Electrical conduction and Joule effect in one-dimensional chains of ...

Electrical conduction in chains of beadsFig. 6 Variation of voltage U as a function of current I . Chain of 5stainless beads (diameter 11 mm). Comparison between experimentaldata (×) and analytical model (−) with the adjusted parameters (Eq. 6).Decreasing current: α ↓ =+1.1×10 −3 and β ↓ = −6.9×10 −4 ;increasingcurrent α ↑ =+2.1 × 10 −3 and β ↑ =+7.0 × 10 −4of beads has a slightly higher resistance to the flow of electriccharge R lc↑ ≃ 2.1 (part DE, Figs. 2 and 6) thanforthe weak decreasing currents R lc↓ ≃ 2.0 (part CD, Fig. 6).This difference is mainly due to the thermal delays associatedto the relatively fast evolution of the electric current. Indeed,for a given small current in the decreasing phase (part CD),the bulk of the beads presents a lower temperature and thus alower measured voltage compared to those of the increasingphase of current (part DE). Thus the chain of beads has anelectromechanical behavior of hysteretic type for which thevoltage–current characteristics present a loop (Fig. 6). Thelarger is the loop, the greater is the thermal energy releasedin the chain and consequently the more the beads warm up.These phenomena are more pronounced for cycles of higherelectric currents, as shown later in Fig. 7.4 Modeling of the hysteretic behavior CD-DEIn the following, the measured hysteretic current–voltagecharacteristics (parts CD and DE) are compared with thoseresulting from an analytical model [7,8,12–14,24].The two beads in contact are considered as two homogeneousand isotropic metal conductors with large dimensionscompared to the contact radius. The electric currentgoes into and leaves by surface ends S 1 and S 2 (Fig. 3).The heat flux only runs out through these same surfaces.For side surfaces, the losses by conduction, convection orradiation are neglected. It should be noted that the thermalconductivities of the air (λ air ≃ 0.026 W m −1 K −1 )andthePVC tubes (λ PVC ≃ 0.16 W m −1 K −1 )aremuchsmallerthan thermal conductivity within the stainless steel (λ steel ≃15 W m −1 K −1 ).Equipotential surfaces in the whole bead are also isothermalsurfaces. The two networks of equipotential andFig. 7 Voltage U,currentI and power P = UI for a chain of 3 stainlessbeads (diameter 11 mm) submitted to four successive cycles ofcurrent. ↗ and ↖:increasingcurrents;↙ and ↘:decreasingcurrentsisotherm present values that depend on the boundary conditionsimposed on the potential and the temperature. Moreover,the equipotential network is independent of the heatflux. Taking into account the previous considerations, phenomenaare considered in a steady-state, and the equationof energy balance leads to the Kohlrausch relation. It providesthe following expression for the potential difference ubetween the contact surface and the surface at temperatureT :u 2 (T, T c ) = 2∫ T cTρ(θ)λ(θ) dθ, (3)where T c , λ and ρ are respectively the temperature of thecontact surface, the thermal conductivity and the electricalresistivity.The conservation equation of the charged particle flow1 −−→ρ(T 0 ) grad[vlc (T 0 )]=ρ(T 1 −−→) grad[u(T )] allows to obtain thepotential difference v lc (between the contact surface and surfaceS 0 )forweakelectricalcurrentswithnoheatflux:∫ T cλ(θ)v lc (T 0 , T c ) = ρ(T 0 ) dθ. (4)U(θ)T 0The dependencies of the electrical resistivity and thermalconductivity on the temperature are considered linear andrespectively of the form:ρ(T ) = ρ r {1 + α(T − T r )}[ m],λ(T ) = λ r {1 + β(T − T r )} [Wm −1 K −1 ], (5)where ρ r ≃ 0.810 −6 mandλ r ≃ 15 W m −1 K −1 arereference values of the electrical resistivity and thermal conductivityat temperature T r ≃ 293 K [22,25]. The values ofcoefficients α and β can be obtained by minimization of thedifference between the voltage–current (U −I ) model (Eq. 6)and the experimental data. It is necessary because of the lack123