# Efficient Series Expansions of the Temperature Field in Dry Surface Grinding for Usual Heat Flux Profiles

Authors: Juan Luis González-Santander
Journal: Mathematical Problems in Engineering
 (2017)

---

## Abstract

In the framework of Jaeger’s model for heat transfer in dry surface grinding, series expansions for calculating the temperature field, assuming constant, linear, triangular, and parabolic heat flux profiles entering into the workpiece, are derived. The numerical evaluation of these series is considerably faster than the numerical integration of Jaeger’s formula and as accurate as the latter. Also, considering a constant heat flux profile, a numerical procedure is proposed for the computation of the maximum temperature as a function of the Peclet number and the depth below the surface. This numerical procedure has been used to evaluate the accuracy of Takazawa’s approximation.


---

## Body

## 1. Introduction

Straight grinding is a machining process that produces a smooth finish on a flat surface of a workpiece. In this process, there are hard abrasive grits stuck to the peripheral area of the grinding wheel, which perform the cutting when it rotates at high speed, removing the surface layer of the workpiece (see Figure1). Also, the workpiece moves at a certain feed rate vd with respect to the wheel and contacts the latter at the grinding area, which is 2l wide (see Table 1 for the nomenclature used). Jaeger’s model [1, Sect. 10.7.VII] is commonly used to calculate the temperature field in dry grinding. In this model, a two-dimensional approach is considered, in which the coordinate system is fixed to the wheel and centered on the middle of the grinding area, as shown in Figure 1. In surface grinding, the cutting depth a is small, whereby the grinding area is assumed to be flat in Jaeger’s model.Table 1
Nomenclature of symbols.
Symbol
Meaning
SI units
k0
Thermal conductivity
Wm-1K-1
k
Thermal diffusivity
m
2
s
-
1
T
Workpiece temperature
K
T∞
Room temperature
K
vd
Feed rate
m
s
-
1
2l
Contact length
m
q
Average heat flux within contact length
W
m
-
2
a
Cutting depth
mFigure 1
Straight grinding setup.The time-dependent temperature fieldTt,x,y of the workpiece in Jaeger’s model satisfies the convective heat equation [1, §1.7(2)](1)∂T∂t=k∂2T∂x2+∂2T∂y2+vd∂T∂x,subjected to the initial condition,(2)T0,x,y=T∞,and to the boundary condition(3)k0∂T∂yt,x,0=-qfxθl-xθl+x,where θx denotes the Heaviside function and fx is the dimensionless heat flux profile going to the workpiece, which is normalized to unity,(4)12l∫-llfxdx=1.The temperature field in the stationary regime is reached when∂T/∂t=0; thus it does not depend on t, and it is denoted as Tx,y. In the stationary regime, the solution of the above equations (1)–(3) is given by [2, Eqn. 6-1](5)Tx,y=T∞+qπk0∫-llfuexp⁡-vdx-u2kK0vd2kx-u2+y2du,where K0z denotes themodified Bessel function of the second kind of zero order [3, Chap. 51].Figure2 shows the usual heat flux profiles fx considered in the literature: constant [4, 5], linear [6–8], triangular [9, 10], and parabolic [11, 12].Figure 2
Usual heat flux profiles considered in the literature.We can rewrite (5) in dimensionless form, setting the following dimensionless quantities: T=πk0T-T∞/(qs), X=x/s, Y=y/s, and L=l/s (Peclet number), where(6)s=2kvdis a characteristic length. Thereby,(7)TX,Y=∫X-LX+LfsX-ue-uK0u2+Y2du.On the one hand, on the surface, that is,Y=0, (7) is reduced to (8)TX,0=∫X-LX+LfsX-ue-uK0udu, and when fx is an analytic function, closed form expressions for the dimensionless temperature on the surface can be obtained [13]:(9)TX,0=∑n=0∞fnsXn!-snJgnuu=X-LX+L,where(10)Jgnx=exxn+1K0xΨnx+K1xΦnx,x≠00,x=0where(11)Ψnx=1n+11-n2n+1F311,1-n,-1-n12-n12x,Φnx=sgnx2n+1F311,-n,-n-n+1212x,polynomials in 1/x. For instance, for a constant heat flux profile,(12)T0X,0=Jg0uu=-X-L-X+L,and according to (10)-(11), we have(13)Jg0x=xexK0x+signxK1x,x≠00,x=0.It is worth noting that thex-axis given in [13] is just in the opposite direction as the one given in Figure 1. For coherence, all the formulas in this paper are referred to the coordinate system given in Figure 1. From the results given in [13], similar formulas to (12) can be derived for other heat flux profiles normalized to unity (4). For instance, considering a linear heat flux profile, we have(14)T1X,0=12LX+LJg0u+Jg1uu=-X-L-X+L.Also, for a parabolic heat flux profile, we obtain(15)T2X,0=34L2X+L2Jg0u-2X+LJg1u+Jg2uu=-X-L-X+L,and, for a triangular one,(16)T∆X,0=2HLX1-λ-H-LX1+λ+2λλ2-1HλLX,where (17)HΛX=Jg0Λ-X1-XΛ-1ΛJg1Λ-X.On the other hand, considering a constant heat flux profile, (18) is reduced to [14](18)T0X,Y=∫X-LX+Le-uK0u2+Y2du.Takazawa [15] provides the following approximation to (18) for the maximum temperature reached at a given depth Y below the surface,(19)T0,maxY,L≈3.1L0.53exp-0.69L-0.37Y.Expression (19) comes from the numerical evaluation of (18) and a parameter fitting of the maximum temperature. Takazawa uses (19) to estimate how the hardness of the workpiece changes beneath its surfaceThe scope of this paper is two-folded. On the one hand, we derive some series expansions to calculate the integral given in (7) for the heat flux profiles considered above (see Figure 2). It turns out that the numerical evaluation of TX,Y by using these series is considerably faster than the numerical integration of (7). On the other hand, we will provide a numerical procedure to calculate T0,maxY,L that allow us to evaluate the accuracy of Takazawa’s approximation (19).This paper is organized as follows. Section2 provides particular expressions of (7) for the different heat flux profiles considered above, that is, constant, linear, triangular, and parabolic. Section 3 is devoted to the calculation of the Taylor series of Kνu2+y2u2+y2-ν/2, whereby, taking ν=0, we get the main factor of the integrand given in (7). In Section 4, we use the result of the previous section to express TX,Y as a series expansion for the different heat flux profiles considered in this paper. From these series expansions, we can recover the results given in (12) and (14)–(16) for the dimensionless temperature on the surface, that is, TX,0. As an example of this consistency test, we derive the latter for the case of a constant heat flux profile. Section 5 describes a quite efficient numerical procedure to calculate T0,maxY,L. In Section 6, we present some numerical simulations in order to compare the performance of the numerical integration of Jaeger’s formula (7) with the series expansions derived in Section 4 for the different heat flux profiles considered. Also, we evaluate the accuracy of Takazawa’s approximation (19) as a function of Y and L. Finally, we collect our conclusions in Section 7.


