Research Papers: Conduction

Solutions for Transient Heat Conduction With Solid Body Motion and Convective Boundary Conditions

[+] Author and Article Information
Robert L. McMasters

Department of Mechanical Engineering, Virginia Military Institute, Lexington, VA 24450mcmastersrl@vmi.edu

James V. Beck

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824beck@egr.msu.edu

J. Heat Transfer 130(11), 111301 (Aug 28, 2008) (8 pages) doi:10.1115/1.2944243 History: Received July 14, 2007; Revised November 27, 2007; Published August 28, 2008

The analytical solution for the problem of transient thermal conduction with solid body movement is developed for a parallelepiped with convective boundary conditions. An effective transformation scheme is used to eliminate the flow terms. The solution uses Green’s functions containing convolution-type integrals, which involve integration over a dummy time, referred to as “cotime.” Two types of Green’s functions are used: one for short cotimes comes from the Laplace transform and the other for long cotimes from the method of separation of variables. A primary advantage of this method is that it incorporates internal verification of the numerical results by varying the partition time between the short and long components. In some cases, the long-time solution requires a zeroth term in the summation, which does not occur when solid body motion is not present. The existence of this zeroth term depends on the magnitude of the heat transfer coefficient associated with the convective boundary condition. An example is given for a two-dimensional case involving both prescribed temperature and convective boundary conditions. Comprehensive tables are also provided for the nine possible combinations of boundary conditions in each dimension.

Copyright © 2008 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 3

Zeroth eigenvalues for XU23 and XU32 cases as a function of Pe. For the XU23 case, BiL=1 and 5 and for the XU32 case, Bi0=1 and 5.

Grahic Jump Location
Figure 4

Minimum Pe for the existence of a zeroth term for the XU33 case, as a function of Bi at the right and left hand boundaries of the body. Unlike the XU23 and XU32 cases, the zeroth term can exist for XU33 at very small values of Pe.

Grahic Jump Location
Figure 1

Schematic diagram of moving body with convective boundary conditions

Grahic Jump Location
Figure 2

Zeroth eigenvalues for XU12, XU22, and XU22 cases as a function of Pe. This figure is also valid for the XU13 case by using the XU12 curve with the Pe on the abscissa replaced by 2BiL+Pe and also for the XU31 case by using the XU21 curve with the Pe on the abscissa replaced by Pe−2Bi0.



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