Maximum Temperature in Dry Surface Grinding for High Peclet Number and Arbitrary Heat Flux Profile

Abstract

Regarding heat transfer in dry surface grinding, simple asymptotic expressions of the maximum temperature for large Peclet numbers are derived. For this purpose, we consider the most common heat flux profiles reported in the literature, such as constant, linear, triangular, and parabolic. In the constant case, we provide a refinement of the expression given in the literature. In the linear case, we derive the same expression found in the literature, being the latter fitted by using a linear regression. The expressions for the triangular and parabolic cases are novel.

Academic Editor:Ruben Specogna

Universidad Catolica de Valencia, 46001 Valencia, Spain

Received 9 November 2015; Accepted 11 January 2016

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.

1. Introduction

The computation of the maximum temperature from the grinding conditions in dry surface grinding is essential for predicting thermal damage [1, Section 6.2]. For grinding regimes in which we find a high Peclet number, normally the use of coolant is required to reduce the risk of thermal damage. However, the pressure to reduce production costs and the considerable increase in environmental awareness have led manufacturers to reduce or eliminate cutting fluids in machining operations. In fact, coolant costs are in the range of 10-17% of total manufacturing costs [2]. Therefore, the prediction of the maximum temperature in order to evaluate the risk of thermal damage in dry grinding for a large Peclet number is not negligible. For this purpose, considering a constant heat flux distribution (with a value of [figure omitted; refer to PDF] Wm^-2 in SI units) within the contact width [figure omitted; refer to PDF] (m) between wheel and workpiece, the maximum temperature [figure omitted; refer to PDF] (K) can be approximated for large Peclet numbers ( [figure omitted; refer to PDF] ) as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] (ms^-1 ) is the constant velocity of the heat source sliding over the workpiece, [figure omitted; refer to PDF] is the thermal diffusivity of the workpiece ( [figure omitted; refer to PDF] ), and [figure omitted; refer to PDF] is its thermal conductivity (Jm^-1 K^-1 s^-1 ). This result is first given by Jaeger in [3, Eq. 33], being reported also by Malkin and Guo in [4, Eq. 2]. It is worth noting, on the one hand, that Jaeger omits the derivation of the asymptotic formula (1), being this given in dimensionless form. On the other hand, Malkin and Guo follow Jaeger's analysis replacing the constant factor [figure omitted; refer to PDF] by the approximation [figure omitted; refer to PDF] and stating that (1) provides a good approximation for [figure omitted; refer to PDF] , although in practice it can be applied down to [figure omitted; refer to PDF] , which includes most of the actual grinding situations (see also [5]).

Despite the fact that (1) has been used widely to predict thermal damage in surface grinding [1, Section 6.2], it seems to be more realistic to assume a linear heat flux distribution with its maximum at the leading edge. Therefore, considering this linear heat flux profile and assuming a geometrical contact length within the wheel and the workpiece [1, Eq. 3-4] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the diameter of the grinding wheel and [figure omitted; refer to PDF] is the depth of cut, the maximum temperature [figure omitted; refer to PDF] can be approximated for large Peclet numbers as [6, Eq. 3] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is now the average heat flux along the contact zone. It is worth noting that in [6] the constant factor [figure omitted; refer to PDF] is calculated by performing a linear regression, without any ab initio derivation.

Since the most common heat flux profiles reported in the literature are constant [7, 8], linear [8, 9], triangular [10, 11], and parabolic [12] (see Figure 1), this paper is intended, on the one hand, to provide a sound proof for the asymptotic formulas (1) and (3) and, on the other hand, to derive new asymptotic formulas for the triangular and parabolic cases.

Figure 1: Linear, triangular, and parabolic heat flux profiles.

