 ) { ⋅ erfc (ℎ̃√𝑘 (𝑡 − 𝑡0 ) +

𝐸 𝐺 (𝑡, 𝑥, 𝑦, 𝑧) = 4𝜋𝑘0 (𝑡 − 𝑡0 )

(10)

} } )} . 2√𝑘 (𝑡 − 𝑡0 ) } } 𝑧

2.2. Instantaneous Strip Source. Let us consider now a strip source of variable heat strength along the 𝑥-axis, according to the following profile:

)

𝑞𝐿 = 𝑞𝐵 𝑓 (𝑥0 ) 𝑑𝑥0 ,

(8)

} } )} . 2√𝑘 (𝑡 − 𝑡0 ) } } The derivation of the solution of (1)–(3) by using (8) follows three steps: ⋅ erfc (ℎ̃√𝑘 (𝑡 − 𝑡0 ) +

𝑑𝐸 = 𝑞𝐿 𝑑𝑦0 ;

2

𝑦󸀠 = 𝑦,

⋅ exp (−

2.1. Instantaneous Line Source. Let us consider now an instantaneous line heat source parallel to the 𝑦-axis. Let us assume in (8) that the energy per unit length of this heat source 𝑞𝐿 (J m−1 ) is constant:

𝑧

(i) Superposition in space to give temperature due to instantaneous line source, which acts on the surface 𝑧 = 0 parallel to the 𝑦-axis. (ii) Superposition of line sources to give temperature due to an instantaneously acting strip source on the surface. (iii) Superposition in time to give temperature due to continuously acting strip source.

−2

(11)

where 𝑞𝐵 (J m ) is the energy that this heat source leaves at the instant 𝑡0 per surface unit of the strip source. Notice that the 𝑓 function in (11) is dimensionless, as in (3). Also, 𝑓 takes into account the heat flux profile entering into the workpiece within the contact zone with the wheel. Substituting now (11) in (10) and integrating over the strip width, 𝑥0 ∈ (−ℓ, ℓ), we have the following temperature field in the workpiece: 𝑇strip (𝑡, 𝑥, 𝑧) =

{ { 𝑞𝐵 −𝑧2 exp ( ) { 2𝜋𝜌𝑐𝑘 (𝑡 − 𝑡0 ) { 4𝑘 (𝑡 − 𝑡0 ) { ̃

̃2

− ℎ̃√𝜋𝑘 (𝑡 − 𝑡0 )𝑒ℎ𝑧+ℎ 𝑘(𝑡−𝑡0 ) ⋅ erfc (ℎ̃√𝑘 (𝑡 − 𝑡0 ) +

} } )} 2√𝑘 (𝑡 − 𝑡0 ) } } 𝑧

4

Mathematical Problems in Engineering ℓ

⋅ ∫ 𝑓 (𝑥0 )

⋅ erfc (ℎ̃√𝑘 (𝑡 − 𝑡0 ) +

−ℓ

2

[ 𝑥 − 𝑥0 + V𝑓 (𝑡 − 𝑡0 ) ] ⋅ exp (− [ ] ) 𝑑𝑥0 . 2√𝑘 (𝑡 − 𝑡0 ) [ ]

∞

⋅ ∑𝑔

(𝑚)

(0) [

𝑥 + V𝑓 (𝑡 − 𝑡0 ) ℓ

𝑚=0

(12) Assuming that the 𝑓 function is analytic, we can expand it in its Taylor series, so that, taking into account also that 𝑓 is dimensionless, we have ∞ (𝑚) 𝑔 (0) 𝑥0 𝑚 𝑥 𝑓 (𝑥0 ) = 𝑔 ( 0 ) = ∑ ( ) ; ℓ 𝑚! ℓ 𝑚=0

(13)

thus, the integral given in (12) can be written as 2

[ 𝑥 − 𝑥0 + V𝑓 (𝑡 − 𝑡0 ) ] ∫ 𝑓 (𝑥0 ) exp (− [ ] ) 𝑑𝑥0 −ℓ 2√𝑘 (𝑡 − 𝑡0 ) ] [

⋅(

1 𝑛=0 𝑛! (𝑚 − 𝑛)!

] ∑

𝑥 + V𝑓 (𝑡 − 𝑡0 )

󵄨𝑥+ℓ ) 𝐹𝑛 (𝑢, 𝑡 − 𝑡0 )󵄨󵄨󵄨𝑢=𝑥−ℓ . (17)

2.3. Continuous Strip Source. Let 𝑞 (W m−2 ) be the energy per time unit and per surface unit that the strip heat source leaves over the workpiece surface. Therefore, during an infinitesimal interval of time 𝑑𝑡0 , we have 𝑞𝐵 = 𝑞𝑑𝑡0 .

(18)

Substitution of (18) in (17) and integration over time 𝑡0 ∈ (0, 𝑡) (remember that 𝑡0 < 𝑡) result in

∞

𝑔(𝑚) (0) 𝑚 𝑚=0 ℓ 𝑚!

𝑚 𝑚

𝑛

−2√𝑘 (𝑡 − 𝑡0 )

ℓ

= ∑

} } )} 2√𝑘 (𝑡 − 𝑡0 ) } } 𝑧

(14) 𝑇 (𝑡, 𝑥, 𝑧) = 2

[ 𝑥 − 𝑥0 + V𝑓 (𝑡 − 𝑡0 ) ] ⋅ ∫ 𝑥0𝑚 exp (− [ ] ) 𝑑𝑥0 . −ℓ 2√𝑘 (𝑡 − 𝑡0 ) ] [

𝑡 exp (−𝑧2 /4𝑘𝑡󸀠 ) 𝑞 ∫ [ √𝑘𝑡󸀠 2𝜋𝜌𝑐 0

ℓ

2

̃ ̃ − ℎ̃√𝜋𝑘𝑡󸀠 𝑒ℎ𝑧+ℎ 𝑘𝑡 erfc (ℎ̃√𝑘𝑡󸀠 +

According to the result given in Appendix (A.9), we have 2

ℓ

∫

−ℓ

𝑥0𝑚

=

⋅ (−

𝑥 + V𝑓 (𝑡 − 𝑡0 ) 2√𝑘 (𝑡 − 𝑡0 )

𝑚 ∑( ) 𝑛=0 𝑛

(15)

