The Example: Laser Heating of a Flat Plate

Let’s consider the case of a flat plate of material that is being heated by a laser heat source. The plate is centered at the origin, as shown in the figure below, and we want to heat its surface at varying locations over time. Suppose that the laser (or the workpiece) is mounted on a stage that provides positional control of the focus point. Let’s also assume that there are some optics that shape the beam profile of the laser, so the heat source is spread over a small area around the focus.

Schematic of a laser heat source traversing over a workpiece.

Now, let’s look at several different ways in which we can define the moving focus point to follow a known tool path.

Method 1: Using Variables

The simplest approach is to use a set of variables to define the position of the focus and distribution of the heat load over time. Let’s say that we want to have a 1-kW total heat load moving in a 40-cm radius circular path every 10 s. Furthermore, the heat load has a Gaussian intensity distribution with a 5-cm waist radius. We can define this information using variables, as shown in the screenshot below.

The first four definitions, Rb, P0, Rp, and T0, are actually just constants. If we would like to alter any of these definitions via a parametric sweep later on, we could also define them as Global Parameters. For now, it is simplest to just show them all in one place.

The next two variables, x_focus and y_focus, are not constant: They vary as a function of time, the built-in variable t. We can see that these variables describe a point moving on a circular path about the origin as:

Rp*cos(2*pi*t/T0)

Rp*sin(2*pi*t/T0)

The next variable, R, is a function of time and space. It makes use of the coordinate variables x and y, as well as x_focus and y_focus, which we just saw are functions of time. So at each instant in time and at every point in space, this variable tells us the distance (in the xy-plane) from the laser focal point.

The last variable, HeatFlux, is a function of R and the constants. It defines a Gaussian intensity profile about the focal point such that the total heat flux equals the defined power. It is this variable, HeatFlux, that we enter as a boundary condition, as shown in the screenshot below.

This prescribed heat flux expression then gives the heating profile path visualized below.

A circular heating profile set up via the Variables node.

Method 2: Using Interpolation Functions

So far, this isn’t very complicated; just a few expressions. But we can replace the simple expressions for x_focus and y_focus with something more general. COMSOL Multiphysics provides a variety of built-in functions in the software. For our discussion here, the most useful is the Interpolation function, which lets us read in data from a text file. Let’s suppose we have a text file containing rows of data of time and the x- and y-locations of the laser focus at that time. A sample of such a file is shown below:

Such data can be read in to the Interpolation function using the settings shown below. Note that there is just a single argument here, time, and the two columns of data after that represent the x-focus and y-focus, respectively, in units of centimeters. Between the specified points in time, we want the laser to move linearly. We specify the function names as x_f and y_f, respectively, and make sure to set the arguments correctly.

We can then just alter our previous expressions for the focus to be x_focus = x_f(t) and y_focus = y_f(t) and get the moving load pictured below.

A heating profile read in from a text file.

We can see that this Interpolation function quickly lets us read in some very complicated profile patterns, we just need a way to generate these profiles and the text files. For example, the text file format used here is not too different from the G-code format, so if you have a heating path defined in such a format, it can be pretty simple to convert it to a COMSOL®-friendly input. On the other hand, maybe we would like to import a profile created in the ubiquitous 2D DXF file format. Let’s look at that next…

Method 3: Using Paths Imported from CAD Geometries

Suppose we want the load to move along a path we read in from an external file, as shown in the image below. We would like the laser to move smoothly along this path from one end to the other. Now, we need to do a bit more work.

The S-shaped profile of the laser path is read in from a DXF file as a geometry.

The file that we’ve read in doesn’t have any information about time within it: It is just a path that we expect the laser to follow at a constant speed. Now, each edge of this imported path (and there may be thousands of edges) does have parameters s1 and s2 that vary linearly along the length, but if there are many edges, we probably wouldn’t want to work with these parameters. So instead, how do we compute where the laser is at every point in time along the entire set of lines? One way in which we can do this is by introducing another partial differential equation (PDE) to solve, along the desired set of lines. The PDE we want to solve is:

where refers to the tangential direction to the curve.

