Thermal Diffusion Length and Heating Regimes

Hello! Welcome to the next lesson in our course on high magnetic field physics and construction.

In the previous lesson, we developed a detailed numerical method (the Finite Difference Method) to calculate the full spatio-temporal temperature profile, $T (r, t)$ , inside a conductor during a current pulse. This is the most accurate way to model thermal behavior, but it can be computationally intensive.

Today, we will develop a powerful analytical shortcut. Our goal is to estimate the thermal diffusion length to determine whether heating is adiabatic or quasi-steady for different pulse durations. This concept allows you to make a rapid, first-order assessment of a coil's thermal behavior—a critical skill for initial design phases, just like the work done at facilities like LNCMI. Instead of running a full simulation, you'll be able to tell with a simple calculation whether the heat generated during a pulse has time to move significantly.

1. Thermal Diffusivity: The Speed of Heat

Let's start by revisiting the heat diffusion equation we worked with previously:

\frac{\partial T}{\partial t} = \frac{k}{ρ c _{p}} \nabla^{2} T + \frac{e _{g e n}}{ρ c _{p}}

The group of material constants $α = \frac{k}{ρ c _{p}}$ is a property of the material itself and is called the thermal diffusivity. It has units of m²/s.

To understand its physical meaning, let's watch a short segment from a lecture on the heat diffusion equation.

Lecture 3 : Heat Diffusion Equation

This video from NPTEL IIT Kharagpur provides a clear definition of thermal diffusivity and explains its physical significance.

Please watch the segment from 21:24 to 24:15. The speaker introduces thermal diffusivity, \alpha, discusses its units, and draws an excellent analogy to mass diffusivity and momentum diffusivity (kinematic viscosity), which you may have encountered in other areas of physics and fluid dynamics.

As the video explains, thermal diffusivity measures a material's ability to conduct thermal energy relative to its ability to store it.

A high $α$ (e.g., in copper) means heat propagates quickly through the material.
A low $α$ (e.g., in an insulator like epoxy) means heat diffuses very slowly.

It is the single most important parameter governing transient heat conduction.

2. The Thermal Diffusion Length

Given that $α$ has units of $L^{2} / T$ , a simple dimensional analysis suggests that in a time $t$ , heat will diffuse over a characteristic distance proportional to $α t$ . This is the fundamental idea behind the thermal diffusion length.

Let's see the formal definition in the context of a pulsed heating problem.

Time Varying Heat Conduction in Solids

The book chapter 'Time Varying Heat Conduction in Solids' provides a dedicated analysis of heat conduction under a finite duration heat pulse, which is directly applicable to our work with pulsed magnets.

Please read Section 4.2, 'A finite sample exposed to a finite duration heat pulse'. Pay close attention to the solution for a sudden temperature change (Eq. 39) and the resulting definition of the thermal diffusion length for pulsed excitation in Equation (41).

From the reading, we get the definition of the thermal diffusion length, $μ_{t}$ , for an event of duration $t$ :

μ_{t} = 2 α t

This equation gives us a quantitative estimate for how far a thermal front propagates into a material in a given time $t$ . The pre-factor (here, 2) can vary depending on the specific definition or problem geometry, but the physical scaling, $α t$ , is universal for diffusion processes.

3. Adiabatic vs. Diffusive Regimes: A Critical Distinction

The power of the thermal diffusion length comes from comparing it to a characteristic dimension of our conductor, such as its radius or thickness, which we'll call $L$ . This comparison defines two distinct heating regimes.

The Adiabatic Regime ( $μ_{t} ≪ L$ )

This regime occurs when the pulse is very short, or the thermal diffusivity is very low. The thermal diffusion length is much smaller than the size of the conductor.

Physical Meaning: Heat is generated by the current pulse far more quickly than it can diffuse away. During the pulse, each part of the conductor heats up as if it were thermally isolated from its neighbors.
Consequence: The spatial diffusion term ( $k \nabla^{2} T$ ) in the heat equation is negligible during the pulse. The equation simplifies to a pure heating problem: $ρ c_{p} \frac{\partial T}{\partial t} \approx e_{g e n} (t) = J (t)^{2} ρ_{e}$ This is precisely the assumption made when using the "action integral" to calculate the final temperature, a common technique in pulsed magnet design.

Let's confirm this with a practical resource on magnet design.

Design and Optimization of Force-Reduced High Field ...

This PhD dissertation on pulsed magnet design explicitly discusses the thermal analysis of coils.

Please read the short Section 5.5.1, 'Heat Analysis and Coil Cooling'. Note the key statement: 'the coil heating during the short current pulse can be regarded as being adiabatic.' This justifies the simplification we just discussed and highlights its importance in practice.

The Diffusive (or Quasi-Steady) Regime ( $μ_{t} ≳ L$ )

This regime occurs for longer pulses or materials with high thermal diffusivity. The thermal diffusion length is comparable to or larger than the conductor size.

Physical Meaning: During the pulse, there is enough time for heat to travel significant distances within the conductor—for example, from the hot center to the cooler surface.
Consequence: You cannot neglect the diffusion term. The temperature at any point depends on both local generation and heat flow from other regions. A full numerical solution, like we studied in the previous lesson, is required for an accurate temperature profile.

The image below perfectly illustrates the difference between these regimes.

