Note: A simple model for thermal management in solenoids Open Access

Solenoid coils have many applications in scientific research and industry, including as solenoidal ion sources¹ and to give nuclear spin-precession. For “air cored” devices, the maximum magnetic field is limited by the resistive heating of the windings and simulations are desirable to maximise the performance of a particular coil. The main obstacle to modelling the thermal properties of solenoid windings is uncertainties in the mean thermal conductivity of the windings. As will be shown in this note, the thermal conductivity is dominated by the thermal conductivity of the material (air, or an epoxy with a relatively good thermal conductivity) that fills the gaps between the wires, and since the thickness of these gaps depends critically on the way the coil is wound it cannot be accurately predicted. Here we determine the mean thermal conductivity and mean heat capacity of the windings by fitting the thermally induced rise in the total winding resistance predicted by a model to that measured for the actual windings. We demonstrate the value of the method by determining the maximum safe current that can be passed through the solenoid coils used for spin precession of helium-3 atoms in the Cambridge Helium spin-echo (HeSE) apparatus,² which probes atomic scale dynamics on pico- to nano-second timescales in a spectroscopic technique using the nuclear spin of helium-3 atoms as an internal timer during scattering from a single crystal surface in ultra-high vacuum (UHV).^3–6 

The geometry of solenoid or electromagnet windings is usually relatively simple, enabling a correspondingly simple finite element model to be constructed. A HeSE spin precession coil is typically wound in a number of sections, with a variable number of layers in each section (Figure 1). Since the coil is long compared to its diameter we only consider radial heat flow. Each layer is treated as a “finite element,” with the heat flow to the next layer calculated in terms of the layer spacing, the mean thermal conductivity of the windings, the temperature difference between layers, and the area between them. The resistivity of the wire is taken as a linear function of temperature with temperature coefficient α, which has a value of 3.9 × 10⁻³ K⁻¹ for copper at 293 K.⁸ The time dependence of the temperature, T_i, above the reference temperature T₀ of the ith layer of the nth section of the coil, of length l_n, can be written as

where λ and C_p are the mean thermal conductivity and the heat capacity per unit volume of the windings, respectively, w is the width of the wire, R_{0, l} is the resistance of the winding wire per unit length at the reference temperature, T₀, Δr is the layer thickness, and I is the current in the wire.

The first term in Eq. (1) describes the heat flowing into a layer from each adjacent layer, and the second describes the heat from power dissipation in that layer, which together yield the change in layer temperature with time. We can simplify the expression in Eq. (1) by writing γ = 2ΔrR_{0, l}/λw and β = 2(Δr)²/κ, where the thermal diffusivity of the coils κ = λ/C_p. This yields

We now apply this model to determine the expected failure current of the spin-precession solenoids in the Cambridge HeSE spectrometer.² The solenoids consist of up to 30 layers of 1 mm thick wire wound on a former of 16.7 mm radius. The epoxy used to fill any gaps between winding layers is rated to 180 °C. In order to help design the coil, thermal properties of a short 30 layer test solenoid were measured,⁷ but having built the actual coil, the method described in this note was used to determine the maximum possible current that could be used.

The complete core and outer edge of the solenoid in sections with 16 or more layers were water cooled, so the water temperature T₀, which was assumed to be constant, was taken as the temperature of these boundaries. The temperature of each layer of each section of the solenoid coil as a function of time was then determined by iteration using Eq. (2).

The time dependence of the resistance of the coil after switching on constant currents of 2 and 5 A was determined by measuring the voltage across the coil. Values of the constants β and γ were then found for which the simulated current and time dependence of the coil resistance matched the measurements for these two currents. The parameters best reproducing the experimental data were β = 35 s and γ = 7.15 × 10⁻² K m A⁻², and simulated results of the fractional change in resistance of the coil for these values of β and γ are shown with thin black solid lines in Figure 2(a), for comparison with the experimental results for solenoid currents of 2 A (bottom; thick red dotted line) and 5 A (top; thick blue dotted line). The residuals, defined here as the fractional difference between the simulated and measured resistances, are shown in Figure 2(b) and are small and approximately constant. The simulation reproduces the experimental data well for both values of solenoid current studied and suggests that the assumptions in this model, notably the neglect of lateral heat transport between sections of the coil, are reasonable. The simulation could also reproduce the measured resistance change on coil cooling when the currents were turned off with the same values of β and γ. Broadly, γ determines the maximum temperature attained, while β affects how long the coil takes to reach this temperature.