## 2. Temperature Field for Usual Heat Flux Profiles

Consider now thatfx is an analytic function within the contact area between the workpiece and the wheel; that is, x∈-l,l,(20)fsX-su=∑m=0∞fmsXm!-sum.Therefore, inserting (20) in (7)(21)TX,Y=∑m=0∞fmsXm!-smJmuu=X-LX+L,where we have set(22)Jmx=∫0xume-uK0u2+Y2du,m=0,1,2,….For a constant heat flux profile,(23)f0x=1, thus (21) is reduced to(24)T0X,Y=J0uu=X-LX+L.Taking into account the normalization condition (4), for a linear heat flux profile, we have (25)f1x=1+xl; hence(26)T1X,Y=1LX+LJ0u-J1uu=X-LX+L.For a triangular heat profile(27)fΔx=21+λ1+xl,x∈-l,lmax21-λ1-xl,x∈lmax,l,where lmax∈-l,l denotes the location of the apex in the triangular profile and λ is the following dimensionless parameter (see Figure 2): (28)λ=lmaxl∈-1,1.Notice that when the heat flux occurs in an arbitrary interval, sayx∈a,b, the dimensionless temperature is given by(29)TX,Y=∫X-BX-AfsX-ue-uK0u2+Y2du,where A=a/s and B=b/s are both dimensionless parameters. Since the heat equation is linear, the temperature field that fΔx generates in the workpiece is given by the superposition of both parts of (27). Therefore, taking into account (29), we have (30)TΔX,Y=21+λ∫X-λLX+L1+X-uLe-uK0u2+Y2du+21-λ∫X-LX-λL1-X-uLe-uK0u2+Y2du.We can rewriteTΔX,Y as follows:(31)TΔX,Y=Tλ,LX,Y-T-λ,-LX,Y,where we have set(32)Tλ,LX,Y=2L1+λX+LJ0u-J1uu=X-λLX+L.Finally, for a parabolic heat flux profile,(33)f2x=341+xl2, whereby(34)T2X,Y=34L2X+L2J0u-2X+LJ1u+J2uu=X-LX+L.


## 3. Taylor Expansion of the Integrand

In order to calculateJmx, which is defined as an integral in (22), consider this more general form of the same integral,(35)Im±ν,y,uu=ab=∫abe±uumKνu2+y2u2+y2ν/2du;thereby(36)Jmuu=ab=Im-0,Y,uu=ab.To expandIm±ν,z,a,b in series, we calculate first the Taylor series of Kνu2+y2u2+y2-ν/2. For this purpose, according to [16, Eqn. 1.14.1(4)], we have this formula for the nth derivative(37)Dnz-ν/2Kνz=-12n1zn+νKn+νz.Since the Taylor expansion of an analytic function is(38)fx+h=∑n=0∞fnhn!xn=∑n=0∞fnxn!hn, according to (37), we have(39)Kνx+hx+hν/2=1hν/2∑n=0∞1n!-12hnKn+νhxn(40)=1xν/2∑n=0∞1n!-12xnKn+νxhn.To calculate the radius of convergence of (39), let us set (41)bn=1n!-12hnKn+νhxn, so, applying the ratio test, we have to determinate when R<1, where(42)R=limn→∞bn+1bn=x2hlimn→∞1n+1Kn+ν+1xKn+νx.By using the asymptotic formula [17, Eqn. 10.41.2],(43)Kμz≈π2μez2μ-μ,μ⟶∞,we calculate (42) as(44)R=xhe-1limn→∞n+ν+1n+νn+11+1n+νn+ν=xh,where we have taken the definition of number e=limn→∞⁡1+1/nn [18, Eqn. 1.2]. Therefore, (39) converges absolutely when x<h. Notice that we can convert (39) into (40) exchanging x and h, so (40) converges absolutely when x>h.Finally, takingh=u2 and x=y2 in (39) and (40), bearing in mind what we have said about the radius of convergence, we arrive at(45)Kνu2+y2u2+y2ν/2=1uν∑n=0∞1n!-y22unKn+νu,y<u1yν∑n=0∞1n!-u22ynKn+νy,y>u.The casey=u is not essential for our purpose, since we are going to insert in (35) the expansions given in (45) and then integrate term by term.


## 4. Series Expansion of the Temperature Field

In order to insert (45) in (35) and integrate term by term, let us define the following functions:(46)Im,upper±ν,y,uu=ab=∑n=0∞-y2/2nn!∫abe±uumKn+νuun+νdu,u>y,(47)Im,lower±ν,y,uu=ab=1yν∑n=0∞-2y-nn!Kn+νy∫abe±uu2n+mdu,u<y.We consider hereaftera<b. Also, according to Figure 1, y≥0; thus, we have dropped the absolute value for y in (46) and (47).


### 4.1. Calculation ofIm,upper±

For the calculation of the integral given in (46), define the following function:(48)hm±α,uu=ab=∫abe±uumKαuuαdu,so that, we can rewrite (46) as(49)Im,upper±ν,y,uu=ab=∑n=0∞-y2/2nn!hm±n+ν,uu=ab.First, note that, performing the change of variablesu→-u on the LHS of (48), we obtain (50)hm±α,uu=ab=-1m+1∫-a-be∓uumKαuuαdu=-1m+1hm∓α,uu=-a-b; thus(51)Im,upper±ν,y,uu=ab=-1m+1Im,upper∓ν,y,uu=-a-b.Also, whena,b>0, (52)hm±α,uu=ab=∫abe±uum-αKαudu=∫abe±uum+βKβudu,where we have set β=-α and applied the property Kνz=K-νz [19, Eqn. 5.7.10]. For the calculation of hm±α,u, we found in the literature [13] (53)∫0ze±xxν+nKνxdx=e±zzν+n+1×Kνz2ν+n+11-n2ν+2n+1F311,1-n,-1-n-2ν12-n-ν±12z±K1+νz2ν+2n+1F311,-n,-n-2ν-n+12-ν±12z+∓1n+12ν+nn!Γν+n+1Γ2ν+n+1Γ2ν+2n+2.Therefore,(54)hm±α,uu=ab=e±uu1+m-αKαu1+m-2α1-mPmαu1+2m-α±Kα-1uQmαu1+2m-αu=ab,a,b>0,where we have defined the following polynomials in 1/u:(55)Pmαu=F311,1-m,2α-m-112-m+α±12u=∑s=0∞1-ms2α-m-1s1/2-m+αs±12us,Qmαu=F311,-m,2α-m12-m+α±12u=∑s=0∞-ms2α-ms1/2-m+αs±12us.Note that the degree of polynomialPmα is m+1 and of polynomial Qmα is m. According to Section 2, we need to calculate Jm for m=0,1,2; thus we are going to derive particular expressions of Im,upper± for ν=0 and m=0,1,2, when a,b>0. Thereby, taking ν=0, (49) is reduced to(56)Im,upper±0,y,uu=ab=∑n=0∞-y2/2nn!hm±n,uu=ab.In the casen=0, according to (54)-(55), we have(57)h0±n,uu=ab=e±uu1-nKnu±Kn-1u1-2nu=ab,a,b>0.Similarly, form=1, (54) is reduced to(58)h1±n,uu=ab=e±uu1-nKnuu-Kn-1u1∓u3-2nu=ab,a,b>0.Finally, form=2, we arrive at(59)h2±n,uu=ab=e±uu1-n×Knu3-2nu±2u±Kn-1u3-2nu2+41-n1∓u3-2n5-2nu=ab,a,b>0.


