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A Theorem for Finding Maximum Temperature in Wet Grinding
We consider the solutions found in the literature for heat transfer in surface grinding, assuming a constant heat transfer coefficient for the coolant acting on the workpiece surface and a constant or linear heat flux profiles entering into the workpiece. From the integral form of the time-dependent temperature field reached in the workpiece, assuming the previous conditions, we prove that the maximum temperature always occurs in the stationary regime on the workpiece surface within the contact zone between the wheel and the workpiece. This result assures a very rapid method for the theoretical computation of the maximum temperature.
Surface grinding is an industrial machining process aiming to remove excess material of a workpiece by means of an abrasive wheel which rotates at high speed over its surface (see Figure 1). Most of the energy used in the grinding process is converted into heat, and it is accumulated within the contact zone between the wheel and the workpiece . The high temperatures reached can thermally damage the quality of the workpiece, burning the workpiece, causing metallurgical phase transformations, softening (tempering) the surface layer with possible rehardening, onsetting residual tensile stresses, and causing cracks [1, 2]. Therefore, the determination of the temperature field evolution inside the workpiece is of great industrial importance [3–7]. In order to avoid thermal damage, coolant is usually delivered to the porous grinding zone at a high velocity, so friction is reduced and cooling by convection occurs. Some recommendations for the optimization of the coolant usage are found in .
1.1. Heat Transfer Model in Surface Grinding
In surface grinding, the heat transfer inside the workpiece is usually modeled [9–11] by a strip heat source infinitely long and of width (m in SI units), which moves at a speed (m s−1) over a semi-infinite solid surface (see Figure 1). Notice that, without losing generality, we can consider because reversing the motion of the workpiece relative to the grinding wheel, , is equivalent to change the direction of the -axis, .
Setting the Cartesian coordinate system fixed to wheel, as shown in Figure 1, the temperature field of the workpiece must satisfy the convective heat equation (notice that if represents the velocity of the motion of the workpiece, then ) [12, ]:where is the thermal diffusivity (m2 s−1). Since initially the workpiece is at room temperature (K), (2) is subjected to the following initial condition:
From a theoretical point of view, heat transfer in wet surface grinding has been modelled assuming a constant heat transfer coefficient (W m−2 K−1) on the workpiece surface [11, 13]. Despite the fact that this is not a very realistic assumption, because outside and inside the grinding zone are expected to have different values of , it is known that most of the energy removed by convection due to coolant occurs within the grinding zone . Therefore, considering that the actual inside the grinding zone is equal to the one outside of it (constant assumption), it is expected that the temperature field does not differ very much from a nonconstant assumption. Moreover, the assumption of constant makes the problem to be analytically tractable. Assuming as well a dimensionless heat flux profile within the contact area between wheel and workpiece, we have the following boundary condition:where denotes the Heaviside function, (W m−1 K−1) is the thermal conductivity, and (W m−2) is the average heat flux entering into the workpiece along the contact width ; thus
1.2. The Solutions
This paper will focus on the issue from a theoretical point of view by using the solution of (2)–(4) given by DesRuisseaux  for a constant heat flux profile and the solution given by Sauer  for a linear heat flux profile. In dry grinding, experiments suggest that the heat flux profile is linear (triangular with its apex at the inlet of the wheel-workpiece contact arc) in the case of upcutting . Moreover, in wet grinding, according to , a constant heat flux profile is expected to occur in downcutting, while a linear heat flux profile is presumed to occur in upcutting. However, a linear heat flux profile may prevail in the literature, although a constant profile is used as well .
For both heat flux profiles, Green’s method is used to arrive to the desired solution. Assuming a Cartesian coordinate system fixed to the wheel (see Figure 1), the Green function is interpreted in this case as the temperature field rise with respect to the room temperature, evaluated at in a semi-infinite body () in which, at the position over the surface , an instantaneous point heat source of energy (J) appears at the instant , where the surface () is subject to a constant heat transfer coefficient : where the radiation coefficient (m−1) is defined as
The derivation of the solution of (2)–(4) by using (6) follows three steps:(i)Superposition in space to give temperature due to instantaneous line source, which acts on the surface parallel to the -axis.(ii)Superposition of line sources to give temperature due to a instantaneously acting strip source on the surface.(iii)Superposition in time to give temperature due to continuously acting strip source.
Let us consider now on the following dimensionless variables, which, taking into account (1), satisfy
Similarly, for a linear heat flux profileSauer in  provides the following solution:where
Notation. We will drop the superscripts or when we refer to the temperature field regardless of the heat flux profile entering into the workpiece, so that we will write simultaneously (10) and (14) as
1.3. Maximum Temperature Search Approach
In order to avoid thermal damage, we need to control the maximum temperature in grinding, calculated from the temperature fields expressions given in (10) and (14). Therefore, let us state the following theorem, which will be very useful for the numerical computation of .
