Temperature vs. Magnetic Pressure at the Surface of semi infinite Plate
Markus J. Löffler Markus Schneider * High Voltage and Pulsed Power Laboratory, EPPL Pulsed Power Technology Division, EPPL * Westphalian Energy Institute French German Research Institute of Saint Louis (ISL) Westfälische Hochschule 68301 Saint Louis, France Neidenburger Str. 43, D 45877 Gelsenkirchen, Germany [email protected] markus.loeffler@w hs.de
Abstract —Transient high magnetic pressures are characteristic material directly under the surface. In combination with strong for pulsed power applications such as electromagnetic forming, magnetically induced mechanical pressures the high thermal electromagnetic acceleration and other applications involving gradients cause additional stress peaks in the surface region of high electric currents. Typically, the current pulses are switched- the conductor. As a consequence small cracks may form in this on rapidly (microsecond timescale) leading to very high current region. Once formed, they grow due to magnetically induced densities at the surface of the current-carrying conductors due to forces. current-field interactions (skin effect): Electromagnetic diffusion is too slow to enable a spatially homogeneous current distribution The temperature at the surface can be calculated numerical inside metal conductors on this time scale. The very high current ly (e.g. [2]). However, numerical results don’t give a physical densities locally generate high ohmic power leading to Joule law or a rule of thumb allowing to understand the relation be heating losses and increasing the conductor’s surface tempera- tween the physical parameters of the material and the surface ture rapidly. Moreover, high local thermal stresses are induced in temperature. Moreover, since both, magnetic pressure and the region close to the surface. The combination of magnetically surface temperature are caused by magnetic flux density – itself and thermally induced stresses and Joule heating can lead to being directly related to the respective current density – a rela severe damage of the conductor, including phase transitions and tion between both quantities is of interest. deformations. II. THEORETICAL MODEL The paper presents a formula that allows estimating the surface temperature of a semi-infinite conductor being exposed to a tran- A. Assumptions sient magnetic pressure at its surface. While this is a textbook A semi infinite plate shall be considered, see Fig. 1. problem, if adiabatic conditions are assumed, the approach taken here considers thermal diffusion inside the conducting material. The presented formula is valid if the conductor’s physical con- stants do not depend on temperature and magnetic flux density.
Keywords - skin effect, thermal shock, magnetic pressure, surface temperature, railgun, electromagnetic forming.
I. INTRODUCTION Magnetic flux densities in the order of 0.1 to 100 T are used e.g. for high velocity forming, for mass acceleration as well as for conductive or inductive heating, welding or hardening of conductive materials. Moreover, flux density pulses above 10 T are applied to investigate physical properties of materials. In any case multiple or single turn coils respectively loops are required to generate the respective magnetic flux. At high current densities as they appear in pulsed power applications thermal fatigue can limit the lifetime of coils even if they are manufactured properly from a mechanical point of view [1]. This is due to the rapid ohmic heating of the conduc tor surface leading to a thermal quenching by the cold bulk FIG . 1: SEMI INFINITE PLATE .
*Research conducted in the scope of the European Pulsed Power Laboratories known as the EPPL The plate consists of a mechanically stable homogeneous x k m material with density ρ, thermal conductivity λ, specific heat B zHx, tL = B0 erfc (6) 2 t cp, electrical conductivity κ, and magnetic permeability =4 ⋅π⋅10 7 H/m (also valid for magnetically saturated iron like materials). These values are considered to be spatially homo The referring current density can be calculated with geneous and to be independent from T. The surface of the plate jx x, t 0 is placed in the y z area. In x and in z direction its dimensions H L 1 jyHx, tL = õä 0 (7) are infinite. The volume outside of the plate is assumed to have m no thermal and electrical conductivity. At time t=0 s a constant jz Hx, tL B z Hx, tL magnetic flux density B0 is switched on. The B vector points in positive z direction. leading to 2 x2 k m Although the assumptions do not reflect physical reality the 4 t formula derived hereinafter can be used as a rule of thumb if 2 (8) the physical parameters of the plate are interpreted as average B0 2 x k m ‰ jyHx, tL = values with sufficiently small deviations of the real parameters m x 4 t p e.g. due to ohmic temperature increase. The assumption of a semi infinite plate reflects reality if the electromagnetic diffu sion length The PDE describing the diffusion of temperature T(x,t) at local heat power density q(x,t) is given by 2 t ∑T Hx, tL ∑ T Hx, tL dem = 2 (1) r c p = qHx, t L + l (9) k m ∑t ∑ x2
is essentially smaller than the smallest radius of curvature The boundary conditions assumed here correspond to of the conductor. Assuming an instantaneous switch on of standard conditions of heating a semi infinite volume with magnetic field and current represents a ∂B/∂t→∞ at t=0 s; in thermally isolated surface: Pulsed Power Technology the gradient always keeps finite. As argued below the assumption of a step function has no essential T Hx, 0L = T0 impact on the material behavior considered in this paper. ∑ T 0, t = 0 (10) B. Calculus ∑ x H L The Partial Differential Equation (PDE) describing the dif T Hx Ø ¶, tL = T0 fusion of the magnetic field into a conductor is given by 0 0 The ohmic power density is 0 ∑ 0 2 = mk (2) 2 ∑ B x,t t jy x, t zH L ∑ B x, t H L (11) 2 z H L qHx, tL = ∑x k With (8) this yields The boundary conditions for the assumed geometry are x2 k m B 0, t = B 2 t 2 zH L 0 2 ‰ B0 (12) q x, t = p with p = B x Ø ¶, t = 0 (3) H L mag,0 mag,0 zH L p t 2 m B x > 0, 0 = 0 T zH L where pmag,0 represents the magnetic pressure at the conductor’s surface. Substitute (12) into (9) gives The essential part of the PDE can be rewritten: x2 k m 2 ∑ B z x, t ∑ B z x, t ∑T x, t B2 1 ‰ 2 t ∑2 T x, t H L = mk H L H L 0 H L (13) 2 2 t r c p = + l x x ∑ (4) ∑t m p t ∑ x2 t ∑ J x2 N t Like (4) this can be transformed to Further transformations give ∑2 T h h ∑T h 4 p 2 H L H L mag,0 h2 ∑ B zHhL mk ∑ B zHhL x + = ‰ (14) 0 = + h where h = (5) ∑h2 r2 ∑h p r c r2 2 2 ∑h p ∑h t where r, the ratio between electromagnetic and electro This leads to the result [3] thermal diffusion length, is given by IV. FURTHER INTERPRETATION dth k l m r = = (15) From (17) the temperature difference between the initial dem r c p and the final temperature at the surface (x=0) can be calculated analytically: with the thermal diffusion length ArcTanh 1 2 r2 4 êp B F DTs t = T 0, t T0 = pmag,0 (18) l t @ D H L r c dth = 2 (16) p 1 2 r2 r c p
Note that the temperature difference does not depend on t. For metals one obtains r<0.2 (except Nickel with r=0.56) Under the assumptions given above the temperature at the what means that the electromagnetic diffusion is always at least surface instantaneously jumps to a constant value according to 5 times faster than the thermal diffusion. (18).