### 4.2. Calculation ofIm,lower±

For the calculation of the integral given in (47), we apply the definition of thelower incomplete gamma function [3, Eqn. 45:3:1]: (60)γν,x=∫0xtν-1e-tdt, so that, straightforwardly, we obtain(61)∫abe-uu2n+mdu=γ2n+m+1,uu=ab,and, performing the change of variables u→-u, we have(62)∫abeuu2n+mdu=-1m+1γ2n+m+1,uu=-a-b.Taking into account (61) and (62), we rewrite (47) as (63)Im,lower±ν,y,uu=ab=∓1m+1yν∑n=0∞-2y-nn!Kn+νyγ2n+m+1,uu=∓a∓b, which, for ν=0, is reduced to(64)Im,lower±0,y,uu=ab=∓1m+1∑n=0∞-2y-nn!Knyγ2n+m+1,uu=∓a∓b,maxa,b<y.


### 4.3. Calculation ofIm±

Applying the property given in (51), we have six different cases for the calculation of Im± in terms of Im,upper± and Im,lower±. For the sake of clarity, here we use the following simplified notation: Im±ab=Im±ν,y,uu=ab.(65)Im±ab=Im,upper±ab,a>y-1m+1Im,upper∓-a-b,b<-yIm,lower±ay+Im,upper±yb,b>y,a<y-1m+1Im,upper∓-a-y+Im,lower±-yy+Im,upper±yb,a<-y,b>yIm,lower±ab,maxa,b<y-1m+1Im,upper∓-a-y+Im,lower±-yb,a<-y,b<ythat can be reduced to only three cases,(66)Im±ab=Im,upper±ab,a>y-1m+1Im,upper∓-a-b,b<-y-1m+1Im,upper∓min-a,-y-y+Im,lower±maxa,-yminb,y+Im,upper±ymaxy,b,a<y,b>-y.Recalling thaty≥0, note that in (65) the integration limits of Im,upper are always both positive, so that we can use (56) in combination with (57), (58), or (59), for its calculation.


### 4.4. Calculation ofTX,Y

Now, we collect the previous results in order to calculate the dimensionless temperature fieldTX,Y for the different heat flux profiles considered in the Introduction. First, consider a constant heat flux profile; thereby, according to (24) and (36), we have(67)T0X,Y=I0-0,Y,uu=X-LX+L.Notice that, takingm=0 in (51), we have(68)I0,upper±ν,y,uu=ab=-I0,upper∓ν,y,uu=-a-b.Also, from (56) and (57), we obtain the following series for positive integration limits a,b:(69)I0,upper±0,y,uu=ab=∑n=0∞-y2/2nn!e±uu1-nKnu±Kn-1u1-2nu=ab,a,b>0.According to (68), we can extend (69) for negative values of the integration limits a,b, defining the following function:(70)Iupper0Y,uu=ab=I0,upper-0,Y,uu=ab=sgna∑n=0∞-Y2/2nn!e-sgnauu1-nKnu-sgnaKn-1u1-2nu=ab,ab>0.Similarly, from (64), define(71)Ilower0Y,uu=ab=I0,lower-0,Y,uu=ab=∑n=0∞KnY-2Ynn!γ2n+1,uu=ab.For a linear heat flux profile, according to (26) and (36), we have(72)T1X,Y=1LX+LI0-0,Y,u-I1-0,Y,uu=X-LX+L,and, applying (51), we have(73)1LX+LI0,upper±0,Y,u-I1,upper±0,Y,uu=ab=1L-X+LI0,upper∓0,Y,u-I1,upper∓0,Y,uu=ab.Now, taking into account (56), (57) and (58), we can rewrite (73) as(74)1LX+LI0,upper±0,Y,u-I1,upper±0,Y,uu=ab=1L∑n=0∞-y2/2nn!e±uu1-n×KnuX+L1-2n-u3-2n+Kn-1u1∓u3-2n±X+L1-2nu=ab,a,b>0.Again, according to (73), we can extend (74) for negative values of the integration limits a,b, defining(75)Iupper1X,Y,uu=ab=1LX+LI0,upper-0,Y,u-I1,upper-0,Y,uu=ab=1L∑n=0∞-Y2/2nn!e-sgnauu1-nKnusgnaX+L1-2n-u3-2n+Kn-1u1+sgnau3-2n-X+L1-2nu=ab,ab>0.Similarly, from (64), define(76)Ilower1Y,uu=ab=1LX+LI0,lower-0,Y,u-I1,lower-0,Y,uu=ab=1L∑n=0∞-2Y-nn!KnYX+Lγ2n+1,u-γ2n+2,uu=ab.By using the recursion formula [3, Eqn 45:5:1](77)γν+1,x=νγν,x-xνe-x,we reduce (76) to(78)Ilower1Y,uu=ab=1L∑n=0∞-2Y-nn!KnYX+L-2n-1γ2n+1,u+u2n+1e-uu=ab.For the parabolic case, we can follow similar steps as in the linear case, arriving at(79)Iupper2X,Y,uu=ab=34L2∑n=0∞-Y2/2nn!e-sgnauu1-nKnusgnaX+L21-2n-2u3-2n15-2n+X+L+sgnau25-2n+Kn-1u-X+L21-2n+21+sgnau3-2nX+L-21-n5-2n-u25-2nu=ab,ab>0,(80)Ilower2Y,uu=ab=34L2∑n=0∞KnY-2Ynn!γ2n+1,uX+L2+22n+1n+1-X-L+2X+L-n-1-uu2n+1e-uu=ab.Finally, taking into account (66) and using the above results for Iupperm and Ilowerm, the dimensionless temperature field TmX,Y for m=0,1,2, is calculated as follows:(81)TmX,Y=ImX,Y,uu=X-LX+L,where(82)Imab=Iuppermab,a>Yorb<-YIuppermmina,-Y-Y+Ilowermmaxa,-Yminb,Y+IuppermYmaxY,b,a<Y,b>-Y.Therefore,(83)TmX,Y=IuppermX-LX+L,X>Y+LIuppermminX-L,-Y-Y+IlowermmaxX-L,-YminX+L,Y+IuppermYmaxY,X+L,X<Y+L,For the case of a triangular heat flux profile, according to (32) and (36) and taking into account (72) and (81), we have(84)Tλ,LX,Y=2L1+λX+LI0-0,Y,u-I1-0,Y,uu=X-λLX+L=21+λI1X,Y,uu=X-λLX+L.It is worth noting thatI1ab is calculated properly with the formulas given for Iupper1 and Ilower1 in (75) and (78) when a<b. Therefore, care has to be taken when we compute the dimensionless temperature for the triangular case as TΔX,Y=Tλ,LX,Y-T-λ,-LX,Y, as stated in (31).Notice also that, in order to calculate the dimensionless temperatureTX,Y, we need to compute the series given in (70), (75), and (79) for Iupperm and (71), (78), and (80) for Ilowerm. These series are all alternating series, which converges slowly when u≈Y. However, we can accelerate the convergence of these alternating series by using Cohen-Villegas-Zagier algorithm [20]. This algorithm approximates an alternating series S=∑k=0∞-1kak as a weighted sum of the first n values of ak by using a “Padé type approximation,” as long as ak is a reasonable well-behaved sequence. We will see in Section 6 that we need a small number of terms n in order to get a good accuracy.


