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Research Papers: Conduction

# Boundary Control of Temperature Distribution in a Rectangular Functionally Graded Material Plate

[+] Author and Article Information
Hossein Rastgoftar

Department of Mechanical Engineering, Shiraz University, Shiraz, Fars 71348-51154, Iran

Departments of Mechanical and Electrical Engineering, Shiraz University, Shiraz, Fars 71348-51154, Iran

J. Heat Transfer 133(2), 021304 (Nov 03, 2010) (6 pages) doi:10.1115/1.4002437 History: Received July 11, 2009; Revised May 21, 2010; Published November 03, 2010; Online November 03, 2010

## Abstract

In this paper, an analytical method and a partial differential equation based solution to control temperature distribution for functionally graded (FG) plates is introduced. For the rectangular FG plate under consideration, it is assumed that the material properties such as thermal conductivity, density, and specific heat capacity vary in the width direction, and the governing heat conduction equation of the plate is a second-order partial differential equation. Using Lyapunov’s theorem, it is shown that by applying controlled heat flux through the boundary of the domain, the temperature distribution of the plate will approach a desired steady-state distribution. Numerical simulation is provided to verify the effectiveness of the proposed method such that by applying the boundary transient heat flux, in-domain distributed temperature converges to its desired steady-state temperature.

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Copyright © 2011 by American Society of Mechanical Engineers
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## Figures

Figure 1

Geometry and coordinates of the FGM plate

Figure 2

Desired temperature distribution in rectangular FGM plate with spatially varying property

Figure 3

Final (steady-state) temperature distribution T in °C after application of boundary control heat fluxes for Δx=0.01, Δy=0.01, and Δt=0.01 (plate with spatially varying thermal properties)

Figure 4

Final (steady-state) temperature distribution T in °C after application of boundary control heat fluxes for Δx=0.01, Δy=0.005, and Δt=0.01 (plate with spatially varying thermal properties)

Figure 5

Final (steady-state) temperature distribution T in °C after application of boundary control heat fluxes for Δx=0.005, Δy=0.01, and Δt=0.005 (plate with spatially varying thermal properties)

Figure 6

Final (steady-state) temperature distribution T in °C after application of boundary control heat fluxes for Δx=0.005, Δy=0.005, and Δt=0.01 (plate with spatially varying thermal properties)

Figure 7

Final (steady-state) temperature distribution T in °C after application of boundary control heat fluxes for Δx=0.005, Δy=0.005, and Δt=0.005 (plate with spatially varying thermal properties)

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