When the solenoid coils were manufactured, a crude estimate of the average thermal conductivity ⟨λ⟩ for a layer of thickness Δr was made using⁷ 

where, for example, t_Cu represents the thickness of copper in the layer. Equation (3) yields ⟨λ⟩ = 3.1 W m⁻¹ K⁻¹, however the thickness of epoxy particularly is dependent on the manufacturing process and not accurately known, and air gaps that the epoxy does not fill will also lower the thermal conductivity, making the actual value of ⟨λ⟩ uncertain. The values for β and γ determined above now allow the actual values of the thermal conductivity and thermal diffusivity of the coil to be determined, yielding λ = 1.05 W m⁻¹ K⁻¹, a third of the value previously estimated, and κ = 2.74 × 10⁻⁷ m² s⁻¹, around three orders of magnitude lower than the value for bulk copper. As copper is the medium for most of the heat storage, the mean heat capacity of the coil, represented by the value of λ/κ = C_p, is expected to agree well with those for bulk copper, and indeed the simulations yield C_p = 3.8 × 10⁶ J K⁻¹ m⁻³, in good agreement with the value for bulk copper⁸ of 3.4 × 10⁶ J K⁻¹ m⁻³.

The assumption in this simulation that β and γ are independent of temperature is justified by the good agreement between the simulation results and the data in Figure 2 for constant values of β and γ.

Once β and γ have been determined for the solenoid coil, the dynamic temperature evolution can be calculated using Eq. (2). In Table I the mean maximum temperature in the coil, ⟨T⟩, calculated from the simulated total resistance change, is compared to the corresponding maximum temperature of any single element of the coil, T_max, for a range of values of solenoid current, for simulations where the entire coil starts at T₀ = 16 °C. The table also gives, for comparison, the maximum temperature for a coil made of wire with temperature independent resistance. The importance of layer by layer simulations when considering temperature evolution in a solenoid can now be fully appreciated: if only the average properties of the coil are considered then far higher temperatures in the central layers and their potential for damaging the coil will be overlooked. An estimate of the errors on the predicted temperatures was found by repeating the simulation for the range of values of β and γ within which an acceptable fit to the experimental data (defined by an increase of less than 2 × 10⁴ in the residuals as judged by eye) is obtained. Values of ⟨T⟩ and T_max increase by less than 1% at 2 A and less than 10% at 16 A. Figure 3 compares the simulated temperature in each layer of the central, hottest section of the spin-echo solenoid coils where the maximum 30 layers of wire are present for solenoid currents of 10 A (left) and 12 A (right) at the times indicated in the caption for which a constant solenoid current has been applied. Coil temperatures do not exceed the thermal limit of the coils (dashed red line) at steady state for 10 A solenoid current. However, the simulation predicts that the estimated failure temperature of 180^○C is reached after 800 s for a 12 A solenoid current, suggesting that the maximum safe value of solenoid current to be continuously supplied to the solenoid coils is 10 A.

The predicted maximum temperature achieved in the coils for finite α (⁠$Tmax,α=αCu$⁠) shows a sharp rise above 10 A – a quasi thermal runaway has started to occur in which a small increase in current gives a very large increase in the maximum temperature. It has been reported^9,10 that such a HeSE coil was melted at relatively modest currents, which we ascribe to such a quasi thermal runaway. A comparison of $Tmax,α=αCu$ with the predictions for wire with a temperature independent resistance (T_{max , α = 0} in Table I) shows that the quasi thermal runaway is due to the increase in resistance of the wire with temperature. For a reasonably pure metal, the winding resistance and the heat generated for a particular current increase roughly linearly with absolute temperature. The heat dissipation, however, increases linearly with temperature above ambient, so for small currents and small temperature the resistance is roughly constant and the temperature rise will be proportional to the square of the current. Once the temperature rise becomes significant compared to the absolute ambient temperature, the resistance and power increase significantly as the windings heat up and this tends to offset the increase in heat dissipation due to the greater temperature above ambient. For these simulations, as a rule of thumb, if the predicted temperature rise for constant resistance wire is roughly 100 °C, the temperature rises due to the temperature dependent resistance remain modest, but for a predicted temperature rise of 150 °C for constant resistance wire a quasi thermal runaway occurs with real wire, the key parameter being the ratio of the temperature rise to the absolute temperature.

The current at which the quasi thermal runaway starts depends critically on the mean thermal conductivity of the windings which will be sensitive to the details of the coil manufacture and may also degrade with time as thermal cycling produces cracks in the epoxy between the layers. Before running such a solenoid at high currents it is therefore important to determine when this quasi runaway will occur using such a method as presented in this note. Degradation of the thermal conductivity of a coil can be detected by changes in the resistance rise of the coil for a particular current. Given the ease with which a quasi thermal runaway can be produced, it would seem advisable to have a voltage limit on the current source such that as the coil heats and the resistance rises the current, and hence power, would be reduced with rising temperature. The coil used here had straight forward boundary conditions: the surfaces were water cooled to a constant temperature. For an air cooled solenoid the constant of proportionality between the heat loss and the surface temperature would be needed in the model as an additional unknown, but this could be fitted if the surface temperature of the coil was also measured.

In summary, we have described a simple iterative simulation to model the dynamic temperature evolution in a solenoid coil and to predict the maximum temperature reached in any coil element, which will normally exceed the average temperature of the coil. The model is calibrated against the measured thermally induced resistance change of the coil and so gives a reliable estimate of the thermal performance of the coil. The model predicts that a quasi thermal runaway condition is created if the absolute temperature of the windings, and hence the resistance, increases by more than ∼50% of the absolute ambient temperature.