### 4.5. Temperature on the Surface

As mentioned before in the Introduction,TX,0 can be calculated in closed form for the heat flux profiles considered in this paper. All these results can be obtained from the results of Section 4 taking Y=0. As a consistency test, we are going to derive here the expression given in (12) for a constant heat flux profile. For this purpose, note that when y=0, the series given for I0,lower±, that is, (47), vanishes; thus, according to (35), we have (85)I0±ν,0,uu=ab=I0,upper±ν,0,uu=ab.Now, takingy=0 in (69) and applying the property Kνz=K-νz [19, Eqn. 5.7.10], (85) is reduced to (86)I0±0,0,uu=ab=ue±uK0u±K1uu=ab,a,b>0.According to the limiting forms [17, Eqn. 10.3.2-3], (87)Kνz≈12Γνz2-ν,z⟶0,Reν>0,K0z≈-log⁡z,z⟶0, we have(88)limu→0+ue±uK0u±K1u=±1,so that(89)I0±0,0,uu=0z=ze±zK0z±K1z∓1,z>0,(90)limz→0+I0±0,0,uu=0z=0.Now, according to (85), apply (51), exchange the integration limits, and take into account (89), to arrive at (91)I0±0,0,uu=0-z=-I0∓0,0,uu=0z=-ze∓zK0z∓K1z∓1,z>0; thus, performing the change z→-z,(92)I0±0,0,uu=0z=ze±zK0-z∓K1-z∓1,z<0.Collecting the results (89), (90), and (92), we arrive at(93)I0±0,0,uu=0z=ze±zK0z±sgnzK1z∓1,z≠00,z=0.Finally, according to (67) and applying again (51), the dimensionless temperature on the surface is (94)T0X,0=I0-0,0,uu=X-LX+L=I0+0,0,uu=-X-L-X+L, which is equivalent to (12), taking into account (93). Therefore, for the computation of the temperature field with the series expansions given above, we will use directly the closed form formulas given in (12) and (14)–(16).


## 4.1. Calculation ofIm,upper±

For the calculation of the integral given in (46), define the following function:(48)hm±α,uu=ab=∫abe±uumKαuuαdu,so that, we can rewrite (46) as(49)Im,upper±ν,y,uu=ab=∑n=0∞-y2/2nn!hm±n+ν,uu=ab.First, note that, performing the change of variablesu→-u on the LHS of (48), we obtain (50)hm±α,uu=ab=-1m+1∫-a-be∓uumKαuuαdu=-1m+1hm∓α,uu=-a-b; thus(51)Im,upper±ν,y,uu=ab=-1m+1Im,upper∓ν,y,uu=-a-b.Also, whena,b>0, (52)hm±α,uu=ab=∫abe±uum-αKαudu=∫abe±uum+βKβudu,where we have set β=-α and applied the property Kνz=K-νz [19, Eqn. 5.7.10]. For the calculation of hm±α,u, we found in the literature [13] (53)∫0ze±xxν+nKνxdx=e±zzν+n+1×Kνz2ν+n+11-n2ν+2n+1F311,1-n,-1-n-2ν12-n-ν±12z±K1+νz2ν+2n+1F311,-n,-n-2ν-n+12-ν±12z+∓1n+12ν+nn!Γν+n+1Γ2ν+n+1Γ2ν+2n+2.Therefore,(54)hm±α,uu=ab=e±uu1+m-αKαu1+m-2α1-mPmαu1+2m-α±Kα-1uQmαu1+2m-αu=ab,a,b>0,where we have defined the following polynomials in 1/u:(55)Pmαu=F311,1-m,2α-m-112-m+α±12u=∑s=0∞1-ms2α-m-1s1/2-m+αs±12us,Qmαu=F311,-m,2α-m12-m+α±12u=∑s=0∞-ms2α-ms1/2-m+αs±12us.Note that the degree of polynomialPmα is m+1 and of polynomial Qmα is m. According to Section 2, we need to calculate Jm for m=0,1,2; thus we are going to derive particular expressions of Im,upper± for ν=0 and m=0,1,2, when a,b>0. Thereby, taking ν=0, (49) is reduced to(56)Im,upper±0,y,uu=ab=∑n=0∞-y2/2nn!hm±n,uu=ab.In the casen=0, according to (54)-(55), we have(57)h0±n,uu=ab=e±uu1-nKnu±Kn-1u1-2nu=ab,a,b>0.Similarly, form=1, (54) is reduced to(58)h1±n,uu=ab=e±uu1-nKnuu-Kn-1u1∓u3-2nu=ab,a,b>0.Finally, form=2, we arrive at(59)h2±n,uu=ab=e±uu1-n×Knu3-2nu±2u±Kn-1u3-2nu2+41-n1∓u3-2n5-2nu=ab,a,b>0.


