Calculation of some integrals arising in heat transfer in grinding.(Research Article)(Report)

1. Introduction

In the resolution of the heat equation that models the heat transfer during the grinding process [1], there appear integrals [2] of the following type:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

which have not been tabulated yet [3]. The first integral arises in the evaluation of the temperature on the surface of the workpiece, while the second integral is used for the evaluation of the field temperature inside the workpiece. On the one hand, despite the fact that these integrals can be computed by using the uniqueness of the solution to the Laplace equation, as done in [2], this is cumbersome, since the integrals to be calculated are subproducts of a wider framework. This framework is the resolution of the heat equation in the stationary regime in two different ways. This paper presents a straightforward proof, based on elementary integral calculus and complex variables. On the other hand, the computation of these improper integrals, which show a parametric dependence on a, b, and c for [I.sub.1], and x and y for [I.sub.2], is expensive in order to get the field temperature of the wokpiece being ground. The goal of this paper is the resolution of (1.1) and (1.2), so that the evaluation of the temperature field may be faster.

This paper is organized as follows. Sections 2 and 3 are devoted to the calculation of [I.sub.1] and [I.sub.2], respectively. Section 4 applies [I.sub.1] and [I.sub.2] to the solution of the stationary regime in dry continuous grinding within the Samara-Valencia model [1].

2. First Integral

In order to calculate the integral (1.1), let us solve the integral (2.1) in two different ways. Afterwards, comparing the results obtained, we will find the sought-for solution. Let us define:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

where a, b, and c are parameters within the integral.

2.1. First Calculation

Applying Fubini's theorem to (2.1), we can exchange the integration order, obtaining

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

Let us develop the exponential within (2.2), as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

Calling the inner integral in (2.2) [??] and taking into account (2.3), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

Let us perform the change of variables [sigma] = 4[b.sup.2] s in (2.4), and let us define the variable z as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

so that (2.4) becomes

[MATHEMATICAL EXPRESSION NOT …