𝑢 + V𝑓 𝑡 2√𝑘𝑡

󵄨𝑥+ℓ 𝐹𝑛 (𝑢, 𝑡 − 𝑡0 )󵄨󵄨󵄨𝑢=𝑥−ℓ ,

2

)𝛾(

𝑛 + 1 𝑢 + V𝑓 𝑡 ] ) (16) ,[ 2 2√𝑘𝑡

and where 𝛾(𝑎, 𝑧) denotes the lower incomplete gamma function [19, Eqn. 45:1:2]. Substituting back these results, we arrive at { { 𝑞𝐵 −𝑧2 𝑇strip (𝑡, 𝑥, 𝑧) = ) exp ( { 4𝑘 (𝑡 − 𝑡0 ) 2𝜋𝜌𝑐√𝑘 (𝑡 − 𝑡0 ) { { ̃ ℎ ̃2 𝑘(𝑡−𝑡0 ) ℎ𝑧+

− ℎ̃√𝜋𝑘 (𝑡 − 𝑡0 )𝑒

(19)

1 𝑛! − 𝑛)! (𝑚 𝑛=0

) ∑

where we have performed the change of variables 𝑡󸀠 = 𝑡 − 𝑡0 . Considering now the dimensionless variables 𝑥 , 𝑠 𝑧 𝑍= , 𝑠

𝑋=

𝑚−𝑛

where 𝐹𝑛 (𝑢, 𝑡) = sign𝑛+1 (

𝑚 𝑚

𝑛

𝑚

)

ℓ

𝑧 )] √ 2 𝑘𝑡󸀠

−2√𝑘𝑡󸀠 󵄨𝑥+ℓ ) 𝐹𝑛 (𝑢, 𝑡󸀠 )󵄨󵄨󵄨󵄨𝑢=𝑥−ℓ ] 𝑑𝑡󸀠 , ⋅( 󸀠 𝑥 + V𝑓 𝑡

𝑚+1

2

⋅ [ ∑ 𝑔(𝑚) (0) (

𝑥 + V𝑓 𝑡󸀠

𝑚=0

[ 𝑥 − 𝑥0 + V𝑓 (𝑡 − 𝑡0 ) ] exp (− [ ] ) 𝑑𝑥0 2√𝑘 (𝑡 − 𝑡0 ) [ ]

(−2√𝑘 (𝑡 − 𝑡0 ))

∞

󸀠

ℓ 𝐿= , 𝑠

(20)

̃ 𝐻 = ℎ𝑠, 𝜏= T=

V𝑓 √ 𝑡

, 2√𝑘 𝜋𝜌𝑐V𝑓 2𝑞

𝑇,

where 𝑠 is a characteristic length 𝑠=

2𝑘 , V𝑓

(21)


5 performing the change of variables 𝑠 = 2𝑤2 , we obtain the following expression for the case of dry grinding:

we finally arrive at 𝜏

T (𝜏, 𝑋, 𝑍) = ∫ [exp ( 0

⋅ erfc (𝐻𝑤 + 𝑚

⋅∑

2 2 −𝑍2 ) − √𝜋𝐻𝑤𝑒𝐻𝑍+𝐻 𝑤 4𝑤2

∞ (𝑚) 𝑔 (0) 𝑍 )] [ ∑ 2𝑤 𝐿𝑚 [𝑚=0 𝑚−𝑛

(−2𝑤)𝑛 (𝑋 + 2𝑤2 )

𝑛=0

𝑛! (𝑚 − 𝑛)!

2

−𝑍2 1 𝜋 2𝜏 √ exp ( ) 𝑋, 𝑍) = T(0) ∫ (𝜏, dry 2 2 0 4𝑠 𝑋−𝐿+𝑠 𝑑𝑠 𝑋+𝐿+𝑠 , ) − erf ( )] ⋅ [erf ( √2𝑠 √2𝑠 √𝑠

(22)

which is the result given in [5, Eqn.22]. Undoing the change of variables and performing the limit 𝜏 → ∞, we get the stationary regime for the dry case [5, Eqn.7]:

󵄨𝑋+𝐿 F𝑛 (𝑢, 𝑤)󵄨󵄨󵄨𝑢=𝑋−𝐿 ] 𝑑𝑤, ]

∞

where

√ lim T(0) dry (𝜏, 𝑋, 𝑍) = 𝜋 ∫ exp (

𝜏→∞

F𝑛 (𝑢, 𝑤) = sign𝑛+1 (

2 𝑢 𝑛+1 𝑢 + 𝑤) 𝛾 ( ,[ + 𝑤] ) . 2𝑤 2 2𝑤

=∫

In this section, we derive, from general expression (22), particular expressions for the most common heat flux profiles considered in the literature (i.e., constant, linear, triangular, and parabolic). For the constant and linear cases, we obtain expressions already given in the literature, as mentioned before.

𝜏→∞

𝑢 + 𝑤) . 2𝑤

𝜏

0

2

𝑋−𝐿

∞

2

𝑒−𝑤 𝐾0 (√𝑤2 + 𝑍2 ) 𝑑𝑤

2

0

⋅ [erf (

𝑍 ) 2𝑤

(30)

𝑋+𝐿 𝑋−𝐿 + 𝑤) − erf ( + 𝑤)] 𝑑𝑤, 2𝑤 2𝑤

which is the expression given by [4]. Recently, in [21], the expression given in (30) particularized on the surface (𝑍 = 0) can be expressed in closed form as lim T(0) (𝜏, 𝑋, 0) ∞

(−𝐻/√2)

𝑛

󵄨𝑋+𝐿 Yu𝑛/2 (𝑢)󵄨󵄨󵄨𝑋−𝐿 , = √𝜋 ∑ 𝑛=0 Γ ((𝑛 + 1) /2)

(25)

(31)

where Yu] (𝑥) 𝑥 |𝑥|] 𝑒𝑥 { [𝐾] (|𝑥|) + sign (𝑥) 𝐾]+1 (|𝑥|)] , 𝑥 ≠ 0, (32) = { 2] + 1 𝑥 = 0. {0,

−𝑍2 ) 4𝑤2

− √𝜋𝐻𝑤𝑒𝐻𝑍+𝐻 𝑤 erfc (𝐻𝑤 + ⋅ [erf (

(29)