## 4.2. Calculation ofIm,lower±

For the calculation of the integral given in (47), we apply the definition of thelower incomplete gamma function [3, Eqn. 45:3:1]: (60)γν,x=∫0xtν-1e-tdt, so that, straightforwardly, we obtain(61)∫abe-uu2n+mdu=γ2n+m+1,uu=ab,and, performing the change of variables u→-u, we have(62)∫abeuu2n+mdu=-1m+1γ2n+m+1,uu=-a-b.Taking into account (61) and (62), we rewrite (47) as (63)Im,lower±ν,y,uu=ab=∓1m+1yν∑n=0∞-2y-nn!Kn+νyγ2n+m+1,uu=∓a∓b, which, for ν=0, is reduced to(64)Im,lower±0,y,uu=ab=∓1m+1∑n=0∞-2y-nn!Knyγ2n+m+1,uu=∓a∓b,maxa,b<y.


## 4.3. Calculation ofIm±

Applying the property given in (51), we have six different cases for the calculation of Im± in terms of Im,upper± and Im,lower±. For the sake of clarity, here we use the following simplified notation: Im±ab=Im±ν,y,uu=ab.(65)Im±ab=Im,upper±ab,a>y-1m+1Im,upper∓-a-b,b<-yIm,lower±ay+Im,upper±yb,b>y,a<y-1m+1Im,upper∓-a-y+Im,lower±-yy+Im,upper±yb,a<-y,b>yIm,lower±ab,maxa,b<y-1m+1Im,upper∓-a-y+Im,lower±-yb,a<-y,b<ythat can be reduced to only three cases,(66)Im±ab=Im,upper±ab,a>y-1m+1Im,upper∓-a-b,b<-y-1m+1Im,upper∓min-a,-y-y+Im,lower±maxa,-yminb,y+Im,upper±ymaxy,b,a<y,b>-y.Recalling thaty≥0, note that in (65) the integration limits of Im,upper are always both positive, so that we can use (56) in combination with (57), (58), or (59), for its calculation.


## 4.4. Calculation ofTX,Y

Now, we collect the previous results in order to calculate the dimensionless temperature fieldTX,Y for the different heat flux profiles considered in the Introduction. First, consider a constant heat flux profile; thereby, according to (24) and (36), we have(67)T0X,Y=I0-0,Y,uu=X-LX+L.Notice that, takingm=0 in (51), we have(68)I0,upper±ν,y,uu=ab=-I0,upper∓ν,y,uu=-a-b.Also, from (56) and (57), we obtain the following series for positive integration limits a,b:(69)I0,upper±0,y,uu=ab=∑n=0∞-y2/2nn!e±uu1-nKnu±Kn-1u1-2nu=ab,a,b>0.According to (68), we can extend (69) for negative values of the integration limits a,b, defining the following function:(70)Iupper0Y,uu=ab=I0,upper-0,Y,uu=ab=sgna∑n=0∞-Y2/2nn!e-sgnauu1-nKnu-sgnaKn-1u1-2nu=ab,ab>0.Similarly, from (64), define(71)Ilower0Y,uu=ab=I0,lower-0,Y,uu=ab=∑n=0∞KnY-2Ynn!γ2n+1,uu=ab.For a linear heat flux profile, according to (26) and (36), we have(72)T1X,Y=1LX+LI0-0,Y,u-I1-0,Y,uu=X-LX+L,and, applying (51), we have(73)1LX+LI0,upper±0,Y,u-I1,upper±0,Y,uu=ab=1L-X+LI0,upper∓0,Y,u-I1,upper∓0,Y,uu=ab.Now, taking into account (56), (57) and (58), we can rewrite (73) as(74)1LX+LI0,upper±0,Y,u-I1,upper±0,Y,uu=ab=1L∑n=0∞-y2/2nn!e±uu1-n×KnuX+L1-2n-u3-2n+Kn-1u1∓u3-2n±X+L1-2nu=ab,a,b>0.Again, according to (73), we can extend (74) for negative values of the integration limits a,b, defining(75)Iupper1X,Y,uu=ab=1LX+LI0,upper-0,Y,u-I1,upper-0,Y,uu=ab=1L∑n=0∞-Y2/2nn!e-sgnauu1-nKnusgnaX+L1-2n-u3-2n+Kn-1u1+sgnau3-2n-X+L1-2nu=ab,ab>0.Similarly, from (64), define(76)Ilower1Y,uu=ab=1LX+LI0,lower-0,Y,u-I1,lower-0,Y,uu=ab=1L∑n=0∞-2Y-nn!KnYX+Lγ2n+1,u-γ2n+2,uu=ab.By using the recursion formula [3, Eqn 45:5:1](77)γν+1,x=νγν,x-xνe-x,we reduce (76) to(78)Ilower1Y,uu=ab=1L∑n=0∞-2Y-nn!KnYX+L-2n-1γ2n+1,u+u2n+1e-uu=ab.For the parabolic case, we can follow similar steps as in the linear case, arriving at(79)Iupper2X,Y,uu=ab=34L2∑n=0∞-Y2/2nn!e-sgnauu1-nKnusgnaX+L21-2n-2u3-2n15-2n+X+L+sgnau25-2n+Kn-1u-X+L21-2n+21+sgnau3-2nX+L-21-n5-2n-u25-2nu=ab,ab>0,(80)Ilower2Y,uu=ab=34L2∑n=0∞KnY-2Ynn!γ2n+1,uX+L2+22n+1n+1-X-L+2X+L-n-1-uu2n+1e-uu=ab.Finally, taking into account (66) and using the above results for Iupperm and Ilowerm, the dimensionless temperature field TmX,Y for m=0,1,2, is calculated as follows:(81)TmX,Y=ImX,Y,uu=X-LX+L,where(82)Imab=Iuppermab,a>Yorb<-YIuppermmina,-Y-Y+Ilowermmaxa,-Yminb,Y+IuppermYmaxY,b,a<Y,b>-Y.Therefore,(83)TmX,Y=IuppermX-LX+L,X>Y+LIuppermminX-L,-Y-Y+IlowermmaxX-L,-YminX+L,Y+IuppermYmaxY,X+L,X<Y+L,For the case of a triangular heat flux profile, according to (32) and (36) and taking into account (72) and (81), we have(84)Tλ,LX,Y=2L1+λX+LI0-0,Y,u-I1-0,Y,uu=X-λLX+L=21+λI1X,Y,uu=X-λLX+L.It is worth noting thatI1ab is calculated properly with the formulas given for Iupper1 and Ilower1 in (75) and (78) when a<b. Therefore, care has to be taken when we compute the dimensionless temperature for the triangular case as TΔX,Y=Tλ,LX,Y-T-λ,-LX,Y, as stated in (31).Notice also that, in order to calculate the dimensionless temperatureTX,Y, we need to compute the series given in (70), (75), and (79) for Iupperm and (71), (78), and (80) for Ilowerm. These series are all alternating series, which converges slowly when u≈Y. However, we can accelerate the convergence of these alternating series by using Cohen-Villegas-Zagier algorithm [20]. This algorithm approximates an alternating series S=∑k=0∞-1kak as a weighted sum of the first n values of ak by using a “Padé type approximation,” as long as ak is a reasonable well-behaved sequence. We will see in Section 6 that we need a small number of terms n in order to get a good accuracy.