[figure omitted; refer to PDF]

This paper is organized as follows. In Section 2, the heat transfer model for surface grinding is presented in order to provide the framework in which the asymptotic formulas of the maximum temperature for large Peclet numbers have to be derived. Section 3 carries out the calculation of these asymptotic expressions considering constant, linear, triangular, and parabolic heat flux profiles. Also, in the constant case, a refinement of (1) is also calculated. Section 4 is devoted to evaluating numerically the accuracy of the asymptotic formulas in order to justify the range in which they are valid. Finally, the conclusions are collected in Section 5.

2. Heat Transfer Model in Surface Grinding

In dry surface grinding, friction due to contact between wheel and workpiece is modelled by an infinite strip heat source of width [figure omitted; refer to PDF] that slides over the workpiece surface at the plane [figure omitted; refer to PDF] and moves at a constant velocity [figure omitted; refer to PDF] [3]. Assuming a two-dimensional model (see Figure 2), the temperature field of the workpiece [figure omitted; refer to PDF] with respect to the room temperature in the stationary regime must satisfy the following equation [13, Eqns. 1.6( [figure omitted; refer to PDF] ) & 1.7( [figure omitted; refer to PDF] )]: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Considering a dimensionless heat flux profile [figure omitted; refer to PDF] within the contact area between wheel and workpiece, (4) is subjected to the following boundary condition: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the Heaviside function and [figure omitted; refer to PDF] is the average heat flux entering into the workpiece along the contact width [figure omitted; refer to PDF] ; thus [figure omitted; refer to PDF]

Figure 2: Heat transfer modelling in surface grinding.

[figure omitted; refer to PDF]

Setting the dimensionless quantities [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a characteristic length: [figure omitted; refer to PDF] the solution of (4) and (5) is expressed as [14] [figure omitted; refer to PDF] with [figure omitted; refer to PDF] being the zeroth-order modified Bessel function of the second kind . On the surface ( [figure omitted; refer to PDF] ), (9) reduces to [figure omitted; refer to PDF]

Assuming that [figure omitted; refer to PDF] is an analytic function [figure omitted; refer to PDF] , then it can be expanded in its Taylor series as [figure omitted; refer to PDF]

Substituting (11) in (10), we obtain [figure omitted; refer to PDF] where we have defined the following function: [figure omitted; refer to PDF]

According to [14], we can calculate the integral given in (13) as [figure omitted; refer to PDF] where the following polynomials in [figure omitted; refer to PDF] in terms of hypergeometric functions are defined as [figure omitted; refer to PDF] as well as the constant [figure omitted; refer to PDF]

In [14], the solution given in (12) is used for computing the maximum temperature in dry surface grinding for the most common heat flux profiles reported in the literature, provided that this maximum must be found in the stationary regime and on the workpiece surface, within the contact zone. Despite the fact that this computation is very rapid, asymptotic expressions of the maximum temperature for large Peclet number ( [figure omitted; refer to PDF] ) for any dimensionless heat flux profile [figure omitted; refer to PDF] can be calculated. This is quite useful not only because these expressions are extremely rapid to compute, but mainly because they provide a starting point for the development of a model of thermal damage in surface grinding as the one elaborated by Malkin and Guo considering a constant heat flux profile [1, Section 6.2].

3. Large Peclet Number Approximations

3.1. Asymptotic Formula for [figure omitted; refer to PDF] within the Grinding Zone

Notice that, taking into account the definition of the generalized hypergeometric function [15, Eq. 2.1.2] [figure omitted; refer to PDF] and the definition of the Pochhammer symbol [figure omitted; refer to PDF] we have the following asymptotic behavior of [figure omitted; refer to PDF] : [figure omitted; refer to PDF] and [figure omitted; refer to PDF] [figure omitted; refer to PDF] where in both sums given in (19) and (20) only survives the first term (i.e., [figure omitted; refer to PDF] ), when we perform the limit [figure omitted; refer to PDF] .