𝜏→∞

Therefore, (22) reduces to T(0) (𝜏, 𝑋, 𝑍) = √𝜋 ∫ [exp (

𝑋+𝐿

− 𝜋𝐻𝑒𝐻𝑍 ∫ 𝑤𝑒𝐻 𝑤 erfc (𝐻𝑤 +

2

= √𝜋 erf (

𝑒−𝑤 𝐾0 (√𝑤2 + 𝑍2 ) 𝑑𝑤,

lim T(0) (𝜏, 𝑋, 𝑍) = ∫

(24)

𝑢 1 𝑢 + 𝑤) 𝛾 ( , [ + 𝑤] ) 2𝑤 2 2𝑤

2

𝑋+𝐿

(28)

where 𝐾0 (𝑥) is the zeroth-order modified Bessel function of the second kind [20, Sect. 9.6]. Therefore, taking into account (29) in (26), we obtain the stationary regime for the wet case as

3.1. Constant Heat Flux Profile. For a constant heat flux profile satisfying (4), we have

Thus, in (22), we have only the summand which corresponds to 𝑚 = 0, so, from (23) and using the property given in Appendix (A.12), we have

−𝑍2 ) 4𝑤2

𝑋+𝐿 𝑋−𝐿 ⋅ [erf ( + 𝑤) − erf ( + 𝑤)] 𝑑𝑤 2𝑤 2𝑤

(23)

3. Particular Cases

F0 (𝑢, 𝑤) = sign (

0

𝑋−𝐿

𝑥 𝑓0 (𝑥) = 𝑔0 ( ) = 1. ℓ

(27)

𝑍 )] 2𝑤

(26)

𝑋+𝐿 𝑋−𝐿 + 𝑤) − erf ( + 𝑤)] 𝑑𝑤, 2𝑤 2𝑤

where the superscript (0) denotes that we are considering a constant heat flux profile. Particularizing (26) to 𝐻 = 0 and

It is worth noting that (31) does not converge for high Biot numbers, that is, 𝐻 > 1. 3.2. Linear Heat Flux Profile. For a linear heat flux profile satisfying (4), we have 𝑥 𝑥 𝑓1 (𝑥) = 𝑔 ( ) = 1 + ; ℓ ℓ

(33)

6


thus, (22) is reduced to

Parabolic profile

−𝑍2 1 𝜏 (𝜏, 𝑋, 𝑍) = ∫ 𝑑𝑤 [exp ( 2 ) 𝐿 0 4𝑤 2

2

− √𝜋𝐻𝑤𝑒𝐻𝑍+𝐻 𝑤 erfc (𝐻𝑤 +

2.5 2.0 Triangular profile

𝑍 )] 2𝑤

Constant profile

f(x)

T

(1)

3.0

(34)

+ 2𝑤 exp (− [

0.5 𝜆 −1.0

𝑋+𝐿

−0.5

2 󵄨󵄨𝑋+𝐿 𝑢 󵄨 . + 𝑤] )󵄨󵄨󵄨 󵄨󵄨𝑢=𝑋−𝐿 2𝑤

3.3. Triangular Heat Flux Profile. For a triangular heat flux profile satisfying (4) (see Figure 3), we have 𝑥 𝑓Δ (𝑥) = 𝑔Δ ( ) ℓ

(35)

𝑥 2 { ( + 1) , 𝑥 ∈ (−ℓ, ℓmax ) { {𝜆 + 1 ℓ ={ { { 2 ( 𝑥 − 1) , 𝑥 ∈ (ℓ , ℓ) , max {𝜆 − 1 ℓ

Performing in (34) the change of variables 𝑟 = 2𝑤2 and the limit 𝜏 → ∞, we obtain the following expression for the stationary regime, which is given in [7, Eqns.3.46-47]: lim T(1) (𝜏, 𝑋, 𝑍) =

𝜏→∞

𝜆=

𝑋+𝐿+𝑟 𝑋−𝐿+𝑟 + 𝐿 + 𝑟) [erfc ( ) − erfc ( )] √2𝑟 √2𝑟

−

(𝑋 + 𝐿 + 𝑟)2 ) 2𝑢

− exp (−

(𝑋 − 𝐿 + 𝑟)2 )} 𝑑𝑟 2𝑟

+√

ℓmax ∈ [−1, 1] . ℓ

(39)

Although we cannot expand (38) in its Taylor series, we can redo the calculation of Section 2.2, performing the integral given in (12) as follows: ℓ

2

∫ 𝑓 (𝑥0 ) exp (− [𝐴 + 𝐵𝑥0 ] ) 𝑑𝑥0

𝜋𝐻 ∞ −𝑍2 𝐻𝑟 ) {(𝑋 + 𝐿 + 𝑟) ) experf ( ∫ exp ( √2𝑟 4𝐿 0 2𝑟 ⋅ [erfc (

(38)

where we have defined the following dimensionless parameter:

𝜋 −𝑍2 1 ∞ ) {√ (𝑋 ∫ exp ( 2𝐿 0 2𝑟 2𝑢

+ exp (−

1.0

Figure 3: Heat flux profiles considered.

where the superscript (1) indicates a linear heat flux profile and where we have applied (25). Also, according to (A.14), the following result has been applied:

= − exp (− [

0.5 x/󰪓

𝑢 , + 𝑤] )] 2𝑤 𝑢=𝑋−𝐿

2 󵄨󵄨𝑋+𝐿 𝑢 󵄨 󵄨𝑋+𝐿 + 𝑤] )󵄨󵄨󵄨 F1 (𝑢, 𝑤)󵄨󵄨󵄨𝑢=𝑋−𝐿 = 𝛾 (1, [ 󵄨󵄨𝑢=𝑋−𝐿 2𝑤

Linear profile

1.0

𝑢 ⋅ [√𝜋 (𝑋 + 𝐿 + 2𝑤2 ) erf ( + 𝑤) 2𝑤 2

1.5

(36)

=

𝑋+𝐿+𝑟 𝑋−𝐿+𝑟 ) − erfc ( )] √2𝑟 √2𝑟

(40)

ℓ 𝑥 2 2 ∫ ( 0 − 1) exp (− [𝐴 + 𝐵𝑥0 ] ) 𝑑𝑥0 , 𝜆 − 1 ℓmax ℓ

where we have set 𝐴=

(𝑋 − 𝐿 + 𝑟)2 − exp (− )]} 𝑑𝑟, 2𝑟

𝑥 + V𝑓 (𝑡 − 𝑡0 ) 2√𝑘 (𝑡 − 𝑡0 )

, (41)

where the following function has been defined: experf (𝑥) = exp (𝑥 ) erfc (𝑥) .