## 4.5. Temperature on the Surface

As mentioned before in the Introduction,TX,0 can be calculated in closed form for the heat flux profiles considered in this paper. All these results can be obtained from the results of Section 4 taking Y=0. As a consistency test, we are going to derive here the expression given in (12) for a constant heat flux profile. For this purpose, note that when y=0, the series given for I0,lower±, that is, (47), vanishes; thus, according to (35), we have (85)I0±ν,0,uu=ab=I0,upper±ν,0,uu=ab.Now, takingy=0 in (69) and applying the property Kνz=K-νz [19, Eqn. 5.7.10], (85) is reduced to (86)I0±0,0,uu=ab=ue±uK0u±K1uu=ab,a,b>0.According to the limiting forms [17, Eqn. 10.3.2-3], (87)Kνz≈12Γνz2-ν,z⟶0,Reν>0,K0z≈-log⁡z,z⟶0, we have(88)limu→0+ue±uK0u±K1u=±1,so that(89)I0±0,0,uu=0z=ze±zK0z±K1z∓1,z>0,(90)limz→0+I0±0,0,uu=0z=0.Now, according to (85), apply (51), exchange the integration limits, and take into account (89), to arrive at (91)I0±0,0,uu=0-z=-I0∓0,0,uu=0z=-ze∓zK0z∓K1z∓1,z>0; thus, performing the change z→-z,(92)I0±0,0,uu=0z=ze±zK0-z∓K1-z∓1,z<0.Collecting the results (89), (90), and (92), we arrive at(93)I0±0,0,uu=0z=ze±zK0z±sgnzK1z∓1,z≠00,z=0.Finally, according to (67) and applying again (51), the dimensionless temperature on the surface is (94)T0X,0=I0-0,0,uu=X-LX+L=I0+0,0,uu=-X-L-X+L, which is equivalent to (12), taking into account (93). Therefore, for the computation of the temperature field with the series expansions given above, we will use directly the closed form formulas given in (12) and (14)–(16).


## 5. Maximum Temperature Beneath the Surface

As aforementioned in the Introduction, we present in this section a numerical method to compute the maximum temperature as a function of the Peclet numberL and the dimensionless depth Y below the surface, that is, T0,maxY,L. For this purpose, note that the integrand of (18) is positive, since ∀z>0, K0z>0 [19, Note 38]. Therefore,(95)T0X,Y=∫X-LX+Le-uK0u2+Y2du>0,X∈R.Also, directly from (95), we have(96)limX→±∞T0X,Y=0.The behavior ofT0X,Y given in (95) and (96) is shown graphically in Figure 3. Moreover, for each Y, T0X,Y exhibits and unique extreme value that matches the maximum value. Therefore, we can search for the maximum temperature at a given depth Y looking for the extreme value of T0X,Y as a function of X, solving for Xmax the equation(97)∂T0Xmax,Y∂X=0.Figure 3
T
0
(
X
,
Y
) for different dimensionless depths Y and Peclet number L=3.By using Leibniz’s theorem for differentiation of integrals [17, Eqn. 1.5.22] (98)ddx∫αxβxfx,ydy=fx,βxβ′x-fx,αxα′x+∫αxβx∂f∂xdy. Equation (97) reads as (99)e-LK0Xmax+L2+Y2=eLK0Xmax-L2+Y2, which has to be solved numerically. Taking as initial iteration point Xmax0=0, Newton’s method for root searching converges quite rapidly. Notice that Xmax=XmaxY,L; thus the maximum temperature T0,max is calculated as a function of the dimensionless depth Y and the Peclet number L,(100)T0,maxY,L=T0XmaxY,L,Y.In Section6, we compare this numerical approach with Takazawa’s approximation (19).


## 6. Numerical Results

In this section, we compare the performance of the numerical evaluation of the temperature field by using the integral form (7) with the series expansion derived in Section 4, namely, (83) and (84). Also, we analyze the accuracy of Takazawa’s approximation for the maximum temperature (19), by using the numerical method derived in Section 5. For both purposes, we need to define first the characteristic dimensionless depth of the temperature field in dry grinding.


### 6.1. Characteristic Dimensionless Depth

According to the one-dimensional approximation, the depth of thermal penetration for dry grinding is given by(101)δp=2g-1pk2lvd,where the function g-1p is the inverse function of the following function:(102)gx=e-x2-πxerfcx,and p is the percentage at which the temperature falls at depth δp with respect to the surface temperature (i.e., at δ0.1 below the surface, the temperature is 10% of the surface temperature). Equation (101) provides a very good approximation to the two-dimensional depth of thermal penetration for moderate values of p, that is, 0.3≲p≤1, and the heat flux profiles presented in Figure 2. This depth of thermal penetration δp is useful to set the characteristic domain of the temperature field, and thereby to compare the performance of the integral form with the series expansion of the temperature field.From (101), we define the characteristic dimensionless depth as (103)YpL=δps=2g-1pL,where we remember that s is the characteristic length given in (6). This characteristic dimensionless depth YpL is very useful to analyze the accuracy of Takazawa’s approximation for the maximum temperature (19).