This PDE, along with the boundary conditions of u = 0 at one end of the path and u = 1 at the other end, will give us a field along the path that varies linearly from 0 to 1, which will be proportional to the total arc length of all of the edges. We can set this up using the Coefficient Form Edge PDE interface, as shown in the screenshots below. All coefficient terms other than the Diffusion Coefficient, c, are set to zero.

Left: Settings of the Coefficient Form Edge PDE interface needed to compute the path. Right: The diffusion coefficient term, c, is a constant; all other coefficients are set to zero.

Then, two Dirichlet boundary conditions set the field, u, at either ends, and we solve this PDE in a stationary step, prior to solving the heat transfer problem but still within the same study.

Two Dirichlet boundary conditions are used at the start and end of the profile path to constrain the field.

Next, we introduce a single minimum component coupling operator into the model, with the source selected as the edges to follow. This minimum operator is used to define the focal point coordinates, e.g.:

x_focus = minop1(abs(u-t/T0),x)

y_focus = minop1(abs(u-t/T0),y)

Note that the minimum operator is given two arguments. When we call the operator with two arguments, it will return the value of the second argument where the first argument is at a minimum. Thus, at each time, t, it will return the x- and y-locations of a point on the edge that is t/T0 fraction of the way from one end to the other.

Left: The minimum operator is defined over the path that the heat source follows. Right: The laser heat source following the path defined by an imported DXF file.

And what if we would like the laser to traverse different parts of the path at different speeds? We would just need to adjust the coefficient, c, along those sets of edges. Suppose we want the laser to move three times faster along the curved boundaries than along the straight lines: just make c three times larger. Note that the absolute value doesn’t matter, it is just the ratio of the coefficient magnitudes that matters. The one drawback to this approach arises when the path crosses itself. In that situation, you would need to subdivide the path into two, or more, groups of paths; solve a PDE on each; and do a bit more bookkeeping for the variables.

Closing Remarks on Modeling Moving Loads and Constraints

In this blog post, we have looked at three different approaches of modeling a moving load. To try them yourself, click the button below to head to the Application Gallery. There, you can download the MPH-files for the models featured above (must have a COMSOL Access account and valid software license).

Several other examples within the Application Gallery also make use of these techniques, including:

• Laser Heating of a Silicon Wafer
• Traveling Load

Although in this blog post we only considered loads, note that we can also apply these techniques to constraints, as described in How to Make Boundary Conditions Conditional in Your Simulation.

Do you have further questions about using COMSOL Multiphysics for your modeling applications? Please let us know!

Understanding Ohm’s Law and Resistive Heating

One of the first physical laws that we learn as engineers is Ohm’s law: The current through a device equals the applied voltage difference divided by the device electrical resistance, or I = V/R_e, where the electrical resistance, R_e, is a function of the device geometry and material’s electrical conductivity.

Shortly after learning that law, we probably also learned about the dissipated power within the device, which equals the current times the voltage difference, or Q = IV, which we could also write as Q = I²R_e or Q = V²/R_e. Maybe a little bit later in our studies, we also learned about thermal conductivity and equivalent device thermal resistance, R_t, which let us compute, in a lumped sense, the rise in temperature of a device via ΔT = QR_t relative to ambient conditions. We can then determine the absolute device temperature using .

We start our discussion from this point and consider a completely lumped model of a device. (Yes, we’re starting so simple that we don’t even need to use the COMSOL Multiphysics® software for this part!) Let’s consider a lumped device with electrical resistance of R_e = 1 Ω and thermal resistance of R_t = 1 K/W. We can drive this device with either a constant voltage and compute the temperature as or drive it via constant current where the peak temperature is .

We choose an ambient temperature of 300 K, or 27°C, which is about room temperature. Let’s now plot out the device lumped temperature as a function of increasing voltage (from 0 to 10 V) and current (from 0 to 10 A), as shown in the image below. Unsurprisingly, we see a quadratic increase in temperature.

Device temperature as a function of applied voltage (left) and applied current (right), assuming constant properties.

We might think that we can use the curve to predict a wider range of operating conditions. Suppose we want to drive the device up to its failure temperature, where the material melts or vaporizes. Let’s say that this material will vaporize when its temperature gets up to 700 K (427°C). Based on this curve, some simple math would indicate that the maximum voltage is 20 V and the maximum current is 20 A, but this is quite wrong!