ℓmax 𝑥 2 2 ∫ ( 0 + 1) exp (− [𝐴 + 𝐵𝑥0 ] ) 𝑑𝑥0 𝜆 + 1 −ℓ ℓ

+

2𝑟 (𝑋 + 𝐿 + 𝑟)2 [exp (− ) 𝜋 2𝑟

2

−ℓ

𝐵= (37)

−1 2√𝑘 (𝑡 − 𝑡0 )

.


7

By using formulas (A.13) and (A.15) given in Appendix, some algebra leads to the following result: ℓ

thus, (22) is reduced to T(2) (𝜏, 𝑋, 𝑍) =

2

∫ 𝑓 (𝑥0 ) exp (− [𝐴 + 𝐵𝑥0 ] ) 𝑑𝑥0 −ℓ

2

𝜑 (𝐴 + 𝐵ℓmax ) 1 𝜑 (𝐴 − 𝐵ℓ) = 2[ +2 ℓ𝐵 𝜆+1 𝜆2 − 1

(42)

2

− (43)

Substituting (42) into (12), integrating over time 𝑡0 ∈ (0, 𝑡), and performing the change of variables 𝑡󸀠 = 𝑡 − 𝑡0 , we get the time-dependent temperature field due to a triangular heat flux profile as 𝑇Δ (𝑡, 𝑥, 𝑧) =

2 󸀠 2𝑞 𝑡 exp (−𝑧 /4𝑘𝑡 ) ∫ { √𝑘𝑡󸀠 𝜋𝜌𝑐 0 2

̃ ℎ ̃ 𝑘𝑡 ̃ ℎ𝑧+ − √𝜋ℎ𝑒 erfc (ℎ̃√𝑘𝑡󸀠 +

√𝑘𝑡󸀠 ℓ

+ −

{

󸀠

𝜆+1

𝜆−1

4 𝜏 −𝑍2 T (𝜏, 𝑋, 𝑍) = ∫ [exp ( 2 ) 𝐿 0 4𝑤 Δ

2

⋅ 𝑤[ +

2

𝑍 )] 2𝑤

2

𝑢 + 𝑤) 2𝑤

(48)

𝑍 )] 2𝑤

𝜑 ((𝑋 + 𝐿) /2𝑤 + 𝑤) 𝜆+1

𝑋+𝐿

2 𝑢 𝑑𝑤. + 𝑤] )] 2𝑤 𝑢=𝑋−𝐿

4. Maximum Temperature

which, in dimensionless variables (20), is rewritten as

2

−𝑍2 3 𝜏 [exp ( ) ∫ 4𝐿2 0 4𝑤2

⋅ [√𝜋 (2𝑤2 + (𝐿 + 2𝑤2 + 𝑋) ) erf (

⋅ exp (− [

} 𝑑𝑡󸀠 ,


where the superscript (2) denotes that we are considering a parabolic heat flux profile. By using the properties given in Appendices (A.12), (A.14), and (A.17), we arrive at

+ 2𝑤 (2𝐿 + 2𝑋 + 2𝑤2 − 𝑢)

−1

𝜑 ((𝑥 − ℓ + V𝑓 𝑡󸀠 ) /2√𝑘𝑡󸀠 )

⋅ F2 (𝑢, 𝑤)] 𝑑𝑤, ]𝑢=𝑋−𝐿

2

2𝜑 ((𝑥 − 𝜆ℓ + V𝑓 𝑡󸀠 ) /2√𝑘𝑡󸀠 ) 𝜆2

𝑋+𝐿

− √𝜋𝐻𝑤𝑒𝐻𝑍+𝐻 𝑤 erfc (𝐻𝑤 + (44)

(47)

2𝑤 𝑋 + 2𝑤2 2𝑤2 (1 + ) F1 (𝑢, 𝑤) + 2 𝐿 𝐿 𝐿

T(2) (𝜏, 𝑋, 𝑍) =

𝑧 )} √ 2 𝑘𝑡󸀠

𝜑 ((𝑥 + ℓ + V𝑓 𝑡󸀠 ) /2√𝑘𝑡󸀠 )

𝑍 )] 2𝑤

2 1 𝑋 + 2𝑤2 (𝑋 + 2𝑤 ) [ ⋅ ( + ) F0 (𝑢, 𝑤) + 2 𝐿 2𝐿2 [

where we have defined 𝜑 (𝑥) = √𝜋𝑥 erf (𝑥) + exp (−𝑥2 ) .

2


𝜑 (𝐴 + 𝐵ℓ) ], 𝜆−1

−

⋅

3 𝜏 −𝑍2 ∫ [exp ( 2 ) 2 0 4𝑤

(45)

2𝜑 ((𝑋 − 𝜆𝐿) /2𝑤 + 𝑤) 𝜆2 − 1

From a physical point of view, the maximum temperature Tmax must be reached at the stationary regime 𝜏 → ∞, because the longer the heat source is acting, the greater the temperature in the workpiece is. Moreover, the location of maximum temperature 𝑋max must be on the surface 𝑍 = 0, within the contact zone between wheel and workpiece, 𝑋max ∈ [−𝐿, 𝐿]. It is worth noting that recently the latter has been mathematically proved in [22] for constant and linear heat flux profiles. Therefore, since T(𝜏, 𝑋, 𝑍) is a differentiable function in 𝑋, a convenient way to evaluate maximum temperature is to solve numerically all points 𝑋∗ that satisfies 𝜕T (𝜏, 𝑋∗ , 0) = 0, 𝑋∗ ∈ [−𝐿, 𝐿] , 𝜏→∞ 𝜕𝑋

𝜑 ((𝑋 − 𝐿) /2𝑤 + 𝑤) − ] 𝑑𝑤. 𝜆−1

lim

(49)

and then the maximum is given by 3.4. Parabolic Heat Flux Profile. For a parabolic heat flux profile satisfying (4) (see Figure 3), we have

Tmax = max [ lim T (𝜏, 𝑋∗ , 0)] . ∗

𝑥 3 𝑥 2 𝑓2 (𝑥) = 𝑔2 ( ) = (1 + ) ; ℓ 4 ℓ

In Section 6, we will check numerically that for the most common heat flux profiles proposed in the literature

(46)

𝑋

𝜏→∞

(50)

8


(constant, linear, triangular, and parabolic) the root 𝑋∗ is unique, so 𝑋∗ = 𝑋max . For a constant heat flux profile, (49) reads ∞