### 6.2. Temperature Field Computation

Table2 shows three sets of parameters (in SI units) for the numerical simulations. Data set 1 considers a carbon steel workpiece AISI 1020 [21] and data set 2 considers aluminum oxide, Al2O3 (sapphire), as workpiece material [22]. Also, data set 3 considers a titanium alloy VT20 workpiece, whose thermal properties are given in [23]. The grinding regime for this simulation can be found in [24]. These simulation parameters have been chosen in order to cover a wide range of Peclet numbers L. The characteristic dimensionless depth Y0.1L of the temperature field is given as well for the three data sets.Table 2
Simulation parameters in SI units.
Data 1
Data 2
Data 3
Workpiece
Material
Sapphire
Steel
Titanium
k
0
46
61
13
k
1.51
×
1
0
-
5
1.78
×
1
0
-
5
4.23
×
1
0
-
6
Grinding regime
2
l
1.4
×
1
0
-
3
2.5
×
1
0
-
3
2.66
×
1
0
-
3
q
1.4
×
1
0
7
3.5
×
1
0
7
5.89
×
1
0
7
v
d
3.3
×
1
0
-
2
0.15
0.53
T
∞
300
300
300
Dimensionless parameters
L
0.76
5.27
83.4
Y
0.1
L
1.68
4.42
17.6Figure4 shows the temperature field for data set 1, where we have considered as plot area(104)D=-1.5l,1.5l×0,δ0.1.Figure 4
Temperature field for data set 1 and a constant heat flux profile.In order to compare the performance of the numerical integration of (7) and the numerical evaluation of the series expansions given in Section 4 for the calculation of the temperature field, we have set the time ratio τ as the ratio between the computation time of the numerical integration tintegration and the series expansion tseries, (105)τ=tintegrationtseries.t
i
n
t
e
g
r
a
t
i
o
n
≈
10
–
2
5
s for plotting the temperature field (i.e., Figure 4) running an 8 kernel i7 CPU computer with MATHEMATICA™ on Linux platform. However, this time could be increased by a factor of 2–5 times in the numerical control of the grinding machines; thus a time ratio τ≥2 would be worthwhile.Also, to compare both temperature fields, we have considered the following distance between two nonnegative scalar fieldsf and g, integrable in a certain connected open set D⊆Rn [25](106)ΔDf,g=∫Dfx→-gx→dV∫Dfx→+gx→dV,forg≠00,f=g=0.This distanceΔD is applicable to the temperature field since, according to (95); the latter is a positive quantity. In [25], it is proved that 0≤ΔDf,g≤1, wherein ΔDf,g=0 means that f and g are overlapped within the integration domain D, and ΔDf,g=1 means that both functions are relatively infinitely far one from each other. Table 3 shows the distances ΔD between the temperature fields computed by the numerical integration of (7) and the numerical evaluation of the corresponding series (83) and (84), taking as integration domain (104).Table 3
Relative distancesΔD and time ratios τ between the integration method and the series method, where n denotes the number of terms taken in the series approximation.
Heat flux
Data 1
Data 2
Data 3
Constant
Δ
D
2.34
×
1
0
-
8
4.26
×
1
0
-
7
2.86
×
1
0
-
5
τ
7.43
7.47
7.45
n
8
8
10
Linear
Δ
D
3.41
×
1
0
-
8
7.57
×
1
0
-
7
2.56
×
1
0
-
5
τ
6.40
6.15
5.45
n
8
8
10
Triangular
Δ
D
2.34
×
1
0
-
8
7.01
×
1
0
-
7
2.56
×
1
0
-
5
τ
17.16
17.10
11.73
n
8
8
10
Parabolic
Δ
D
4.26
×
1
0
-
8
1.07
×
1
0
-
6
2.33
×
1
0
-
5
τ
5.06
5.50
1.63
n
8
8
10To evaluate numerically the alternating seriesS=∑k=0∞-1kak given in (70), (75), and (79) for Iupperm and (71), (78), and (80) for Ilowerm, we have taken n=8 terms in the Cohen-Villegas-Zagier algorithm [20] for data sets 1 and 2 and n=10 for data set 3. In particular, we have used the 2A algorithm, which consists first of computing and storing in memory the polynomial (107)Ansin2t=12nn!Dnsinn2t, and then we apply the following procedure.Algorithm 1 (2A).
LetAnx=∑k=0nbkxk. Set d=-An-1; c=-d; s=0. For k=0 up to k=n-1, repeat: c=bk-c; s=s+cak. Output: s/d.It is worth noting that there is a typo in the algorithm published in [20], where d is given as An-1.Table3 shows the time ratios for the three data sets. We can see that the computation of the corresponding series expansions of Section 4 is considerably faster than the numerical integration. This is especially true for a triangular heat flux profile, for which we have set λ=0.5 (similar results are found for different values of λ). Also, both methods provide very similar temperature fields, since ΔD≪1. This is especially true for low Peclet numbers.


### 6.3. Takazawa’s Approximation Analysis

We can compare the exact value given in (100) with the approximation given by Takazawa in (19). Figure 5 shows this comparison for different values of the Peclet number. We can appreciate that, regardless of the value of the Peclet number, Takazawa’s approximation deviates from the exact value for large Y.Figure 5
T
0
,
m
a
x
Y for intermediate Peclet numbers.In order to measure the accuracy of Takazawa’s approximation (19), we can compute the relative distance ΔD of (19) with respect to the numerical value (100) as a function of the Peclet number L, taking as integration domain D=0,Y0.1L. Figure 6 presents this computation, showing that the accuracy of Takazawa’s approximation is in general poor, especially for low Peclet numbers. Therefore, despite the fact Takazawa’s formula being extremely easy to compute, probably it is better to use the numerical method given in (100), since the numerical evaluation of XmaxY,L is very rapid, and the computation of TX,Y is quite fast by using the series expansion method described in Section 4. For instance, Figure 5 is plotted in ≈0.87 s by using MATHEMATICA and an Intel i7 CPU.Figure 6
Accuracy of Takazawa’s approximation.


## 6.1. Characteristic Dimensionless Depth

According to the one-dimensional approximation, the depth of thermal penetration for dry grinding is given by(101)δp=2g-1pk2lvd,where the function g-1p is the inverse function of the following function:(102)gx=e-x2-πxerfcx,and p is the percentage at which the temperature falls at depth δp with respect to the surface temperature (i.e., at δ0.1 below the surface, the temperature is 10% of the surface temperature). Equation (101) provides a very good approximation to the two-dimensional depth of thermal penetration for moderate values of p, that is, 0.3≲p≤1, and the heat flux profiles presented in Figure 2. This depth of thermal penetration δp is useful to set the characteristic domain of the temperature field, and thereby to compare the performance of the integral form with the series expansion of the temperature field.From (101), we define the characteristic dimensionless depth as (103)YpL=δps=2g-1pL,where we remember that s is the characteristic length given in (6). This characteristic dimensionless depth YpL is very useful to analyze the accuracy of Takazawa’s approximation for the maximum temperature (19).