Thermal Response in Pulsed Magnetic Fields: Adiabatic vs. Quasi-Adiabatic Regimes — This figure shows a thermal model (a) and the resulting temperature evolution. In panel (b), the 'Adiabatic' curve shows temperature tracking the field pulse (H) exactly. The 'Quasi-adiabatic' curve shows a delayed response due to thermal diffusion. Panel (c) shows the temperature vs. field, where the non-adiabatic case results in a hysteretic loop, meaning the sample is hotter during the field ramp-down than the ramp-up at the same field value.

4. A Practical Example

Let's apply this to a typical scenario. Consider a pulsed magnet wound with copper wire of diameter 2 mm (radius $L = 1$ mm). The magnet is powered by a capacitor bank that produces a current pulse with a primary duration of $t_{p} = 10$ ms. Is the heating during this pulse adiabatic?

First, we need the thermal diffusivity of copper. Near room temperature:

Thermal conductivity, $k \approx 400$ W/m·K
Density, $ρ \approx 8960$ kg/m³
Specific heat, $c_{p} \approx 385$ J/kg·K

So, $α = \frac{k}{ρ c _{p}} \approx \frac{400}{8960 \times 385} \approx 1.16 \times 1 0^{- 4}$ m²/s.

Now, we calculate the thermal diffusion length for the pulse duration $t_{p} = 0.01$ s:

μ_{t} = 2 α t_{p} = 2 (1.16 \times 1 0^{- 4} m^{2} / s) \times (0.01 s) \approx 2.15 mm

Analysis:
We compare $μ_{t}$ to the wire radius $L = 1$ mm. Since $μ_{t} = 2.15$ mm > $L = 1$ mm, the condition $μ_{t} ≪ L$ is not met.

Conclusion: For a 10 ms pulse, the heating is not adiabatic. It is firmly in the diffusive regime. Heat has ample time to diffuse across the entire cross-section of the wire during the pulse. To find the true temperature profile, you would need to account for heat flow, likely using a numerical model.

Test your understanding!

You are evaluating a new conductor material, a Zylon-strengthened copper composite, for a very high-stress, short-pulse magnet. The conductor has a square cross-section of 4 mm x 4 mm, and its effective thermal diffusivity is $α = 0.8 \times 1 0^{- 4}$ m²/s.

You want to operate in a regime where the heating can be confidently modeled as adiabatic to simplify your stress analysis. A common rule of thumb is to consider the heating adiabatic if the diffusion length is less than 20% of the characteristic dimension ( $μ_{t} < 0.2 L$ ). What is the maximum pulse duration $t_{p}$ you can use?

Show answer

Identify the characteristic length, $L$ : The shortest distance for heat to escape is from the center to the surface, which is half the thickness. So, $L = 4 mm /2 = 2$ mm = 0.002 m.
Set up the adiabatic condition: We want $μ_{t} < 0.2 L$ .
$2 α t_{p} < 0.2 L$
Solve for $t_{p}$ :
$α t_{p} < 0.1 L$ $α t_{p} < (0.1 L)^{2}$ $t_{p} < \frac{( 0.1 L ) ^{2}}{α}$
Substitute the values:
$t_{p} < \frac{( 0.1 \times 0.002 m ) ^{2}}{0.8 \times 1 0 ^{- 4} m ^{2} / s}$ $t_{p} < \frac{( 2 \times 1 0 ^{- 4} m ) ^{2}}{0.8 \times 1 0 ^{- 4} m ^{2} / s} = \frac{4 \times 1 0 ^{- 8} m ^{2}}{0.8 \times 1 0 ^{- 4} m ^{2} / s}$ $t_{p} < 5 \times 1 0^{- 4} s = 0.5 ms$

The maximum pulse duration to remain safely in the adiabatic regime is approximately 500 microseconds. For any longer pulse, diffusive effects become significant.

Conclusion

Today, we've formalized the intuitive idea that heat needs time to travel. By defining the thermal diffusion length, you've gained a powerful tool for quickly assessing the thermal behavior of a pulsed magnet conductor without resorting to a full simulation.

Key Takeaways:

Thermal Diffusivity ( $α$ ) is the material property that governs the speed of transient heat conduction.
The Thermal Diffusion Length ( $μ_{t} = 2 α t$ ) estimates the characteristic distance heat travels in a time $t$ .
Comparing $μ_{t}$ to the conductor's characteristic size $L$ determines the heating regime:
- Adiabatic ( $μ_{t} ≪ L$ ): Occurs for short pulses. Heat is generated faster than it diffuses. The temperature rise is purely local.
- Diffusive ( $μ_{t} ≳ L$ ): Occurs for longer pulses. Heat flow within the conductor is significant during the pulse.

This concept is a cornerstone of thermal analysis, providing the first-pass answer to the crucial question: "Does the heat stay put, or does it move?"

Next Lesson Preview:
Our analysis today relied on the thermal diffusivity, $α$ , which is a composite of thermal conductivity, density, and specific heat. For pulsed magnets, these properties—along with electrical resistivity and mechanical strength—are what determine a material's suitability. In our next lesson, we will begin our exploration of materials science by evaluating the electrical and mechanical properties of key conductors like copper (Cu) and its high-strength alloys like Copper-Silver (CuAg) and Copper-Niobium (CuNb), including their crucial dependence on temperature.

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