2 2 1 0 = ∫ [ − √𝜋𝐻𝑒𝐻 𝑤 erfc (𝐻𝑤)] 𝑤 0 ∗ 2 󵄨󵄨𝑋 +𝐿 𝑢 󵄨󵄨 𝑑𝑤. ⋅ exp (− [ + 𝑤] )󵄨󵄨 󵄨󵄨𝑢=𝑋∗ −𝐿 2𝑤

(51)

Similarly, for a linear heat flux profile, we have ∞

2

𝜕T (𝜏, 𝑋max , 0) (56) = 𝜂 ≈ 0. 𝜕𝜏 In this section, we are going to provide approximated explicit solutions of (56) for the different heat flux profiles considered. Each of these approximate solutions will be used as starting iteration point for the numerical root searching of (56). 5.1. Constant Heat Flux Profile. For a constant heat flux profile, according to (26), (56) is written as

2

0 = ∫ [1 − √𝜋𝐻𝑤𝑒𝐻 𝑤 erfc (𝐻𝑤)] 0

2 2

𝑢 ⋅ [√𝜋 (𝑋 + 𝐿 + 2𝑤 ) erf ( + 𝑤) 2𝑤 ∗

−

is reached (i.e., the relaxation time of the transient regime), we can solve the following equation for 𝜏 ∈ (0, ∞):

2

𝜂 = √𝜋 [1 − √𝜋𝐻𝜏𝑒𝐻 𝜏 erfc (𝐻𝜏)] (52)

𝑋∗ +𝐿

2 𝑢 𝑋∗ + 𝐿 − 𝑢 𝑑𝑤. exp (− [ + 𝑤] )] 𝑤 2𝑤 𝑢=𝑋∗ −𝐿

For a triangular profile, we obtain ∞

2

∞

2

2

√𝜋𝑧𝑒𝑧 erfc (𝑧) ≈ 1 + ∑ (−1)𝑚

0 = ∫ [1 − √𝜋𝐻𝑤𝑒𝐻 𝑤 erfc (𝐻𝑤)] 0

⋅[

(57) 𝑋max + 𝐿 −𝐿 𝑋 + 𝜏) − erf ( max + 𝜏)] . 2𝜏 2𝜏 Let us solve (57) approximately. On the one hand, we find in the literature [20, Eqn.7.1.23] the following asymptotic expansion: ⋅ [erf (

𝑚=1

(2𝑚 − 1)!! 𝑚 , (2𝑧2 )

∗

erf ((𝑋 + 𝐿) /2𝑤 + 𝑤) 𝜆+1 ∗

+

2 erf ((𝑋 − 𝜆𝐿) /2𝑤 + 𝑤) 𝜆2 − 1

−

erf ((𝑋∗ − 𝐿) /2𝑤 + 𝑤) ] 𝑑𝑤, 𝜆−1

𝑧 󳨀→ ∞; (53)

thus, expanding up to 𝑚 = 1 and taking 𝑧 = 𝐻𝜏, we have 1 , 𝜏 󳨀→ ∞. (59) 2𝐻2 𝜏2 On the other hand, expanding in Taylor series the following function up to the first order, we have 2 2

1 − √𝜋𝐻𝜏𝑒𝐻 𝜏 erfc (𝐻𝜏) ≈

erf (𝜏 +

and for a parabolic one ∞

2

± 𝐿 −𝜏2 𝑋max ± 𝐿 𝑋 𝑒 , ) ≈ erf (𝜏) + max √𝜋𝜏 2𝜏

2

0 = ∫ [1 − √𝜋𝐻𝑤𝑒𝐻 𝑤 erfc (𝐻𝑤)] thus,

𝑢 ⋅ [√𝜋 (𝑋 + 𝐿 + 2𝑤 ) erf ( + 𝑤) 2𝑤 2

2

(𝑋∗ + 𝐿 − 𝑢) +( + 2𝑤) 2𝑤

√𝜋 [erf ( (54)

𝑋∗ +𝐿

2 𝑢 ⋅ exp (− [ 𝑑𝑤. + 𝑤] )] 2𝑤 𝑢=𝑋∗ −𝐿

−𝐿 𝑋max + 𝐿 𝑋 + 𝜏) − erf ( max + 𝜏)] 2𝜏 2𝜏

(61) 2𝐿 −𝜏2 ≈ 𝑒 , 𝜏 󳨀→ ∞. 𝜏 Substituting (59) and (61) in (57), we get the following approximated equation: 2

𝜏3 𝑒𝜏 ≈

5. Relaxation Time The stationary regime is asymptotically reached at 𝜏 → ∞; that is, 𝜕T (𝜏, 𝑋, 𝑍) lim = 0. 𝜏→∞ 𝜕𝜏

(60)

𝜏 󳨀→ ∞;

0

∗

(58)

(55)

In order to avoid thermal damage, the most important point is the location of the maximum temperature 𝑋max . Therefore, for estimating how rapidly the stationary regime

𝐿 . 𝐻2 𝜂

(62)

By using the Lambert W function [23], (62) can be solved explicitly, obtaining the following approximated expression for the relaxation time: 3 2 𝐿 2/3 𝜏 ≈ √ W ( [ 2 ] ). 2 3 𝐻𝜂

(63)

Notice that (63) does not depend on 𝑋max , so we do not have to compute (57) if we want an estimation of the relaxation time for a constant heat flux profile.


9

5.2. Linear Heat Flux Profile. Considering now a linear heat flux profile, according to (34), (56) is written as 𝜂𝐿 = {√𝜋 (𝑋max + 𝐿 + 2𝜏2 ) [erf (

𝑋max + 𝐿 + 𝜏) 2𝜏

𝜑(

−𝐿 𝑋 − erf ( max + 𝜏)] 2𝜏 − 2𝜏 [exp (− [ − exp (− [

Equation (69) can be solved approximately expanding in the Taylor series the following function up to first order, taking into account the 𝜑 function definition (43) and asymptotic formulas (60) and (65): 𝜒 𝜒 𝜒 + 𝜏) = √𝜋 ( + 𝜏) erf ( + 𝜏) 2𝜏 2𝜏 2𝜏 + exp (− [

2

𝑋max + 𝐿 + 𝜏] ) 2𝜏

(64) ≈ √𝜋 (

2 2 2 𝑋max − 𝐿 + 𝜏] )]} [1 − √𝜋𝐻𝜏𝑒𝐻 𝜏 2𝜏

⋅ erfc (𝐻𝜏)] .