Theorem 1. The maximum temperature of the time-dependent temperature field iswhere
The proof of Theorem 1 follows three steps:(1)The maximum temperature is reached in the stationary regime, that is to say, when .(2)The maximum temperature is reached on the workpiece surface, that is to say, at .(3)The maximum temperature is located within the grinding zone, that is to say, .
As far as author’s knowledge, the occurrence of the maximum temperature in the stationary regime is taken for granted in the literature and we did not find any formal proof of it. Moreover, we have found the following considerations about the maximum temperature:(i)Jaeger calculates in  the temperature field in the case of dry grinding, considering a constant heat flux profile. He takes for granted that the maximum temperature occurs in the stationary regime on the workpiece surface. Moreover, he provides some approximations for large in order to calculate the surface temperature in the stationary regime, from which he estimates the maximum temperature.(ii)Malkin and Guo in  follow Jaeger approximated formula for maximum temperature.(iii)DesRuisseaux in  presents the temperature field in wet grinding for a constant heat flux profile. He does not consider his solution in the transient regime. Also, he plots the temperature for and for different depths , in order to illustrate that the maximum temperature occurs on the surface and within the grinding zone.(iv)Sauer in  calculates the temperature field in wet grinding for a linear heat flux profile, but again he considers his solution only in the stationary regime. He says that the region in which the maximum temperature is expected lies on Therefore, it might be expected that the maximum temperature occurs outside the grinding zone and at certain depth from the surface.
However, for the case of a constant heat flux profile, we find in the literature the following partial results regarding to Theorem 1:(i)The maximum temperature occurs when [18, Sect. 3.1].(ii)When , the temperature field reaches the stationary regime; that is, [16, Sect. 5.2].(iii)When the temperature field reaches the stationary regime, the maximum temperature occurs at the boundary, that is, on the workpiece surface [18, Sect. 3.2].(iv)In order to locate the maximum temperature in the stationary regime on the workpiece surface, it can be proved for small Biot numbers that always this maximum occurs within the grinding zone [18, Sect. 3.3].
The principal scope of this paper is just to generalize the above results for any Biot number, for both a constant and a linear heat flux profiles. We will see later on that this is a challenging mathematical problem.
This paper is organized as follows. Section 2 is devoted to prove that the maximum temperature is found on the surface in the stationary regime, for both a linear and a constant heat flux profiles. Section 3 proves that the maximum temperature occurs within the grinding zone, regardless as well of the heat flux profile considered: constant or linear. Section 4 provides a numerical example of the temperature field in the stationary regime, considering a constant and a linear heat flux profile. Our conclusions are summarized in Section 5. In the Appendix we collect some auxiliary lemmas in order to support the results given in body of the paper.
2. Maximum Temperature on the Surface in the Stationary Regime
In this section we will prove first that is a monotonically increasing function of (Theorem 4), so that must be asymptotically reached when . This result prevents the occurrence of a temperature peak during the transient regime. Once this is proved, we will prove that when , the temperature field does not evolve over time; that is, the stationary regime is asymptotically reached when (Proposition 8). This fact is the key result to prove that the temperature field in the stationary regime can be interpreted as a harmonic function, so that, according to the maximum principle of harmonics functions, the maximum temperature is found to be on the boundary, that is, on the workpiece surface (Theorem 9).
Proposition 2. The function satisfies
Proof. According to [20, Sect. 2.2], we haveSincewe may rewrite (21) asMoreover, the following inequality is also satisfied: Taking and in (24), we haveso, according to (11), it follows (20), as we wanted to prove.
Proposition 3. The functions satisfies
Proof. According to the definition of given in (12), we may rewrite this function aswhich is positive since , , and the integrand is positive within the integration interval. Similarly, let us rewrite , defined in (15), aswhich is also positive because , , and the integrand is positive within the integration interval.
Theorem 4. The time-dependent temperature field is a monotonically increasing function of :
Corollary 5. The maximum temperature is asymptotically reached when .
Let us prove now that the stationary regime is asymptotically reached at . For this purpose, let us start proving the following propositions.
Proposition 6. The function satisfies
Proof. According to the definition of the given in (11), we haveTaking into account now the asymptotic expansion [21, Eq. 7.12.1] up to first order, we haveso, substituting (34) in (32), we conclude (31), as we wanted to prove.
Proposition 7. The functions satisfy
Proposition 8. The stationary regime is asymptotically reached, when :
The above proposition allows us to prove now the following result.