III. RESULTS The ratio of Ts and pmag,0 is given by Further calculation leads to ArcTanh 1 2 r2 DT s 4 p B F = ê (19) p r c mag,0 p 1 2 r2 (17) Fig. 3 illustrates (19) for different materials. For the most commonly used metals Al and Cu the obtained skin tempera ture increases are about 1.6 °C/MPa and about 1.0 °C/MPa Application to Aluminum (κ=37.7 MS/m, ρ=2700 kg/m 3, 1 1 1 1 respectively. λ=235 W⋅K m , cp=897 J⋅kg K ), assuming T0=18°C and pmag,0 =400 MPa, Fig. 2 shows the profile of the temperature together with the profiles of the flux and current densities at t=50 µs:
FIG . 3: RATIO OF TEMPERATURE INCREASE AND MAGNETIC PRESSURE AT THE SURFACE . FIG . 2: DISTRIBUTION OF FLUX DENSITY B, CURRENT DENSITY J , AND TEMPERATURE T IN AN AL PLATE . To avoid melting of the surface the skin temperature has to be kept smaller than the melting temperature of the material. At the surface of the conductor the flux density is about From (19) the maximum allowed magnetic pressure is given by 30 T, the current density is about 15 kA/mm 2 and the tempera ture reaches about 660°C corresponding to the melting temper p 1 2 r2 ` 4 ature of Al. Below the surface, the drawn curves are rapidly pmag,0 = r c p HTmelt T0L (20) decreasing. However, the obtained values confirm a rule of ArcTanh 1 2 r2 thumb for electromagnetic forming stating that Al should not B F be treated with magnetic pressures larger than 350 MPa [4].
Fig. 4 visualizes (20) for different metals. Taking again Al pressure of 360 MPa leads to a maximum temperature increase and Cu as examples it can be seen that the maximum allowed at the surface of a copper conductor (r=0.093) of about 284°C magnetic pressures at the surface are about 400 MPa and about rather than of 365°C as predicted with (19). However, (19) and 1100 MPa respectively. (20) give a first estimate for these technical important relations. Therefore, they are useful for the understanding of principal limits of high current applications.
V. CONCLUSION
Depending on the conducting material being used high cur rents generating high magnetic flux densities and high magnet ic pressures can lead to a melting of the conductors’ surfaces. To avoid melting the respective amplitudes have to be restrict ed. Especially with respect of the classical conductor materials Al and Cu the magnetic pressure has to be kept in the order of 400 MPa respectively 1100 MPa. If higher magnetic pressures shall be generated and if surface melting of the material shall be avoided the duration of current pulses has to be adapted in a suited way: The higher the magnetic pressure the higher the pulse duration. However, due to steep temperature gradients in the material the lifetime of the conducting structure will be restricted. When developing repetitively working Pulsed Power devices for commercial applications the skin temperature as well as the gradient has to be kept in proper limits. These limits strongly depend on the conductor material.
FIG . 4: MAXIMUM ALLOWED MAGNETIC PRESSURE REFERENCES AT THE SURFACE . [1] K. Svendsen, S.T. Hagen, “Thermo Mechanical Fatigue Life Estimation The findings presented so far were derived for theoretical of Induction Coils”, Int. Sc. Colloquium Modelling for Electromagnetic assumptions mentioned in II.A, in particular a step function Processing, Hannover, Oct. 27 29,2008, pp. 127 132. was used for the flux density. In reality the rise of a magnetic [2] T. Emilia Motoasca, Peter M. vand den Berg, Hans Blok, Martin D. Verweij: “Inclusion of Temperature Effects in a Model of field or of a current has a finite gradient. This leads to a slower Magnetoelasticity”, IEEE Trans. On Magnetics, Vol. 42, No. 3, March increase of the power density in the conductor skin than given 2006, pp 369 377. in (12). Hence, in reality the temperature increase will be [3] Günther Lehner: Elektromagnetische Feldtheorie für Ingenieure und weaker due to better cooling conditions. To give an example: a Physiker. 1996. ISBN 3 540 60373 5. numerical simulation with a sinusoidal field pulse of 30 µs rise [4] http://www.magneform.com/pres.html . 07.09.2012. time and an amplitude corresponding to a maximum magnetic