𝜒2 2𝜏

±𝐿 𝑋 exp (− [𝜏 + max ] ) ≈ 𝑒−𝜏 − (𝑋max ± 𝐿) 𝑒−𝜏 , 2𝜏 2

𝜏3 𝑒𝜏 ≈

2

(65)

𝜏 󳨀→ ∞;

𝑋max + 𝐿 + 𝜏] ) 2𝜏 2 2 𝑋max − 𝐿 + 𝜏] ) ≈ 2𝐿𝑒−𝜏 , 2𝜏

(66)

Now, substituting (59), (61), and (66) in (64), we arrive at 2

(71)

𝑋max + 𝐿 𝐻2 𝜂

(67)

+𝐿 3 2 𝑋 𝜏 ≈ √ W ( [ max2 ] 2 3 𝐻𝜂

2/3

).

4𝐿2 𝜂 = {2𝜏 [(𝑋max + 𝐿 + 2𝜏2 ) 3 ⋅ exp (− [

2 𝑋max + 𝐿 + 𝜏] ) − (𝑋max + 3𝐿 + 2𝜏2 ) 2𝜏

⋅ exp (− [

2 𝑋max − 𝐿 + 𝜏] )] + √𝜋 [2𝜏2 2𝜏

(68)

2

+ (𝑋max + 𝐿 + 2𝜏2 ) ] [erf (

5.3. Triangular Heat Flux Profile. Let us consider now a triangular heat flux profile, so, according to (45), (56) reads as 𝜑 ((𝑋max + 𝐿) /2𝜏 + 𝜏) 𝜂𝐿 =[ 4 𝜆+1 2𝜑 ((𝑋max − 𝜆𝐿) /2𝜏 + 𝜏) + 𝜆2 − 1 𝜑 ((𝑋max − 𝐿) /2𝜏 + 𝜏) − ] 𝜏 [1 𝜆−1 − √𝜋𝐻𝜏𝑒𝐻 𝜏 erfc (𝐻𝜏)] .

(72)

As in the constant case, (72) does not depend on 𝑋max ; moreover, it does not depend on 𝜆 either.

that can be solved explicitly as

2 2

2𝐿 𝐻2 𝜂

5.4. Parabolic Heat Flux Profile. Finally, for a parabolic heat flux profile, according to (48), (56) is expressed as

𝜏 󳨀→ ∞.

𝜏3 𝑒𝜏 ≈

𝜏 󳨀→ ∞.

3 2 2𝐿 2/3 𝜏 ≈ √ W ( [ 2 ] ). 2 3 𝐻𝜂

2

− exp (− [

2

) 𝑒−𝜏 ,

that can be solved explicitly as

thus, exp (− [

2

Taking into account (59) and (70) in (69), after some algebra, we arrive at the following approximated equation: 2

2

(70)

𝜒 + 𝜏) erf (𝜏) 2𝜏

+ (1 +

In order to solve (64) approximately, let us expand in Taylor series the following function up to first order:

2 𝜒 + 𝜏] ) 2𝜏

− erf (

(73)

𝑋max + 𝐿 + 𝜏) 2𝜏

2 2 𝑋max − 𝐿 + 𝜏)]} [1 − √𝜋𝐻𝜏𝑒𝐻 𝜏 2𝜏

⋅ erfc (𝐻𝜏)] .

(69)

Taking into account asymptotic formulas (59), (61), and (65), some algebra leads (73) to the following approximate equation: 2

(𝑋max + 𝐿) + 4𝑋max 𝜏2 −𝜏2 4𝑋max −𝜏2 4𝐿 𝑒 ≈ 𝜂≈ 𝑒 , 3 𝐻2 𝜏 𝐻2 𝜏3 𝜏 󳨀→ ∞,

(74)

10

Mathematical Problems in Engineering Table 1: Simulation parameters in SI units.

600

Data 2

Data 3

Workpiece

Material 𝑘0 𝑘

VT20 13 4.23 × 10−6

Steel 60.5 1.77 × 10−5

Sapphire 46 1.51 × 10−5

Grinding regime

ℎ 2ℓ 𝑞 V𝑓 𝑇0

27.29 × 104 2.66 × 10−3 5.89 × 107 0.53 300

4.1 × 105 1.4 × 10−3 1.4 × 107 3.3 × 10−2 300

1.3 × 105 2.5 × 10−3 1.8 × 107 3.3 × 10−2 300

𝐻 𝐿 T 𝜂

0.335 83.41 23.02 10−3

7.29 0.65 79.23 10−3

2.59 1.36 114.3 10−3

Dimensionless parameters

500 450 400 350

−1.5

−1.0

−0.5

0.5

1.0

x/󰪓 Heat flux profile Constant Linear

Parabolic Triangular

Figure 4: Surface temperature for a VT20 titanium alloy workpiece (data 1).

Table 2: Location and value of maximum temperature. Data 1

Data 2

Data 3

Constant profile

𝑥max /ℓ 𝑇max [K]

−0.973 477.6

−0.301 329.1

−0.501 407.6

Linear profile


0.609 537.3

0.687 341.3

0.605 442.4

Triangular profile


0.459 551.6

0.530 343.6

0.459 450.4

Parabolic profile


0.748 612.8

0.799 353.4

0.751 481.4

which can be solved as 18𝑋 1 𝜏 ≈ √ W ( 4 max ). 2 𝐻 𝐿2 𝜂2

550 T(∞, x, 0) (K)

Data 1

(75)