Introducing Material Nonlinearities into the Simple Lumped Model

At this point, you’re probably ready to point out the simple mistake that we’ve made: Electrical resistance is not constant with temperature. For most metals, the electrical conductivity goes down with an increasing temperature and since resistivity is the inverse of conductivity, the device resistivity goes up. So, let’s introduce a temperature dependence for the resistivity:

This is known as a linearized resistivity model, where ρ₀ is the reference resistivity at , the reference temperature, and α_e is the temperature coefficient of electrical resistivity.

Let’s choose ρ₀ = 1 Ω, = 300 K, and α_e = 1/200 K. Now, the resistance is 1 Ω at a device temperature of 300 K and 2 Ω at a temperature rise of 200 K above the set temperature. The equations for lumped temperature as a function of voltage and current now become:

and

These equations are a bit more complicated (the first is a quadratic equation in terms of T) but still possible to solve by hand. The plots of temperature as a function of increasing voltage and current are displayed below.

Device temperature as a function of applied voltage (left) and applied current (right) with the electrical resistivity as a function of temperature.

For the voltage-driven problem, as the temperature rises, the resistance rises. Since the resistance occurs in the denominator of the temperature expression, higher resistance lowers the temperature and we see that the temperature will be lower than that for the constant resistivity case. If we drive the device with constant current, the temperature-dependent resistance appears in the numerator. As we increase the current, the resistive heating will be higher than that for the linear material case.

We might be tempted at this point to compute the maximum voltage or current that this device can sustain, but you are probably already realizing the second mistake we’ve made: We also need to incorporate the temperature dependence of the thermal resistance. For metals, it’s reasonable to assume that the electrical and thermal conductivity will show the same trends. Thus, let’s use a nonlinear expression that is similar to what we used before:

Now, our voltage-driven and current-driven equations for temperature become:

and

Although only slightly different from before, these nonlinear equations are now quite a bit more difficult to solve. Simulation software is starting to look more attractive! Once we do solve these equations (let’s set r₀ = 1 K/W, α_t = 1/400 K, and = 300 K), we can plot the device temperature, as shown below.

Device temperature as a function of applied voltage (left) and applied current (right) with the electrical and thermal resistivity as a function of temperature.

Observe that for the current-driven case, the temperature rises asymptotically. Since both the electrical and the thermal resistance increase with an increasing temperature, the device temperature rises very sharply as the current is increased. As the temperature rises to infinity, the problem becomes unsolvable. This is actually entirely expected; in fact, this is how your basic car fuse works. Now, if we were solving this problem in COMSOL Multiphysics, we could also solve this as a transient model (incorporating the thermal mass due to the device density and specific heat) and predict the time that it takes for the device temperature to rise to its failure point.

Things are luckily a bit simpler for the voltage-driven case. Here, we also see a predictable behavior: The rising thermal resistivity drives the temperature higher than when we only considered a temperature-dependent electrical conductivity. Now, the interesting point here is the temperature is still lower than for the constant resistivity case. This also sometimes confuses people, but just keep in mind that one of these nonlinearities is driving the temperature down while the other is driving the temperature up. In general, for a more complicated model (such as one you would build in COMSOL Multiphysics), you don’t know which nonlinearity will predominate.

What other mistake might we have made here? We have used a positive temperature coefficient of thermal resistivity. This is certainly true for most metals, but it turns out that the opposite is true for some insulators, glass being a common example. Usually, the total device thermal resistance is mostly a function of the insulators rather than the electrically conductive domains. In addition, the device’s thermal resistance should incorporate the effects of the cooling to the surrounding ambient environment. So, the effects of free convection (which increases with the temperature difference) and radiation (which has a fourth-order dependence on temperature difference) could also be lumped into this single thermal resistance. For now, though, let’s keep the problem (relatively) simple and just switch the sign of the temperature coefficient of thermal resistivity, α_t = -1/400 K, and directly compare the voltage- and current-driven cases for driving voltage up to 100 V and current up to 100 A.

Device temperature as a function of applied voltage (pink) and applied current (blue) with a negative temperature coefficient of thermal resistivity.