Now taking into account the asymptotic behavior of the Macdonald function [16, Eq. 5.16.5], [figure omitted; refer to PDF] and the results (19) and (20), we can calculate the asymptotic behavior of the function [figure omitted; refer to PDF] given in (14) as [figure omitted; refer to PDF]

Similarly [figure omitted; refer to PDF]

Notice that within the grinding zone [figure omitted; refer to PDF] thus, according to (22)-(24), we conclude that [figure omitted; refer to PDF]

3.2. Constant Profile

Notice that the dimensionless function, [figure omitted; refer to PDF] provides a constant heat flux profile satisfying (6) (see Figure 1). Therefore, substituting (26) in (12), we have [figure omitted; refer to PDF] so, according to (25), the asymptotic behavior for large Peclet number within the grinding zone is [figure omitted; refer to PDF]

Considering both constant and linear heat flux profiles, the maximum temperature in wet grinding is located on the surface within the grinding zone, in the stationary regime, as it is proved in [17]. Also in [17], wet grinding is modelled assuming a constant heat transfer coefficient [figure omitted; refer to PDF] over all the workpiece surface. Therefore, setting [figure omitted; refer to PDF] , we can apply this result of the location of the maximum temperature [figure omitted; refer to PDF] also to dry grinding in order to find an estimation of [figure omitted; refer to PDF] for a large Peclet number. Notice that, in (28), the maximum is reached at [figure omitted; refer to PDF] since we have a monotonic increasing function within the grinding zone, thus [figure omitted; refer to PDF] and in dimension variables, recalling the definitions (7)-(8), we obtain the result given by Jaeger (1). It is remarkable that Jaeger found this approximation from the integral representation of [figure omitted; refer to PDF] , that is to say (13), and not from its solution, given in (14).

We can improve the approximation given in (28) taking into account the following asymptotic expansion of the Macdonald function [18, Eq. 10.40.2]: [figure omitted; refer to PDF] where, according to [18, Eq. 10.17.1], [figure omitted; refer to PDF] so [figure omitted; refer to PDF]

Therefore, the result given in (23) can be refined to the following one: [figure omitted; refer to PDF] and then, taking into account (22), (24), and (34), we arrive at [figure omitted; refer to PDF]

Despite the fact that we cannot solve exactly the maximum of the function given in (35), let us solve it approximately. First, let us perform the change of variables: [figure omitted; refer to PDF] so that we have to find the maximum of the function: [figure omitted; refer to PDF]

Second, notice that, according to (29), [figure omitted; refer to PDF] so, expanding in Taylor series, we have [figure omitted; refer to PDF] and (37) can be approximated as [figure omitted; refer to PDF]

In order to find out the maximum, let us solve [figure omitted; refer to PDF] thus [figure omitted; refer to PDF]

Finally, substituting (42) in (28), we obtain the following refinement of (30): [figure omitted; refer to PDF]

3.3. Linear Profile

The following linear dimensionless function, [figure omitted; refer to PDF] provides null heat flux at the trailing edge [figure omitted; refer to PDF] and also satisfies (6) (see Figure 1). Therefore, according to (12), we have [figure omitted; refer to PDF]

Applying (25) to (45), we arrive at [figure omitted; refer to PDF]

In order to estimate the maximum temperature for large Peclet number, we can apply again the mathematical proof given in [17], in order to assure that such maximum must occur in the stationary regime and on the surface, within the grinding zone. Therefore, since [figure omitted; refer to PDF] is a differentiable function in [figure omitted; refer to PDF] , let us calculate the extrema points of the function given in (46); that is to say, [figure omitted; refer to PDF]

Note that since (47) has got a unique solution, it must correspond to the maximum. Therefore [figure omitted; refer to PDF] and, in dimensional variables, we have [figure omitted; refer to PDF]

It is worth noting that assuming a geometrical contact length within the wheel and the workpiece, that is, (2), (49) is rewritten as [figure omitted; refer to PDF] which is, in fact, formula (3) given by Malkin and Guo, provided that the factor [figure omitted; refer to PDF] has been fitted to [figure omitted; refer to PDF] .

