Journal of Applied Mathematics
Volume 2017 (2017), Article ID 3591713, 5 pages
https://doi.org/10.1155/2017/3591713

Research Article

New Integrals Arising in the Samara-Valencia Heat Transfer Model in Grinding

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; martinez.gonzalez@ucv.es

Received 20 August 2017; Accepted 24 October 2017; Published 14 November 2017

Academic Editor: Ali R. Ashrafi

Copyright © 2017 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.

Abstract.

The Samara-Valencia model for heat transfer in grinding has been recently used for calculating nontabulated integrals. Based on these results, new infinite integrals can be calculated, involving the Macdonald function and the modified Struve function.

1. Introduction

Usually, mathematical developments facilitate the computation of mathematical modeling expressions in many different fields. However, mathematical modeling also yields in many cases a good field to develop new mathematical identities and formulas. This is the case of the mathematical modeling of heat transfer in surface grinding. This machining process consists in material removal from a workpiece by an abrasive wheel that rotates at high speed over its surface [1]. Classically, Jaeger’s model [2, 3] is used for the calculation of the temperature field in dry grinding. DesRuisseaux’s model [4] extends Jaeger’s model to include the effect of surface cooling (wet grinding). More recently, the Samara-Valencia model [5] has been proposed. In this model, the two-dimensional convective heat equation is considered. Also, the heat flux profile entering the workpiece and the action of the coolant are considered in the boundary condition. In [5], this boundary-value problem is transformed into an integral equation that is useful for the numerical evaluation of the heat transfer in intermittent wet grinding [6]. However, in the case of dry grinding, this integral equation can be reduced to a two-dimensional integral ( theorem) [7].

New mathematical identities have been proved in this framework. For instance, comparing Jaeger’s model with Samara-Valencia model, a new Dirac delta representation [8],

and two new nontabulated integrals [9] have been obtained. Also, by using theorem, the following integrals have been calculated [10]:

It is worth noting that (3) has been calculated also in [9] by using a complex integration contour. Recently, in [11], the following generalization of (4) has been calculated as a finite sum of terms containing beta and hypergeometric functions, by using the convolution theorem of the Fourier transform. This generalization reads as follows:

where the following coefficients are defined as

with

The scope of this paper is just to calculate more integrals based on results (2)–(4), which do not seem to be reported in the most common tables of integrals [12–14].

This paper is organized as follows. Section 2 is devoted to the calculation of the new integrals. It is divided into three subsections, each one of them containing one new result. Section 3 collects the conclusions, highlighting the main results obtained in the body of the paper.

2. The Integrals

2.1. First Result

Theorem 1. The following integral holds true:

Proof. Expanding the hyperbolic cosine as [15, Eqn. 8.2] in (2), we have

Now, perform the change of variables , and rewrite the first or the second integral given in (9) as

Therefore, inserting (10) in (9), we can define the following function:

Let us take temporarily , so that we can drop the absolute value in (11). Then, knowing that [16, Eqn. 51:10:2], we have

Recover now the absolute value and take into account again (9)-(10) to obtain (8).

Remark 2. It is worth noting that when , (12) seems to fail. Nonetheless, this is apparent. In order to see it, rewrite the RHS of (12) as follows, knowing that and performing the changes of variables and :

Therefore, taking the limit and applying the Dirac delta representation given in (1), we have

which agrees with the LHS of (12), performing the limit .

We can continue calculating derivatives with respect to in (12) in order to get new integrals, but the integrands we get are increasingly complex and we omit these results here.

2.2. Second Result

Theorem 3. The following integral holds true:

Proof. Let us define

Notice that

thereby, hereafter, we will assume that , dropping the absolute value. Performing in (17) the derivative with respect to , we have

The above integral can be calculated taking in (3); thus,

Now, let us apply the following integral [12, Eqn. 1.12.1()]:

where is the modified Struve function, defined as [17, Eqn. 11.2.2]

Therefore, taking in (21) , knowing that ([18], Eqn. 5.7.10) and also that [16, Eqn. 43:4:2]

we calculate from (20) as

where is an integration constant. In order to calculate this integration constant, notice, on the one hand, that, from (16), we have

The above integral can be calculated taking in (4):

Thus,

On the other hand, taking limits in (24), we have

According to (22), consider the following asymptotic formula:

where we have applied the property of the gamma function ([18], Eqn. 1.2.1) and (23). Similarly, we have

Also, the asymptotic behavior of the Macdonald function for is ([17], Eqn. 10.30.2-3)

and thus

Therefore, taking into account (29)–(32), we calculate the limit given in (28) as

Thereby, inserting (33) in (28) and recalling (27), we calculate the integration constant as follows:

Finally, substituting (34) into (24) and remembering the definition of , we arrive at (15), where we have considered (18).

2.3. Third Result

Theorem 4. The next integral holds true:

Proof. For the third result, integrate both sides of (16) as follows:

In order to calculate the inner integral given in (36), consider and in (2); hence,

Inserting (37) into (36), we can drop the absolute value in the exponential, since the variable of integration is positive (i.e., ); thereby,

To calculate the above integral, in the literature, we find the following integral, termed King’s integral [19, Eqn. 11.3.16]:

Hence, applying to King’s Integral and the asymptotic formula [18, Eqn. 5.16.5]

we obtain

Recalling now the result given in (15) for and taking into account (41), (38) finally reads as (35).

3. Conclusions

Based on integrals (2)–(4), calculated in the framework of the Samara-Valencia heat transfer model in surface grinding by using the theorem, new integrals have been derived. From (2), we have derived integrals (11) and (8). Also, applying (3) and (4), we have derived (15). Finally, integrating the result given in (15) and taking into account (2), the integral given in (35) has been obtained. It is worth noting that these results have been confirmed, evaluating numerically the corresponding integrals.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The author wishes to acknowledge the financial support received from Universidad Católica de Valencia under Grants PRUCV/2015/612 and 2017-160-001.

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