## 6.2. Temperature Field Computation

Table2 shows three sets of parameters (in SI units) for the numerical simulations. Data set 1 considers a carbon steel workpiece AISI 1020 [21] and data set 2 considers aluminum oxide, Al2O3 (sapphire), as workpiece material [22]. Also, data set 3 considers a titanium alloy VT20 workpiece, whose thermal properties are given in [23]. The grinding regime for this simulation can be found in [24]. These simulation parameters have been chosen in order to cover a wide range of Peclet numbers L. The characteristic dimensionless depth Y0.1L of the temperature field is given as well for the three data sets.Table 2
Simulation parameters in SI units.
Data 1
Data 2
Data 3
Workpiece
Material
Sapphire
Steel
Titanium
k
0
46
61
13
k
1.51
×
1
0
-
5
1.78
×
1
0
-
5
4.23
×
1
0
-
6
Grinding regime
2
l
1.4
×
1
0
-
3
2.5
×
1
0
-
3
2.66
×
1
0
-
3
q
1.4
×
1
0
7
3.5
×
1
0
7
5.89
×
1
0
7
v
d
3.3
×
1
0
-
2
0.15
0.53
T
∞
300
300
300
Dimensionless parameters
L
0.76
5.27
83.4
Y
0.1
L
1.68
4.42
17.6Figure4 shows the temperature field for data set 1, where we have considered as plot area(104)D=-1.5l,1.5l×0,δ0.1.Figure 4
Temperature field for data set 1 and a constant heat flux profile.In order to compare the performance of the numerical integration of (7) and the numerical evaluation of the series expansions given in Section 4 for the calculation of the temperature field, we have set the time ratio τ as the ratio between the computation time of the numerical integration tintegration and the series expansion tseries, (105)τ=tintegrationtseries.t
i
n
t
e
g
r
a
t
i
o
n
≈
10
–
2
5
s for plotting the temperature field (i.e., Figure 4) running an 8 kernel i7 CPU computer with MATHEMATICA™ on Linux platform. However, this time could be increased by a factor of 2–5 times in the numerical control of the grinding machines; thus a time ratio τ≥2 would be worthwhile.Also, to compare both temperature fields, we have considered the following distance between two nonnegative scalar fieldsf and g, integrable in a certain connected open set D⊆Rn [25](106)ΔDf,g=∫Dfx→-gx→dV∫Dfx→+gx→dV,forg≠00,f=g=0.This distanceΔD is applicable to the temperature field since, according to (95); the latter is a positive quantity. In [25], it is proved that 0≤ΔDf,g≤1, wherein ΔDf,g=0 means that f and g are overlapped within the integration domain D, and ΔDf,g=1 means that both functions are relatively infinitely far one from each other. Table 3 shows the distances ΔD between the temperature fields computed by the numerical integration of (7) and the numerical evaluation of the corresponding series (83) and (84), taking as integration domain (104).Table 3
Relative distancesΔD and time ratios τ between the integration method and the series method, where n denotes the number of terms taken in the series approximation.
Heat flux
Data 1
Data 2
Data 3
Constant
Δ
D
2.34
×
1
0
-
8
4.26
×
1
0
-
7
2.86
×
1
0
-
5
τ
7.43
7.47
7.45
n
8
8
10
Linear
Δ
D
3.41
×
1
0
-
8
7.57
×
1
0
-
7
2.56
×
1
0
-
5
τ
6.40
6.15
5.45
n
8
8
10
Triangular
Δ
D
2.34
×
1
0
-
8
7.01
×
1
0
-
7
2.56
×
1
0
-
5
τ
17.16
17.10
11.73
n
8
8
10
Parabolic
Δ
D
4.26
×
1
0
-
8
1.07
×
1
0
-
6
2.33
×
1
0
-
5
τ
5.06
5.50
1.63
n
8
8
10To evaluate numerically the alternating seriesS=∑k=0∞-1kak given in (70), (75), and (79) for Iupperm and (71), (78), and (80) for Ilowerm, we have taken n=8 terms in the Cohen-Villegas-Zagier algorithm [20] for data sets 1 and 2 and n=10 for data set 3. In particular, we have used the 2A algorithm, which consists first of computing and storing in memory the polynomial (107)Ansin2t=12nn!Dnsinn2t, and then we apply the following procedure.Algorithm 1 (2A).
LetAnx=∑k=0nbkxk. Set d=-An-1; c=-d; s=0. For k=0 up to k=n-1, repeat: c=bk-c; s=s+cak. Output: s/d.It is worth noting that there is a typo in the algorithm published in [20], where d is given as An-1.Table3 shows the time ratios for the three data sets. We can see that the computation of the corresponding series expansions of Section 4 is considerably faster than the numerical integration. This is especially true for a triangular heat flux profile, for which we have set λ=0.5 (similar results are found for different values of λ). Also, both methods provide very similar temperature fields, since ΔD≪1. This is especially true for low Peclet numbers.


## 6.3. Takazawa’s Approximation Analysis

We can compare the exact value given in (100) with the approximation given by Takazawa in (19). Figure 5 shows this comparison for different values of the Peclet number. We can appreciate that, regardless of the value of the Peclet number, Takazawa’s approximation deviates from the exact value for large Y.Figure 5
T
0
,
m
a
x
Y for intermediate Peclet numbers.In order to measure the accuracy of Takazawa’s approximation (19), we can compute the relative distance ΔD of (19) with respect to the numerical value (100) as a function of the Peclet number L, taking as integration domain D=0,Y0.1L. Figure 6 presents this computation, showing that the accuracy of Takazawa’s approximation is in general poor, especially for low Peclet numbers. Therefore, despite the fact Takazawa’s formula being extremely easy to compute, probably it is better to use the numerical method given in (100), since the numerical evaluation of XmaxY,L is very rapid, and the computation of TX,Y is quite fast by using the series expansion method described in Section 4. For instance, Figure 5 is plotted in ≈0.87 s by using MATHEMATICA and an Intel i7 CPU.Figure 6
Accuracy of Takazawa’s approximation.


## 7. Conclusions

In the framework of Jaeger’s model for heat transfer in dry surface grinding, we consider the integral formula for the dimensionless temperature field of the workpiece, assuming the most common heat flux profiles reported in the literature.On the one hand, series expansions of the temperature field in dry grinding for the heat flux profiles considered have been calculated. By using the Cohen-Villegas-Zagier algorithm for computing alternating series, the numerical evaluation of these series expansions is considerably faster than the numerical integration of the integral form given in Jaeger’s model for a wide range of Peclet numbers. The latter is especially true in the case of a triangular heat flux profile, for which the series method is ≈17 times faster than the numerical integration. Also, these series expansions provide an evaluation of the temperature field as accurate as the integral form within the domain where the temperature rise is more significant. The latter is especially true for low Peclet numbers.On the other hand, considering a constant heat flux profile, a numerical procedure for the computation of the maximum temperature as a function of the Peclet number and the depth below the surface has been proposed. This numerical procedure has been used to evaluate the accuracy of Takazawa’s approximation.



---

*Source: 1856523-2017-12-28.xml*