3.4. Triangular Profile

The dimensionless triangular heat flux profile satisfying (6) is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and the profile apex is located in [figure omitted; refer to PDF] (see Figure 1). Also, we have set the dimensionless parameter: [figure omitted; refer to PDF]

Notice that when the heat flux occurs in an arbitrary interval, say [figure omitted; refer to PDF] instead of [figure omitted; refer to PDF] , the dimensionless surface temperature is given by [figure omitted; refer to PDF] where we have set the following dimensionless parameters: [figure omitted; refer to PDF]

Since (4) is a linear differential equation, the field temperature is given by the superposition of both parts of (51), so according to (12), we have [figure omitted; refer to PDF]

Applying (13) and reordering terms, we arrive at [figure omitted; refer to PDF]

where we have defined [figure omitted; refer to PDF]

On the one hand, notice that calculating the limit in (57) when [figure omitted; refer to PDF] , according to (22) and (24), we can conclude that [figure omitted; refer to PDF]

Similarly, taking into account (23) and (24), we arrive at [figure omitted; refer to PDF]

On the other hand, note that within grinding zone, on the left-hand side of the apex, we have [figure omitted; refer to PDF] and, on the right-hand side of the apex, we have [figure omitted; refer to PDF] thus [figure omitted; refer to PDF]

Taking into account (58), (59), and (62) in (56), after some algebra, we arrive at [figure omitted; refer to PDF]

Despite the fact we do not possess any mathematical proof for the location of the maximum temperature, we have to assume now that this one must be found in the stationary regime and on the surface, within the contact zone, as in the constant and linear cases. In fact, this is a quite natural assumption, physically speaking. Bearing this in mind, notice that, according to (63), the maximum cannot lie on the left-hand side of the apex. Therefore, since the maximum must have null derivative, let us solve [figure omitted; refer to PDF] thus [figure omitted; refer to PDF] and the maximum temperature is then given by [figure omitted; refer to PDF]

Notice that if the apex of the triangular profile tends to the leading edge, that is to say [figure omitted; refer to PDF] , then the profile is linear and (66) becomes (48).

3.5. Parabolic Profile

The dimensionless parabolic heat flux profile satisfying (6) is given by [figure omitted; refer to PDF] where the heat flux at the trailing edge fulfills that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Therefore, according to (12), we obtain [figure omitted; refer to PDF]

Taking into account (25), we have the following asymptotic behavior for large Peclet number within the grinding zone: [figure omitted; refer to PDF]

As in the triangular case, let us consider the fact that the maximum temperature is located on the surface, within the grinding zone, in the stationary regime. Therefore, in order to find out the maximum, let us solve [figure omitted; refer to PDF] so [figure omitted; refer to PDF]

Since the "+" sign in (71) leads to a value out of the interval [figure omitted; refer to PDF] , we have to choose the "-" sign. Substitution of the latter value in (69) yields [figure omitted; refer to PDF]

It is worth noting that substitution of (71) in (69) taking the "+" sign leads to the same result as (72), but with a " [figure omitted; refer to PDF] ", so anyway it is clear that (72) is the approximation to the maximum.

4. Numerical Results

In this section we examine the goodness of the asymptotic approximations given in Section 3 for large Peclet number. Figure 3 shows the dimensionless temperature in the stationary regime on the surface [figure omitted; refer to PDF] , considering a constant heat flux profile (27), as well as the asymptotic approximations (28) and (35). The graphs are plotted within the grinding zone [figure omitted; refer to PDF] and taking [figure omitted; refer to PDF] . Note that the maximum lies nearby the trailing edge within the grinding zone. Also, the first and second approximations, (28) and (35), respectively, overlap everywhere except for the zone nearby the maximum, where the second approximation has got a vertical asymptote. Note as well in Figure 3 that the location of the maximum [figure omitted; refer to PDF] can be better estimated by the second approximation [figure omitted; refer to PDF] , given in (42), than by the first approximation [figure omitted; refer to PDF] , given in (29). However, the maximum temperature is better estimated by using the first approximation (28) at [figure omitted; refer to PDF] , that is to say, the approximation given in (43). These considerations justify the procedure followed for the refinement of the maximum temperature approximation performed in Section 3.2.

