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RESEARCH PAPERS: Conduction

The Effect of Internal Temperature Gradients on Regenerator Matrix Performance

[+] Author and Article Information
K. L. Engelbrecht

Department of Mechanical Engineering,  University of Wisconsin, 1500 Engineering Drive, Madison, WI 53706kengelbrecht@wisc.edu

G. F. Nellis, S. A. Klein

Department of Mechanical Engineering,  University of Wisconsin, 1500 Engineering Drive, Madison, WI 53706

J. Heat Transfer 128(10), 1060-1069 (Mar 20, 2006) (10 pages) doi:10.1115/1.2345428 History: Received May 27, 2005; Revised March 20, 2006

Background . One-dimensional regenerator models treat the solid material as a lumped capacitance with negligible temperature gradients. Advanced regenerator geometries operating at low temperatures or active magnetic regenerators which use a liquid heat transfer fluid may have temperature gradients in the solid regenerator that significantly affect performance. It is advantageous to utilize a one-dimensional, or lumped, model of the regenerator that is coupled with a correction factor in order to account for the impact of the internal temperature gradients. Previous work relative to developing such a correction factor is shown here to be inadequate or only valid over a limited range of dimensionless conditions. Method of Approach . This paper describes a numerical model of a sphere subjected to a time varying fluid temperature (representing a passive process) or time varying internal heat generation induced by a magnetic field (representing an active magnetic process). The governing equations are nondimensionalized and the efficiency of the sphere is presented as a function of the Fourier number and Biot number. Results . An approximate correction (or degradation) factor is obtained based on these results that is valid over a wide range of dimensionless conditions and therefore useful to regenerator designers. The degradation factor correlation was developed for a sinusoidal variation in the fluid temperature, however, the same results can be applied to different functional forms of the time variation using the concept of an effective cycle time that is weighted by the magnitude of the driving temperature difference. Conclusions . The heat transfer degradation factor presented here can be applied to one-dimensional regenerator models in order to accurately account for the transient performance of a matrix with finite thermal conductivity. This degradation factor allows regenerator models to approximately account for internal temperature gradients without explicitly modeling them and therefore remain computationally efficient while improving the range of applicability and accuracy.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Reduced temperature as a function of reduced time for various values of reduced radius with Bi=2.5 and Fo=0.25

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Figure 2

Total efficiency as a function of FoBi for various values of Fo. Also shown is the efficiency in the limit of infinite thermal conductivity.

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Figure 3

Reduced temperature as a function of r̃ for various values of t̃ for Fo=0.1 and FoBi=1.0. Note that the cycle time is insufficient to allow the entire sphere to participate in the energy storage process and so only the outer edge temperature is affected by the fluid.

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Figure 4

Efficiency predicted by the numerical model and using lumped capacitance model corrected with the Jeffreson and Hausen degradation factor. Efficiency is shown as a function of the product of the Biot and Fourier number for various values of the Fourier number.

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Figure 5

Degradation factor predicted by the numerical model as a function of the product of the Biot and Fourier number for various values of the Fourier number

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Figure 6

Total efficiency computed using the corrected degradation factor presented in Eqs. 55,56 and predicted by the numerical model as a function of the product of the Biot and Fourier number for various values of the Fourier number

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Figure 7

Reduced temperature as a function of reduced time for various values of reduced radius for a sphere with (a)Bi=0.2, Fo=5.0, (b)Bi=5.0, Fo=5.0, and (c)Bi=5.0, Fo=0.2

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Figure 8

Total efficiency for the active and passive processes as a function of the product of the Biot and Fourier number for various values of the Fourier number. Note that the behavior of these processes is identical when normalized in this manner.

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Figure 9

Total efficiency for the passive process with a sinusoidal and step-wise variation in the fluid temperature as a function of the product of the Biot and Fourier number for various values of the Fourier number. Note that the efficiency of the step-wise variation is always higher than the sinusoidal variation due to the larger fluid-to-surface temperature difference.

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Figure 10

Total efficiency for a sinusoidal, step-wise, and linear variation all with Fo=0.1 as a function of the product of the Biot and Fourier number. Also shown are the total efficiency of sinusoidal processes with Fo=0.157 and Fo=0.0785 which are the appropriate effective Fourier numbers to characterize the step-wise and linear processes, respectively. Note that the sinusoidal processes with appropriately modified Fourier number more closely match the behavior of the nonsinusoidal variations.

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Figure 11

Total efficiency for step-wise variation as a function of the product of the Biot and Fourier number for various values of Fo. Also shown are the total efficiency of sinusoidal processes with the Fo modified according to Eq. 66. Note that the sinusoidal processes with appropriately modified Fourier number closely match the behavior of the nonsinusoidal variations.

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Figure 12

Total efficiency for a linear variation as a function of the product of the Biot and Fourier number for various values of Fo. Also shown are the total efficiency of sinusoidal processes with the Fo modified according to Eq. 66. Note that the sinusoidal processes with appropriately modified Fourier number closely match the behavior of the nonsinusoidal variations.

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