6. Numerical Results Table 1 shows three sets of parameters (SI units) for the numerical simulations. Data set 1 considers a titanium alloy VT20 workpiece, whose thermal properties are given in [24]. The grinding regime for this simulation can be found in [25]. Data sets 2 and 3 consider plain carbon steel and aluminum oxide, Al2 O3 (sapphire), as workpiece material, respectively. The thermal properties and the grinding regimes for these data sets can be found in [26]. Figure 3 presents the different heat flux profiles considered. All of them have been normalized according to (4); thus, they have got the same heat flux on average. Therefore, we can compare how the distribution of the heat flux within the contact zone affects the graph of the workpiece surface temperature. Also, for all the numerical simulations, we have taken a value of 𝜆 = 0.7 for the triangular profile (45). Table 2 shows the value of maximum temperature 𝑇max and its location 𝑥max /ℓ within the friction zone. For this purpose, we have computed the roots of (51)–(54) by using Brent’s method [27], taking as starting interval [−𝐿, 𝐿]. Notice

that, for every data set, the maximum temperature is ordered depending on the heat flux profile. This order, from lowest to highest, corresponds to a constant, linear, triangular, and parabolic heat flux profile, respectively. This qualitative behavior of maximum temperature agrees with the heat flux distribution of each profile (see Figure 3), in which maximum temperature is higher where input of heat is also higher. Therefore, the location of maximum temperature agrees with the location of maximum heat flux. Thus, considering the same input data, the ordering of these locations from left to right with respect to the type heat flux of profile is as follows: constant, triangular, linear, and parabolic; that is, the more the heat flux profile is skewed to the leading edge, the nearer the maximum temperature is located in that edge. It is worth noting that the numerical evaluation of maximum temperature by the root searching of (51)–(54) is much faster than applying directly Mathematica’s NMaximize command to the surface temperature in the stationary regime, that is, lim𝜏→∞ T(𝜏, 𝑋, 0). For instance, taking data set 1 as input parameters and a constant heat flux profile, NMaximize command is ≈ 125 times slower than the numerical root finding of (51). Moreover, NMaximize command fails if we take a triangular heat flux profile for all the data sets of Table 1. Figures 4–6 show the surface temperature (𝑍 = 0) in the stationary regime (𝜏 → ∞) for the three data sets given in Table 1, respectively. For the numerical evaluation of these plots, we have used the formulas given in Section 3: (26) for constant heat flux profile, (34) for linear profile, (45) for triangular profile, and (48) for a parabolic one. It is worth noting that the numerical integration of the linear and the parabolic cases is quite efficient if we use the double exponential strategy [28]. The shapes of the graphs are quiet similar to the different data sets of input parameters, except at the leading and trailing edges (𝑥/ℓ = ±1). This is because the heat source velocity V𝑓 in Figure 4 is much faster than in Figures 5 and 6. Also, Figures 4–6 show that the skewness of the temperature curves follows the corresponding heat flux profile.


11 550

350

500 TΔ (t, x, 0) (K)

T(∞, x, 0) (K)

340 330 320

−1.0

−0.5

0.5

1.0

−1.5

−1.0

−0.5

x/󰪓

0.5

1.0

x/󰪓 t (s)

Heat flux profile Constant Linear

400 350

310 −1.5

450


Figure 5: Surface temperature for a carbon steel workpiece (data 2).

1.897 × 10−3 ∞

4.701 × 10−6 3.474 × 10−5 2.567 × 10−4

Figure 7: Evolution of the surface temperature in a titanium alloy VT20 workpiece (data set 1). 450

400

TΔ (t, x, 0) (K)

T(∞, x, 0) (K)

340

350

330 320 310

−1.5

−1.0

−0.5

0.5

1.0

x/󰪓 Heat flux profile Constant Linear

−1.5


t (s)

Table 3: Relaxation times for the different heat flux profiles considered.

Constant profile Linear profile Triangular profile Parabolic profile

Data 1

Data 2

2.00 × 10−1

6.06 × 10

−1

1.11 × 10

1.92 × 10−1

Exact

1.66 × 10−3

1.04 × 10−1

1.82 × 10−1

Approx.

1.08 × 10−4

5.60 × 10−3

1.77 × 10−2

−3

1.90 × 10

−1

1.05 × 10

1.83 × 10−1

Approx.

6.42 × 10−4

1.36 × 10−1

2.19 × 10−1

Exact

1.25 × 10−3

1.05 × 10−1

1.83 × 10−1

Approx.

5.31 × 10−4

2.10 × 10−1

2.79 × 10−1

Approx.

Exact

5.90 × 10

−4

0.5

1.0

2.599 × 10−4 1.92 × 10−3 1.419 × 10−2

1.049 × 10−1 ∞

Figure 8: Evolution of the surface temperature in a carbon steel workpiece (data set 2).

Data 3 −2

Exact

−3

−0.5 x/󰪓

Figure 6: Surface temperature for a sapphire workpiece (data 3).

𝑡 [s]

−1.0

9.89 × 10

Table 3 shows the relaxation times for the different heat flux profiles considered. For this purpose, we have used the relaxation times approximations given in (63), (68), (72), and (75) for the different heat flux profiles, as starting

iteration points for the root searching of (57), (64), (69), and (73), respectively. In general, the approximation formulas provide the order of magnitude of the exact root. Also, the convergence to the exact value is extremely rapid (≈0.01 s). Figures 7–9 show the temperature evolution on the workpiece surface, taking a triangular heat flux profile for the three data sets of Table 1. The black solid line in these figures corresponds to the stationary regime and the gray solid line to the relaxation time (see Table 3). In the case of Figures 8 and 9, the plot corresponding to the relaxation time and the stationary regime are overlapped. For these cases, this is due to the fact that the relaxation time is greater than the time that the heat source needs to cover the friction zone width; that is, 𝑡 > 2ℓ/V𝑓 . The plots are exponentially discretized over time because the transient regime occurs very rapidly. Notice that the first stages of the temperature evolution resemble the

12

Mathematical Problems in Engineering performing the change of variables 𝑢 = 𝑎 + 𝑏𝑥, we get

440

TΔ (t, x, 0) (K)

420 400 380

𝑉𝑚 (𝑎, 𝑏) =

320 −1.0

−0.5

𝑏𝑚+1

∫

𝑎+𝑏ℓ2

2

(𝑢 − 𝑎)𝑚 𝑒−𝑢 𝑑𝑢.

𝑎+𝑏ℓ1

0.5

1.0

𝑚 𝑎+𝑏ℓ2 𝑚 2 ( ) (−𝑎)𝑘 ∫ 𝑢𝑘 𝑒−𝑢 𝑑𝑢. ∑ 𝑚+1 𝑏 𝑎+𝑏ℓ1 𝑘=0 𝑘

1

𝛼

−4

4.541 × 10 3.356 × 10−3 2.48 × 10−2

1.832 × 10 ∞

(A.3)

Now, consider the integral

x/󰪓 t (s)

(A.2)

Taking into account Newton’s formula [19, Eqn. 6:14:1], we can rewrite (A.2) as

360 340

−1.5

1

𝑉𝑚 (𝑎, 𝑏) =

2

𝐼 (𝛼, 𝑘) = ∫ 𝑢𝑘 𝑒−𝑢 𝑑𝑢.

