Mathematical Problems in Engineering

Volume 2010, Article ID 535801, 14 pages

http://dx.doi.org/10.1155/2010/535801

## Calculation of Some Integrals Arising in Heat Transfer in Grinding

^{1}Universidad Católica de Valencia, 46002 Valencia, Spain^{2}Universidad de Pinar del Río, 20200 Pinar del Río, Cuba^{3}Samara State Aerospace University, 443071 Samara, Russia^{4}Instituto de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, 46022 Valencia, Spain

Received 19 May 2010; Accepted 20 August 2010

Academic Editor: Massimo Scalia

Copyright © 2010 J. L. G. Santander et al. 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

We calculate some integrals involved in the temperature field evaluation of a workpiece during the grinding process. For the case of dry continuous grinding, this calculation allows a faster computation of the temperature field on surface and inside the workpiece.

#### 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: 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 , and for , and and for , 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 and , respectively. Section 4 applies and 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: where , , and are parameters within the integral.

##### 2.1. First Calculation

Applying Fubini's theorem to (2.1), we can exchange the integration order, obtaining Let us develop the exponential within (2.2), as follows: Calling the inner integral in (2.2) and taking into account (2.3), we have Let us perform the change of variables in (2.4), and let us define the variable as so that (2.4) becomes Knowing the following integral representation of the zero order modified Bessel function [4]: and substituting (2.7) in (2.6), we get Remembering the definition of given in (2.5), and substituting (2.8) in (2.2), we find the following expression for : Now, performing in (2.9) the change of variables we obtain In the appendix, it is shown that, Straightforwardly from (2.11), (2.10) can be rewritten as

##### 2.2. Second Calculation

Let us consider now the inner integral in (2.1): Performing the following change of variables in (2.13) we obtain that We can rewrite (2.14) in terms of the error function [4], so that Therefore, substituting (2.15) in (2.1), we obtain that Let us perform now the change of variables in (2.16), so that (2.16) becomes

##### 2.3. Comparison

Comparing (2.12) and (2.17), and the remembering the definition given in (1.1), we finally get, where the function Jg has been defined in (2.11).

#### 3. Second Integral

In order to calculate the second integral defined in (1.2), let us perform the translation of coordinates, thus, Let us define the complex integral along the contour , as presented in Figure 1: Observe that we can divide the integration path as follows: where we have defined: and where and are the semicircumferences appearing in Figure 1. Taking absolute values and performing in (3.5) the changes of variables and along and , respectively, we obtain that Taking limits in (3.6), Taking into account the asymptotic formula [4], we can take limits in (3.7), so that Observe that from (3.4) and (3.10), the following limit is satisfied: Therefore, taking limits in (3.3) and applying the results (3.11), (3.8), we can conclude that Since the integrand in (3.2) contains an unique singularity inside the contour at the point, applying the residue theorem [5], we can contract the contour to the contour the latter surrounding the neighbourhood of the point as shown in Figure 1. Therefore, according to (3.12), where Taking a circular contour around , we can perform in (3.15) the change of variables Taking now into account the asymptotic formula [4], we can take limits in (3.16), Substituting the value of given in (3.13), and simplifying, Finally, from (3.14) and (3.19) we conclude that

#### 4. Application to the Grinding Process

##### 4.1. Samara-Valencia Model

The Samara-Valencia model setup is depicted in Figure 2. The workpiece moves at a constant speed and is assumed to be infinite along and , and semiinfinte along . The plane is the surface being ground. The contact area between the wheel and the workpiece is an infinitely long strip of width located parallel to the axis and on the plane . Both the wheel and the workpiece are assumed rigid.

The Samara-Valencia model [1] solves the convection heat equation, subject to the initial condition and the boundary condition, where and The first term of (4.3) models the application of coolant over the workpiece surface considering as the heat transfer coefficient. The second term, , represents the heat flux entering into the workpiece. This heat flux is generated on the surface by friction between the wheel and the workpiece. The solution of the Samara-Valencia model (4.1)–(4.3) may be presented as the sum of two terms, where Notice that contains the friction function , and contains the temperature field on the surface and the heat transfer coefficient .

##### 4.2. Stationary Regime in Dry Continuous Grinding

Let us consider a wrokpiece being ground with thermal diffusivity and thermal conductivity In the stationary regime, let us define, For the case of dry grinding, and a constant flux entering into the workpiece, we can rewrite (4.6) as, [2] where we have used dimensionless variables, and a normalized temperature, Notice that the numerical solution of the temperature field is cumbersome since (4.8) involves the temperature field at the workpiece surface inside the integrand. We will see in the following section that this integral equation can be solved by using .

##### 4.3. Application of

For simplicity, let us define the functions, for so that we can rewrite (1.2), (4.7), and (4.8) into a simpler form, Defining the Fourier transform as we may apply the convolution theorem of the Fourier transform to (4.13) and (4.15) [3], obtaining Performing the derivative with respect to in (4.14), and applying the Fourier transform to (4.19), we eventually obtain where we have defined Multiplying by (4.17) and taking into account (4.20), we get, Let us divide now (4.18) and (4.22) for , But straightforwardly from (4.11), so that (4.23) can be rearranged, leading to, thus, Substituting (4.26) into (4.18) and comparing the result to (4.22), so that, Performing now the Fourier antitransformation in (4.28), and according to (4.7), Equation (4.29) agrees with [6]. Notice that the numerical computation of in (4.30) is now quite more simple.

##### 4.4. Application of

According to the integral representation of the Bessel function (2.7), we can rewrite (4.30) as By Fubini's theorem, we may rearrange (4.31) as The inner integral in (4.32) can be expressed in terms of the error function [4], Performing the change of variables we get so that the temperature field on the surface, is Applying now (2.18), taking , and we may calculate (4.35),

#### Appendix

#### A. The Function

In order to solve the integral (2.10), we can take advantage of the following result, known as *King's integral *[7]:
Let us define the function Jg as follows:
We can distinguish three different cases, , and

##### A.1. Jg for

Straightforwardly from *King's integral*, (A.1), we obtain that

##### A.2. Jg for

We can perform in (A.2) the change of variables , so that
Now, we can apply in (A.4) the result given by *King's integral* (A.1),

##### A.3. Jg for

We can rewrite (A.3) and (A.5) as a single expression, for , where

In order to evaluate Jg at we can perform the following limit Applying now the asymptotic representation of and [4], and taking into account (A.7), we can evaluate (A.8), Finally, according to (A.6) and (A.10), we can express Jg in the following terms:

#### Acknowledgments

The authors wish to thank the financial support received from *Generalitat Valenciana* under grant GVA 3012/2009 and from Universidad Politécnica de Valencia under grant PAID-06-09. This work has been partially supported by the *Structural Funds of the European Regional Development Fund* (ERDF).

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