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Research Papers: Conduction

Solid-State Thermal Rectification With Existing Bulk Materials

[+] Author and Article Information
C. Dames

Department of Mechanical Engineering and Materials Science and Engineering Program, University of California, Riverside, A311 Bourns Hall, Riverside, CA 92521

J. Heat Transfer 131(6), 061301 (Mar 31, 2009) (7 pages) doi:10.1115/1.3089552 History: Received March 30, 2008; Revised September 30, 2008; Published March 31, 2009

A two-terminal thermal device exhibits thermal rectification if it transports heat more easily in one direction than in the reverse direction. Within the framework of classical heat conduction by Fourier’s law, thermal rectification occurs in a two-segment bar if the thermal conductivities of the segments have different dependencies on temperature. The general solution to this problem is a pair of coupled integral equations, which in previous work had to be solved numerically. In this work the temperature dependencies of the thermal conductivities are approximated using power laws, and perturbation analysis at low thermal bias leads to a simple algebraic expression, which shows that the rectification is proportional to the difference in the power-law exponents of the two materials, multiplied by a geometric correction function. The resulting predictions have no free parameters and are in good agreement with the experimental results from the literature. For maximum rectification, the thermal resistances of the two segments should be matched to each other at low thermal bias. For end point temperatures of 300 K and 100 K, using common bulk materials it is practical to design a rectifier with rectification of well over one hundred percent. A new quantity, the normalized thermal rectification, is proposed to better facilitate comparisons of various rectification mechanisms across different temperature ranges.

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Figures

Grahic Jump Location
Figure 1

Basic concept of thermal rectification due to temperature-dependent thermal conductivity k(T). A two-segment bar is made of materials with different trends in their k(T), for example, k1∝T3 and k2∝T−3. (a) For heat flow from left to right, segment 1 is relatively hot while segment 2 is relatively cold, causing both segments to have relatively high k. (b) These trends are reversed for heat flow from right to left, causing both segments to have relatively low k. Thus this structure exhibits thermal rectification. The graphical interpretation is that Tj must be chosen so that the appropriate areas under each g(T) curve are equal.

Grahic Jump Location
Figure 2

Temperature-dependent thermal conductivity of selected materials, showing that the power-law exponents n commonly range from −3 to +3, with extreme values from −3.5 to +5.4 noted here. The data are from Refs. 28-31.

Grahic Jump Location
Figure 3

Heat flux as a function of thermal bias for materials with several different combinations of power-law exponents (n1,n2). This plot is closely analogous to the “I-V curve” of a conventional electrical diode.

Grahic Jump Location
Figure 4

Thermal rectification γ as a function of thermal bias for materials with several different combinations of power-law exponents (n1,n2). Inset: detail at low bias, including the linearized approximation from Eq. 24 (dashed lines).

Grahic Jump Location
Figure 5

Effect of the thermal matching parameter ρ on the rectification for materials with several different combinations of power-law exponents (n1,n2). Main plot: high thermal bias (Δ=1). Inset: low thermal bias (Δ=0.1). The full calculation refers to Eqs. 14,15, while the linearized calculation is Eq. 24.

Grahic Jump Location
Figure 6

Comparison with the experimental measurements of Jezowski and Rafalowicz (16) for a bar made of graphite and quartz. “Full theory” denotes the numerical solution of Eqs. 1,2,3,4,5,6 based on the reported k(T) for bulk graphite and bulk quartz from Ref. 16. Linearized theory and “second-order theory” denote Eqs. 24,23 respectively, which approximate the reported bulk k(T) by power laws with our fits for n1 and n2.

Grahic Jump Location
Figure 7

Comparison with the experimental measurements of Balcerek and Tyc (15) for a bar made of brass and tin. Full theory denotes the numerical solution of Eqs. 1,2,3,4,5,6 based on the reported k(T) for bulk brass and bulk tin from Ref. 15. Linearized theory denotes Eq. 24, which approximates the reported bulk k(T) by power laws with our fits for n1 and n2.

Grahic Jump Location
Figure 8

Modeling temperature-dependent contact resistances in the present framework of a two-segment bar. (a) Contact resistances (1) and (2) in series with a bar of thermal conductivity k3. (b) When the thermal resistance of the central bar (3) is negligible, the system becomes formally equivalent to a two-segment bar (as in Fig. 1).

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