Theorem 9. The maximum temperature is reached at the surface of the workpiece, .
Proof. Let us denote the Cartesian coordinates fixed to the workpiece as (see Figure 2), where, at , both coordinates systems are overlapped. Consider as well the temperature field referred to as , whereApplying chain’s rule, it is easy to see that the convective term in (2) disappears:Taking now similar dimensionless variables as in (8), that is to say,(2) becomesAccording to (40) and (42), it is clear that the relationship between the temperature field referred to the workpiece and the one referred to the wheel is given bySince (38) is satisfied for all , according to (44), we obtainThus, performing the limit in (43), taking into account (45), we haveso is a harmonic function. Therefore, according to the maximum principle of harmonics functions [22, Chap. VI], the maximum temperature of (47) occurs on the boundary, that is, on the workpiece surface, , as we wanted to prove.
3. Maximum Temperature within the Grinding Zone
According to Corollary 5 and Theorem 9, the maximum temperature must be found on the surface of the workpiece in the stationary regime, . Therefore, let us define the function as the surface temperature in the stationary regime:
The idea is to prove that is a monotonically increasing (Theorem 14) or decreasing (Theorem 10) function outside the grinding zone. Therefore, must be found within the grinding zone, , as it is stated in Theorem 1.
Theorem 10. The surface temperature in the stationary regime is a monotonically decreasing function on the right-hand side of the leading edge:
Proof. From (48), we havewhere, according to (12), and, according to (15), Let us set now the following variables:thussince the integration interval in (50) means that . Notice also that implies that ; so taking into account (54), we haveOn the one hand, rewriting now (51) in terms of and , we havedue to (55) and the fact that the function is a monotonically decreasing function for positive arguments [23, Sect. 1.3. Def. 1].
On the other hand, (52) in terms of and reads out asApplying the mean value theorem for derivatives [24, Theorem 4.5], we haveTherefore, , so and (58) is written asTaking into account (59) in (57), we concludeFinally, collecting the results (56) and (60) and taking into account (20), we may writeso integrating (61) over the interval , according to [25, Theorem 3.3.4], then (49) follows directly, as we wanted to prove.
In the above theorem we have just seen that the sign of for is determined by the sign of . This is not the case for ; so before arriving to Theorem 14, we have to prove the following three propositions.
Proposition 11. Let Then is a positive and monotonically decreasing function for positive arguments:
Proof. Particularizing (20) to , we haveso it directly follows (63), since .
In order to prove (64), rewrite as follows, taking into account (11): Define now the functionso Considering that [20, Sect. 2.2]we can say Since according to (70), it follows (64), as we wanted to prove.
Proposition 12. Let If then consider the following.(1)There is an unique crossing point such that(2)The following inequalities are satisfied:
Proof. Notice that so the values or satisfy (74). Nevertheless, these values do not belong to the dominion of . Therefore, let us substitute (72) in (74), that is to say,soThe equation with the “+” sign leads to but this contradicts (73). The equation with the “−” sign leads towhich is positive, according to (73), so it falls within the dominion of .
Notice that (79) is an actual crossing point between the graphs of and as functions of sinceIndeedso, if thenbut this contradicts (73). Therefore, there is a unique crossing point satisfying (74), as we wanted to prove.
In order to prove (75), see that the function for given satisfies the assumptions of Lemma A.1 of the appendix:According to (72), , for all .According to (76), .Also, such that . Indeed, it is easy to prove that the only value for satisfying is given by
Now, since , by definition of absolute maximum, we haveand taking into account (88), we obtain
See now that the functions and , where , satisfy the assumptions of Lemma A.2 of the Appendix, that is to say,(1) such that , according to (74);(2), according to (80);(3)taking , according to (86) , and (90) reads as .
Proposition 13. The following improper integral is independent of , if :
Theorem 14. The surface temperature in the stationary regime is a monotonically increasing function on the left-hand side of the trailing edge:
Proof. Substituting in (58) the values of and given in (53) and taking into account (54), we havewhereTaking into account that , rewrite asso (97) is satisfied and (96) reads out asSubstituting (99) in (52), we haveTaking into account (51) and (100) and recalling the definitions of the and functions, (72) and (62), respectively, let us express (50) as follows:where or and depending on whether we are in the constant or linear case, respectively. Therefore, according to (98) and (102), we haveNotice that the integral given in (101) satisfies the assumptions of Lemma A.3 given in the Appendix:(1)According to Proposition 13 and taking into account (103), we have(2)According to Proposition 11, we have (3)According to Proposition 12 and taking into account (103), we have
Therefore, since , we conclude (95), as we wanted to prove.
