Research ArticlePHYSICS

Heat flowing from cold to hot without external intervention by using a “thermal inductor”

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Science Advances  19 Apr 2019:
Vol. 5, no. 4, eaat9953
DOI: 10.1126/sciadv.aat9953
  • Fig. 1 Sketch of two situations in which objects with different temperatures are thermally connected.

    Arrows represent the direction of the flow of heat from (light/yellow) or to (dark/purple) the respective warmer object. (A) When an initially hot body is thermally connected at time t = 0 to a colder thermal reservoir held at temperature Tr, its temperature Tb is expected to drop monotonically by the loss of heat Q to the colder reservoir and to approach Tr in the limit t → ∞. (B) Sketch of a process in which Tb undershoots the temperature of the reservoir for t > t0, and heat Q is thereafter temporarily transferred from the chilling body to the warmer reservoir. The lowest temperature of the body Tb,min < Tr is reached at t = tmin when the connection can be removed. (C) Two similarly connected finite heat capacities are expected to smoothly approach thermodynamic equilibrium at a mean temperature T¯, with heat flowing in one direction only and always Tb > Tr. (D) Two bodies showing opposite oscillations in temperature, with an alternating direction of the heat flow and a repeated temporary transfer of heat from cold to hot. The roman numerals (i) to (iv) refer to the four quarters of the period of one full oscillation cycle of Tb(t), as elaborated in the text and in the caption of Fig. 2B.

  • Fig. 2 Equivalent electrical network and illustration of the heat flow within the considered thermal connection between a body with heat capacity C at temperature Tb and another body or a thermal reservoir at Tr.

    (A) The electrical network consists of a Peltier element (Π) with internal resistance R and thermal conductance k in a closed circuit with an ideal inductance L. The oscillatory current I is ultimately driven by the voltages supplied by the thermoelectric effect due to the temperature difference between the cold and the hot end of the Peltier element, and the induced voltage LI. across L (see Eq. 1A). (B) Sketch of the individual contributions to the flow of heat (open arrowheads, arrow lengths not to scale) in Eqs. 1B and 1C for situations when heat is flowing from (filled light/yellow arrows) or to (filled dark/purple arrows) the warmer end of the Peltier element, drawn for one oscillation cycle of Tb(t), as depicted in Fig. 1 (B and D). The thermal oscillator acts during a full period of an oscillation cycle of Tb(t) alternately as a thermoelectric generator (i), a cooler (ii), a generator (iii), and a thermoelectric heater (iv). During all these processes, a small amount of electromagnetic power (LII., green double arrows) is exchanged with the inductor, although the total stored magnetic energy 12LI2 is always less than a fraction Δ0/Tr of the initially deposited excess heat ~ CΔ0 (see text and Fig. 5).

  • Fig. 3 Evolution of the temperature difference between a cooling body and a thermal bath or another finite body, which are connected in an experiment using a thermal inductor.

    (A) Normalized temperature difference (Tb(t) − Tr)/Δ0 between a finite body and a thermal reservoir for L = L* = RC/k and ZT between 0.25 (red) and 5 (blue) in steps of 0.25, obtained from solving Eq. 3. The time is in units of τ* = C/k. The black line represents a corresponding relaxation process with a time constant τ*, which would take place if the Peltier circuit were interrupted from the beginning. If the thermal connection is not removed after reaching the respective Tb,min (dashed line), Tb(t) approaches thermal equilibrium with eventually Tb = Tr in all cases. The inset shows the damped oscillations of both Tb(t) and I(t). (B) Temperatures Tb(t) and Tr(t) of two connected finite bodies with equal heat capacities, relative to the mean initial temperature T¯=[Tb(0)+Tr(0)]/2 and normalized to the initial temperature difference Δ0, for ZT = 5 (time in units of τ*). Tav denotes their average value showing local minima around TbTr (the numbers for Tav were calculated for Δ0/Tr = 0.27). The inset shows the evolution of the total entropy gain as a function of time in corresponding normalized units.

  • Fig. 4 Results from experiments with oscillating thermal circuits containing the equivalent of a thermal inductor.

    (A) Temperature Tb(t) data taken for two configurations of superconducting coils with L = 30 and 58.5 H, respectively. In this type of an experiment, the oscillating thermal circuit is entirely passive. The temperature Tb(t) of a copper cube that has been thermally connected to a heat reservoir held at Tr = 295 K and initially heated by Δ0 = Tb(0) − Tr ≈ 82 K substantially undershoots with respect to Tr by ≈1.7 K for L = 58.5 H. The inset shows the respective equivalent electrical network, including a parasitic electrical resistance Rs in series due to electrical leads and connections. The solid lines are Tb(t) data obtained by solving the corresponding relevant differential equations using the parameters from a global fit to the data shown in (B) and with Rs = 21 and 43 milliohms for L = 30.0 and 58.5 H, respectively. (B) Experiments using a gyrator-type substitute of an electric inductor with a nominal Rs ≈ 0. Main panel: Temperature Tb(t) for four different values of nominal inductance L, with a maximum undershoot of Tb(t) with respect to Tr by ≈2.7 K for L = 90.9 H. The Tb(t) data from (A) (green and purple dots) are included for comparison. Inset: Evolution of the electric current flowing through the Peltier element. The solid black lines correspond to a global fit to the four datasets according to the relations given in the main text, with the fitting parameters C, R, k, and ZT.

  • Fig. 5 The different fractions of the rates of energy flow in the oscillating thermal circuit for the experiment with L = 58.5 H shown in Fig. 4A.

    The roman numerals (i) and (ii) refer to the definition provided in the text and in the caption of Fig. 2B. The subsequent stages (iii) and (iv) are not discernible in these data because of the still quite low value ZT = 0.432 (see also Fig. 3A). The thermoelectric contributions αTbI and αTrI (right scales) dominate all the other terms in Eqs. 1B and 1C (left scales).

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/4/eaat9953/DC1

    Section S1. Maximum possible cooling effect

    Section S2. Two finite bodies with different temperatures

    Section S3. Effect of a finite resistance Rs in series

    Section S4. Analogy to a thermal inductor

    Section S5. Pictures of the experimental setups

    Fig. S1. Optimized values for a maximum temperature undershoot.

    Fig. S2. Experimental setup of the oscillating thermal circuit.

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Maximum possible cooling effect
    • Section S2. Two finite bodies with different temperatures
    • Section S3. Effect of a finite resistance Rs in series
    • Section S4. Analogy to a thermal inductor
    • Section S5. Pictures of the experimental setups
    • Fig. S1. Optimized values for a maximum temperature undershoot.
    • Fig. S2. Experimental setup of the oscillating thermal circuit.

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