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Research Papers: Conduction

# Approximation of Transient 1D Conduction in a Finite Domain Using Parametric Fractional Derivatives

[+] Author and Article Information
Sergio M. Pineda, Carlos F. M. Coimbra

School of Engineering, University of California, 5200 North Lake Rd., Merced, CA 95343

Gerardo Diaz

School of Engineering, University of California, 5200 North Lake Rd., Merced, CA 95343gdiaz@ucmerced.edu

J. Heat Transfer 133(7), 071301 (Apr 01, 2011) (6 pages) doi:10.1115/1.4003544 History: Received May 23, 2010; Revised January 11, 2011; Published April 01, 2011; Online April 01, 2011

## Abstract

A solution to the problem of transient one-dimensional heat conduction in a finite domain is developed through the use of parametric fractional derivatives. The heat diffusion equation is rewritten as anomalous diffusion, and both analytical and numerical solutions for the evolution of the dimensionless temperature profile are obtained. For large slab thicknesses, the results using fractional order derivatives match the semi-infinite domain solution for Fourier numbers, $Fo∊[0,1/16]$. For thinner slabs, the fractional order solution matches the results obtained using the integral transform method and Green’s function solution for finite domains. A correlation is obtained to display the variation of the fractional order $p$ as a function of dimensionless time $(Fo)$ and slab thickness $(ζ)$ at the boundary $ζ=0$.

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## Figures

Figure 8

Results obtained with the correlation given in Eq. 29. The behaviors of the order of the fractional derivative p at ζ=0 are displayed for a range of values of parameters λ and Fo.

Figure 7

Range of variable order derivative at ζ=0 for several values of λ and for Fo=0.0625

Figure 6

Error of symmetry given in Eq. 23 for several values of λ for Fo=0.0625

Figure 5

Change in the perturbation ε for several values of dimensionless slab size λ for Fo=0.0625

Figure 4

(a) Half-derivative plus disturbance ε, semi-infinite domain, and integral transform solutions match for Fo=0.0625. ((b)–(d)) Numerical approximations from half-derivative plus disturbance (ε) and integral transform are compared for several values of λ.

Figure 3

Comparison of semi-infinite, integral transform, and half-derivative solutions for Fourier number Fo∊[0.005,0.0625]. All methods match satisfactorily.

Figure 2

Comparison of results with integral transform, semi-infinite domain, and finite differences for one-dimensional transient heat conduction with a boundary condition of the second kind at ζ=0. Solutions are shown for Fo=0.0625, but other Fourier numbers show equal matching between solutions using these methods. Steady and unsteady regimes are depicted to highlight the solution domains.

Figure 1

Description of the problem. Constant heat flux is applied from the left to a slab with initial temperature Ti. Temperature distributions at different times are shown.

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