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

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 SamaraValenciamodel [5] has been proposed. In thismodel, the twodimensional 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 (T(0) theorem) [7]. New mathematical identities have been proved in this framework. For instance, comparing Jaeger’s model with Samara-Valencia model, a newDirac delta representation [8],


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 twodimensional 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 ( (0) 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  (0) 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, 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][13][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.
Remark 2. It is worth noting that when  = 0, (12) seems to fail.Nonetheless, this is apparent.In order to see it, rewrite the RHS of ( 12) as follows, knowing that  > 0 and performing the changes of variables  =  and  = : Therefore, taking the limit  → 0 and applying the Dirac delta representation given in (1), we have lim which agrees with the LHS of (12), performing the limit  → 0.
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.
Journal of Applied Mathematics

Conclusions
Based on integrals (2)-( 4), calculated in the framework of the Samara-Valencia heat transfer model in surface grinding by using the  (0) 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.