4. Numerical Example
For the numerical evaluation of the temperature field, we will use the thermal properties of a VT20 titanium alloy workpiece :
Also, we will take the following values for the grinding regime:
Figures 4 and 5 show the temperature field in the stationary regime for a constant and a linear heat flux profile, respectively. Since the temperature gradient nearby the grinding zone is very high, note that the -axis is zoomed with respect to the -axis in order to obtain a more detailed view of the contour plots. Notice as well that, although the temperature fields are quite different in both graphs, the maximum temperature is found to be on the surface within the grinding zone. For a constant heat flux profile, the value and location on the surface of the maximum temperature isand for a linear heat flux profile is
Note that the maximum temperature is higher in the linear case because the heat flux is skewed to the leading edge, but in the constant case the heat flux is uniform along the grinding zone. Nonetheless, the temperature field is more skewed to the trailing edge in the constant case because of the motion of the workpiece, being compensated the effect of the motion of the workpiece with the skewness of the heat flux profile in the linear case.
It is worth noting that the principal result of this paper (Theorem 1) assures a very rapid method for searching the maximum temperature. Indeed, the numerical search of the maximum in a three-dimensional field within a unbounded region is reduced to the numerical computation of the maximum in a one-dimensional function within a particular interval; that is, .
We have considered the solutions found in the literature to model the heat transfer in surface grinding, assuming a constant and a linear heat flux profile entering into the workpiece and a constant heat transfer coefficient for the coolant acting on the workpiece surface. From these time-dependent temperature fields in dimensionless variables , we have analyzed the location of the maximum temperature in the workpiece, proving that the latter must be located on the workpiece surface, in the stationary regime and within the grinding zone, that is, Theorem 1. As long as the heat transfer model stated in Section 1.1 can be applied to the real grinding process, Theorem 1 has the following consequences:(i)The proof that the maximum temperature is found in the stationary regime prevents to encounter a temperature peak during the transient regime.(ii)Also, the proof about the location of the maximum temperature on the workpiece surface prevents to find an internal thermal damage in the workpiece when its surface seems not to be damaged.(iii)Moreover, the proof that the maximum temperature is located in the grinding zone on the surface and in the stationary regime converts the search of the maximum temperature in a three-dimensional temperature field within an unbounded region into the search of the maximum for a one-dimensional function within a particular interval, which is a much more simple problem from a computational point of view.
These theoretical results are intended to be useful in order to avoid thermal damage in surface wet grinding.
Lemma A.1. Let be differentiable in and continuous in that satisfies If then is an absolute maximum:
Proof. Since the function is continuous in , according to the Extreme-Value Theorem of Continuous Functions [24, Theorem 3.12], the absolute maximum belongs to the interval . Suppose that this absolute maximum is not located in , say in . Thus, by definition of absolute maximum and according to (A.2), we haveIf or , then, according to (A.1), , but this contradicts (A.5). If , then, since is differentiable in , is also a local maximum, so , but this contradicts (A.3). Therefore, the absolute maximum must be located in , as we wanted to prove in (A.4).
Lemma A.2. Let be two continuous and differentiable functions which have a single crossing point, that is to say, Ifthen
Proof. Let us consider the following differentiable function:According to (A.6), we haveSince is a differentiable function, according to (A.7), we haveso, according to (A.12),Therefore, changes sign in the neighborhood of , that is to say, On the one hand, assume now thatHowever, by hypothesis (A.8),Therefore, from (A.16) and (A.17), within the interval a sign change occurs; so applying Bolzano’s theorem, it follows that but this contradicts the hypothesis that the crossing point is unique (A.6). Thus such that , that is, as we wanted to prove in (A.9).
On the other hand, notice that from (A.15) and (A.19) we haveAssume now thatTherefore, from (A.20) and (A.21), within the interval a sign change occurs; so applying Bolzano’s theorem, it follows thatbut this contradicts that the crossing point is unique (A.6). Thus such that , that is, as we wanted to prove in (A.10).
Lemma A.3. Let be three real functions that satisfy the following.(1)The following improper integrals of and are convergent and satisfy (2)The function is positive and monotonically decreasing for positive arguments (3)There is an unique crossing point for and such that
Proposition A.4. Then
Proof. From (A.24), we havethat is,Since is a positive (A.25) and decreasing (A.26) function, then, due to (A.27), we have According to [25, Theorem 3.3.4] we may integrate in (A.33) preserving the sense of the inequality:Similarly, we havethuswhere we have taken into account (A.31) and (A.34). Rewrite now (A.36) aswhich is just (A.29), as we wanted to prove.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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