−1

(A.4)

0

When 𝛼 > 0, perform the change of variables 𝑠 = 𝑢2 , so

Figure 9: Evolution of the surface temperature in a sapphire workpiece (data set 3).

triangular heat flux profile, this being more clear for data set 1 in Figure 7.

2

1 1 𝛼 𝑘+1 2 𝐼 (𝛼, 𝑘) = ∫ 𝑠(𝑘−1)/2 𝑒−𝑠 𝑑𝑠 = 𝛾 ( ,𝛼 ), 2 0 2 2 where 𝑧

𝛾 (], 𝑧) = ∫ 𝑡]−1 𝑒−𝑡 𝑑𝑡

7. Conclusions

(A.6)

0

By using the Green function, we have solved the boundaryvalue problem that models the heat transfer in wet surface grinding, considering an arbitrary heat flux profile and a constant heat transfer coefficient on the workpiece surface. We have particularized this result to the most common heat flux profiles reported in the literature, that is, constant, linear, parabolic, and triangular. In constant case (26) and linear case (34), we recover expressions given in the literature. Nonetheless, the triangular case (45) and parabolic case (48) do not seem to be found in the literature. From the analytical expressions found for the surface temperature in the stationary regime for the different heat flux profiles considered, we propose the root searching of some equations in integral form (see (51)–(53)), for the maximum temperature computation. It turns out that this method is very fast (≈1 s) and also much more rapid than the direct numerical maximization of the surface temperature in the stationary regime. Therefore, it provides a very useful aid to prevent thermal damage in the online monitoring grinding process. Finally, we have provided a very rapid method (≈0.01 s) for the computation of the relaxation time for the different heat flux profiles considered. For the initial guess of this root searching, we have given analytical expressions that in general provide the order of magnitude of the relaxation time.

denotes the lower incomplete gamma function [19, Eqn. 45:3:1]. Notice that, performing the change of variables 𝑠 = −𝑢, we have 𝐼 (−𝛼, 𝑘) = ∫

−𝛼

0

𝛼

2

2

𝑢𝑘 𝑒−𝑢 𝑑𝑢 = (−1)𝑘+1 ∫ 𝑠𝑘 𝑒−𝑠 𝑑𝑢 0

𝑘+1

= (−1)

Let us calculate the integral

Therefore, for 𝛼 ∈ R, we have 𝐼 (𝛼, 𝑘) =

sign𝑘+1 (𝛼) 𝑘+1 2 𝛾( ,𝛼 ). 2 2

ℓ1

𝑚 = 0, 1, . . . ;

(A.8)

Applying (A.8) to (A.3) results in 𝑉𝑚 (𝑎, 𝑏) =

1 𝑚 𝑚 󵄨𝑎+𝑏ℓ ∑ ( ) (−𝑎)𝑘 𝐹𝑘 (𝑥)󵄨󵄨󵄨𝑎+𝑏ℓ21 , 2𝑏𝑚+1 𝑘=0 𝑘

(A.9)

where we have defined 𝐹𝑘 (𝑥) = sign𝑘+1 (𝑥) 𝛾 (

𝑘+1 2 ,𝑥 ). 2

(A.10)

According to the formula [19, Sect. 45:4] (A.11)

and knowing that erf(−𝑥) = − erf(𝑥) [19, Eqn. 40:5:1] and 𝑥 = sign(𝑥)|𝑥| [19, Eqn. 8:0:1], for 𝑘 = 0, (A.10) is reduced to

ℓ2

𝑉𝑚 (𝑎, 𝑏) = ∫ 𝑥𝑚 exp (− [𝑎 + 𝑏𝑥]2 ) 𝑑𝑥,

(A.7)

𝐼 (𝛼, 𝑘) .

1 𝛾 ( , 𝑥) = √𝜋 erf (√𝑥) , 𝑥 ≥ 0, 2

Appendix

(A.5)

(A.1)

1 𝐹0 (𝑥) = sign (𝑥) 𝛾 ( , 𝑥2 ) = √𝜋 erf (𝑥) , 2

(A.12)


13

so ℓ2

𝑉0 (𝑎, 𝑏) = ∫ exp (− [𝑎 + 𝑏𝑥]2 ) 𝑑𝑥 ℓ1

√𝜋 𝑎+𝑏ℓ = erf (𝑥)|𝑎+𝑏ℓ2 . 1 2𝑏

(A.13)

For 𝑘 = 1, applying directly [19, Sect. 45:4], we have 𝐹1 (𝑥) = 𝛾 (1, 𝑥) = 1 − 𝑒−𝑥 ,

(A.14)

so ℓ2

𝑉1 (𝑎, 𝑏) = ∫ 𝑥 exp (− [𝑎 + 𝑏𝑥]2 ) 𝑑𝑥 ℓ1

2 𝑎+𝑏ℓ2 −1 . = 2 [√𝜋𝑎 erf (𝑥) + 𝑒−𝑥 ] 𝑎+𝑏ℓ1 2𝑏

(A.15)

Similarly, for 𝑘 = 2, taking into account (A.12) and (A.14) and applying the recursion formula [19, Eqn. 45:5:1] 𝛾 (] + 1, 𝑥) = ]𝛾 (], 𝑥) − 𝑥] 𝑒−𝑥 ,

(A.16)

we arrive at 3 𝐹2 (𝑥) = sign (𝑥) 𝛾 ( , 𝑥2 ) 2 1 1 = sign (𝑥) [ 𝛾 ( , 𝑥2 ) − |𝑥| exp (−𝑥2 )] 2 2 =

(A.17)

√𝜋 erf (𝑥) − 𝑥 exp (−𝑥2 ) , 2

so ℓ2

𝑉2 (𝑎, 𝑏) = ∫ 𝑥2 exp (− [𝑎 + 𝑏𝑥]2 ) 𝑑𝑥 ℓ1

2 𝑎+𝑏ℓ2 1 1 = 3 [√𝜋 (𝑎2 + ) erf (𝑡) + (2𝑎 − 𝑡) 𝑒−𝑡 ] . 2𝑏 2 𝑎+𝑏ℓ1

(A.18)

Competing Interests The author declares that there are no competing interests regarding the publication of this paper.

Acknowledgments The author wishes to thank the financial support received from Generalitat Valenciana under Grant GVA/2015/007.

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