Thermomagnetic Mechanism for Self-Cooling Cables

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Thermomagnetic Mechanism for Self-Cooling Cables

Luca de’ Medici

Phys. Rev. Applied 5, 024001 – Published 1 February 2016

Abstract

A solid-state mechanism for cooling high-current cables is proposed based on the Ettingshausen effect, i.e., the transverse-thermoelectric cooling generated in magnetic fields. The intense current running in the cable generates a strong magnetic field around it that can be exploited by a small current running in a coating layer made out of a strong “thermomagnetic” material to induce a temperature difference between the cable core and the environment. Both analytical calculations and realistic numerical simulations for the steady state of bismuth coatings in typical magnetic fields are presented. The latter yield temperature drops $≃ 60 K$ and $> 100 K$ for a single- and double-layer coating, respectively. These encouraging results should stimulate the search for better thermomagnetic materials in view of applications such as self-cooled superconducting cables working at room temperature.

Received 18 August 2015

DOI:https://doi.org/10.1103/PhysRevApplied.5.024001

Physics Subject Headings (PhySH)

Electrical conductivity Ettingshausen effect Thermomagnetic effects

Energy applications Superconductors

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Luca de’ Medici^*

European Synchrotron Radiation Facility, 71 Avenue des Martyrs, Grenoble, France and Laboratoire de Physique et Etude des Matériaux, UMR8213 CNRS/ESPCI/UPMC, Paris, France

^*demedici@esrf.fr

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Vol. 5, Iss. 2 — February 2016

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Images

Figure 1
Upper panel: Illustration of the Ettingshausen effect in a slab of rectangular section. A current flowing in the $\hat{z}$ direction in the presence of a magnetic field oriented in the $\hat{y}$ direction generates a temperature gradient in the $\hat{x}$ direction. Lower panel: Sector of a section of the coaxial cable in the proposed setup. The (super)conductor carrying the main current is depicted in gray ( $r < r_{0}$ ) and the thermomagnetic material in beige ( $r_{0} < r < R$ ). Electrical insulation is implied between the layers. The magnetic field $B$ is generated by the main current $I$ , whereas the temperature gradient is the outcome of the Ettingshausen effect due to the current $J$ . The direction of $J$ relative to $I$ (parallel or antiparallel) is chosen according to the sign of the Nernst-Ettingshausen coefficient in the used thermomagnetic material, i.e., so that the temperature gradient is directed outward.
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Figure 2
Maximum relative temperature drop [i.e., for $E_{z} = E_{z}^{\max} (α)$ ] for a model of circular Ettingshausen cooler with constant coefficients ( $ρ = 2.5 m Ω cm$ , $K = 0.062 W / K cm$ , $N = 35 μ V / K T$ , yielding $Z \equiv 0.00125 K^{- 1}$ roughly representative of bismuth at ambient temperature and $B ≃ 12 T$ ) for a magnetic field at the interface between the cable core and the thermomagnetic coating of $B_{0} = 12.6 T$ (e.g., generated by a current of density $10^{5} A / {cm}^{2}$ running in a core of radius $r_{0} = 2 cm$ ) plotted as a function of the thickness $α$ . The results of the numerical solution (squares) and the analytical result Eq. (6) are plotted, validating the minor approximation used in the analytical treatment (see Appendix pp2). The red full dots represent the numerical results for the same parameters but for a magnetic field artificially kept constant ( $B = B_{0}$ ) at all distances. The black line is the result for a rectangular cooler Eq. (1) at the same $B = B_{0}$ . These results show that the order of magnitude of the temperature drop (for zero heat load $q_{in} = 0$ ) is correctly described by Eq. (6) (being approximately 50 K for the present parameters), and that, for constant coefficients, in the cylindrical geometry a slow reduction with the material thickness comes from the decay of the magnetic field that supersedes the advantages of the cylindrical shape over the rectangular one in terms of cooling.
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Figure 3
Numerical simulations at maximum cooling [i.e., for $E_{z} = E_{z}^{\max} (α)$ ] for a circular Ettingshausen cooler using realistic material properties for single-crystal bismuth along the crystal axes of maximum thermomagnetic performance and a magnetic field equal to $B_{0} = 12.6 T$ at the interface between the cable core and the thermomagnetic coating (generated by a current of density $10^{5} A / {cm}^{2}$ running in a core of radius $r_{0} = 2 cm$ ). Data in the main plot show (in units indicated in the legend) the temperature drop $Δ T$ , the total expelled heat for the unit cable length, and the average current density running in the thermomagnetic coating, as a function of the thickness of the cooling layer $α = R / r_{0}$ . The improvement for one realization of a double-layer coating (see text) to $Δ T = 105.5 K$ is indicated by the arrow. Inset: Measured figure of merit for Bi (reproduced from Ref. [5]) showing the saturation field and saturation value for $Z T$ at various temperatures. As long as the cooler is immersed in high enough magnetic field so that the thermomagnetic figure of merit assumes its saturation value, $Δ T$ raises with the cooler thickness.
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Figure 4
Maximum relative temperature drop [i.e., for $E_{z} = E_{z}^{\max} (α)$ ] for a model of circular Ettingshausen cooler with constant coefficients (same parameters as in Fig. 2 of the main text) plotted as a function of the thickness $α$ . Besides the results of the numerical solution (squares), the analytical result Eq. (b10) is plotted for two typical choices of $\bar{Z T}$ , as explained in the text. These results show that the order of magnitude of the temperature drop (for zero heat load $q_{in} = 0$ ) is correctly described by Eq. (b10) and illustrate the reduction with material thickness in the cylindrical geometry compared to the rectangular one at fixed $B = B_{0}$ , due to the decay of the magnetic field that supersedes the advantages of the cylindrical shape in this model with fixed coefficients.
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Figure 5
Typical temperature and heat current density distributions in a numerical simulation. Here shown is the run for the set of parameters giving the maximal temperature drop for a cable with a single bismuth layer coating shown in Fig. 3 ( $B_{0} = 12.6 T$ , $r_{0} = 2 cm$ , $R = 5 cm$ , $α = R / r_{0} = 2.5$ , $E_{z} = E_{z}^{\max} = 0.04 V / cm$ ).
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