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Research Papers

An Accurate and Stable Numerical Method for Solving a Micro Heat Transfer Model in a One-Dimensional N-Carrier System in Spherical Coordinates

[+] Author and Article Information
Weizhong Dai1

College of Engineering and Science,  Louisiana Tech University, Ruston, LA 71272dai@coes.latech.edu

Da Yu Tzou

Department of Mechanical and Aerospace Engineering,  University of Missouri, Columbia, MO 65211tzour@missouri.edu

1

Corresponding author.

J. Heat Transfer 134(5), 051005 (Apr 13, 2012) (7 pages) doi:10.1115/1.4005635 History: Received April 12, 2010; Revised September 07, 2010; Published April 11, 2012; Online April 13, 2012

We consider the generalized micro heat transfer model in a 1D microsphere with N-carriers and Neumann boundary condition in spherical coordinates, which can be applied to describe nonequilibrium heating in biological cells. An accurate Crank–Nicholson type of scheme is presented for solving the generalized model, where a new second-order accurate numerical scheme for the Neumann boundary condition is developed so that the overall truncation error is second order. The scheme is proved to be unconditionally stable and convergent. The present scheme is then tested by three numerical examples. Results show that the numerical solution is much more accurate than that obtained based on the Crank–Nicholson scheme with the conventional method for the Neumann boundary condition. Furthermore, the convergence rate of the present scheme is about 1.8 with respect to the spatial variable, while the convergence rate of the Crank–Nicholson scheme with the conventional method for the Neumann boundary condition is only 1.0 with respect to the spatial variable. The scheme is ready to apply for thermal analysis in N-carrier systems.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 6

Normalized electron temperature change with time on the surface (r = 0.1 μm) of the sphere

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Figure 7

(a) Electron temperature distribution and (b) lattice temperature distribution along the r axis

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Figure 5

Numerical solutions and the corresponding exact solutions at t = 0.5

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Figure 4

Numerical solutions and the corresponding exact solutions at t = 0.2

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Figure 3

Maximum errors of numerical solutions as compared with the exact solution versus time t

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Figure 2

Comparison of numerical solutions with the exact solution for the first example

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Figure 1

Mesh and locations of grid points

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