Figure 3: Dimensionless surface temperature for a constant heat flux profile taking [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

In Figure 4 is plotted the dimensionless temperature on the surface [figure omitted; refer to PDF] for linear, triangular, and parabolic heat flux profiles, (45), (56), and (68), respectively, as well as its asymptotic approximations, (46), (63), and (69), taking [figure omitted; refer to PDF] for all the graphs and [figure omitted; refer to PDF] for the triangular case. We can appreciate that the asymptotic approximations are quite near to the exact solutions within the contact zone [figure omitted; refer to PDF] .

Figure 4: Dimensionless temperature (solid line) and its approximation (dashed line) taking [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

In order to evaluate quantitatively the distance between the exact solution and the asymptotic approximation as a function of [figure omitted; refer to PDF] , we can use the functional defined in [19]. If [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are two nonnegative functions defined in an interval [figure omitted; refer to PDF] , the relative distance between them is defined as [figure omitted; refer to PDF]

It is easy to prove that (see [19]) [figure omitted; refer to PDF] where a value close to [figure omitted; refer to PDF] means relatively near, but a value close to [figure omitted; refer to PDF] means relatively infinitely far. Figure 5 shows the computation of (73), taking as functions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] the exact solution [figure omitted; refer to PDF] given in (12) and the asymptotic approximation [figure omitted; refer to PDF] of the surface temperature for the different heat flux profiles considered, within the interval [figure omitted; refer to PDF] . We can see that, for [figure omitted; refer to PDF] , the asymptotic approximation is relatively quite near to the exact solution, regardless of the heat flux profile considered. Note that second approximation of the constant case (35) is worse than the first approximation (28) because we are considering the distance within the grinding zone [figure omitted; refer to PDF] ; and nearby the maximum, the first approximation fits better (see Figure 3).

Figure 5: Relative distance between [figure omitted; refer to PDF] and its approximation for large Peclet number within the interval [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Nevertheless, the behavior of the maximum is a little bit different from the global behavior of the asymptotic approximation with respect to the exact solution. To see it quantitatively, let us define the relative error as [figure omitted; refer to PDF]

Figure 6 presents the computation of (75) for the different heat flux profiles considered, where the exact value of the maximum [figure omitted; refer to PDF] has been computed according to [14]. It is worth noting that, eliminating the absolute value bars in (75), we can check that all the asymptotic approximations of the maximum temperatures overestimate the exact value, except for the second approximation of the constant case. Notice as well that the behavior of the second approximation for the constant case substantially improves the first approximation, which confirms the derivation given in the previous section.

Figure 6: Relative error of the maximum temperature approximation.

[figure omitted; refer to PDF]

5. Conclusions

We have considered the heat transfer in dry surface grinding in the stationary regime. Assuming the most common heat flux profiles proposed in the literature (constant, linear, triangular, and parabolic), we have derived very simple asymptotic expressions of the maximum temperature for high Peclet numbers. In the constant case, we have found a refinement of the expression found in the literature, which is quite accurate (an error less than [figure omitted; refer to PDF] for [figure omitted; refer to PDF] ). In the linear case, we have arrived at an expression which is reported in the literature by using a linear regression to fit the multiplicative constant. The expressions for the triangular and parabolic cases seem not to be reported in the literature.

Acknowledgments

The author is pleasured to thank Professor Raul Fernandez, from the Mondragon Unibertsitatea, for the suggestions given. The author wishes to thank also the financial support received from Generalitat Valenciana under Grant GVA/2015/007.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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