Heat transfer across a solid

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Heat transfer across a solidHeat transfer across a solid–solid interface obtained
by machining in a lathe
A. Ba ıri
∗， N. Laraqi
Department of Heat Transfer， University of Paris 10， LEEE， EA. 387 1， Chemin Desvalli` eres， 92410 Ville dAvray， France
Received 6 July 2004; received in revised form 10 February 2005; accepted 10 February 2005
Abstract
In this paper the heat transfer across the interface between two cylinders in axial contact for which the surfaces are obtained by machining
in a lathe was studied. The macro-contact generated by this process has a spiral shape. The governing equations being difficult to solve for this
geometric configuration， the spiral was simulated by multiple concentric circular annular contacts. Thus， the problem is two-dimensional and
hence easier to solve. An exact explicit analytical solution was developed in order to calculate the temperature distribution and the thermal
contact resistance Rc between the two cylinders. The Rc is expressed as a function of the ratio of real contact area to the apparent one， and
the number of contacts. A simple correlation is also proposed to calculate the Rc with high accuracy.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Machining; Lathing; Heat transfer; Contact; Interface
1. Introduction
Heat transfer across a solid–solid interface is an impor-
tant phenomenon in a wide range of mechanical technolo-
gies. This phenomenon is usually modeled by the thermal
contact resistance， which has received much attention over
the last decades. Several models [1–5] have been developed
to calculate its evolution as a function of geometric and ther-
mal characteristics of materials. These models are generally
based on a unique or multiple contacts with various arrange-
ment or shape (elliptic， circular， rectangular and square).
In this paper， the thermal phenomenon due to annular con-
tacts was studied. This configuration is encountered in ma-
chine tools when the circular surfaces of cylinders are ob-
tained by machining in a lathe (Fig. 1). The macro-contact
generated by this process has a spiral shape which is due to
the combination of rectilinear and rotative motions between
the tool and the work-piece. In order to study the macro-
constriction phenomenon due to this contact， the spiral is
approximated by multiple concentric circular annular con-
tacts.
∗ Corresponding author. Tel.: 33 1 47 09 70 30; fax: 33 1 47 09 30 67.
E-mail address: abairi###u-paris10.fr (A. Ba ıri).
An analytical solution is developed by using the finiteHan-
kel transform for solving the governing equations in steady
state. In order to calculate the total thermal constriction re-
sistance， we consider that all the contacts are subjected to
the same average contact temperature. The heat flux applied
to each annular contact is assumed to be uniform over the
contact and its value depends on the contact location. The
average contact temperature being the same for all contacts，
we calculate the total constriction resistance of cylinder by
adding the individual constriction resistances in parallel. This
procedure has been recently usedwith success in randomcon-
tacts problems [6]. Themicro-constrictionwas assumed to be
lower than the macro-constriction.
2. Description of the contact
Let us consider two cylinders in axial contact， as shown
in Fig. 2. The real contact area is modeled by multiple an-
nular contacts (a) which simulate the spiral generated by the
machining in a turn. The flux lines converge to the annu-
lar contacts (b). Thus， the temperature field presents a jump，

Tc， at the vicinity of the interface (c). This phenomenon is
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.jmatprotec.2005.02.261