We now see some results that are quite different. Observe that for both the voltage- and current-driven cases, the temperature increases approximately quadratically at low loads, but at higher loads, the temperature increase starts to flatten out due to the decreasing thermal resistivity. The slope, although always positive, decreases in magnitude. The current-driven case starts to asymptotically approach T = 700 K, but the voltage-driven case stays significantly below the failure temperature.

This is an important result and highlights another common mistake. The nonlinear material models we used here for electrical and thermal resistivity are approximations that start to become invalid if we get too close to 700 K. If we anticipate operating in this regime, we should go back to the literature and find a more sophisticated material model. Although our existing nonlinear material models did solve, we always need to check that they are still valid at the computed operating temperature. Of course, if we are not close to these operating conditions, we can use the linearized resistivity model (one of the built-in material models within COMSOL Multiphysics). Then, our model will be valid.

We can hopefully now see from all of this data that the temperature has a very complicated relationship with respect to the driving voltage or current. When nonlinear materials are considered, the temperature might be higher or lower than when using constant properties, and the slope of the temperature response can switch from quite steep to quite shallow just based on the operating conditions.

Have these results thoroughly confused you yet? What if we went back and changed one of the coefficients in the resistance expressions? Certain materials have negative temperature coefficients of electrical and thermal resistivity. What if we used an even more complicated nonlinearity? Would you feel confident in saying anything about the expected temperature variations in even this simple lumped device case, or would you rather check it against a rigorous calculation?

Concluding Thoughts on Common Pitfalls in Electrothermal Analysis

What about the case of a real-world device? One that has a combination of many different materials, different electrical and thermal conductivities as a function of temperature, and complex shapes? Would you model this under steady-state conditions only or in the time domain, to find out how long it takes for the temperature to rise? Maybe — in fact, most likely — there will also be nonlinear boundary conditions such as radiation and free convection that we don’t want to approximate via a single lumped thermal resistance. What can you expect then? Almost anything! And how do you analyze it? Well, with COMSOL Multiphysics, of course!

Next Step

Evaluate how COMSOL Multiphysics can help you meet your multiphysics modeling and analysis goals. Contact us via the button below.

Improving the Efficiency of TPV Systems

During the TPV cell energy production process, fuel burns within an emitting device that intensely radiates heat. Photovoltaic (PV) cells capture this radiation and convert it into electricity, with an efficiency of 1–20%. The required efficiency depends on the intended application of the cell. For example, efficiency is not a major factor when TPVs are used to cogenerate electricity within heat generators. On the other hand, efficiency is critical when TPVs are used as electric power sources for vehicles.

Left: Simplified schematic depicting the electricity generation process of a TPV. Right: An image from a prototype TPV system. Right image courtesy Dr. D. Wilhelm, Paul Sherrer Institute, Switzerland.

To improve the efficiency of TPV systems, engineers need to maximize radiative heat transfer, but this comes with a catch. The more radiation in the system, the less radiation converted to electric power. These losses — as well as conductive heat transfer — raise the temperature of the PV cell. If the temperature increases too much, it can exceed the operating temperature range of the PV cell, causing it to stop functioning.

One option for increasing the operation temperature of a TPV system is to use high-efficiency semiconductor materials, which can withstand temperatures up to 1000°C. Since these materials tend to be expensive, engineers can reduce costs by combining smaller-area PV cells with mirrors that focus radiation onto the cells. Of course, there is a limit to how much the beams can be focused, since the cells overheat if the radiation intensity gets too high.

Engineers designing TPV devices need to find optimal system geometries and operating conditions that maximize performance, minimize material costs, and ensure that the device temperature stays within the operating range. Heat transfer simulation can help achieve these design goals.

Evaluating the Design of a TPV System with Heat Transfer Modeling

This example uses the Heat Transfer Module and the Surface-to-Surface Radiation interface to determine how operating conditions (e.g., the flame temperature) affect the efficiency of a normal TPV system as well as the temperature of the system’s components. The goal is to maximize surface-to-surface radiative heat fluxes while minimizing conductive heat fluxes. In this model, the effects of geometry changes are also evaluated.

The model geometry includes an emitter, mirrors, insulation, and a PV cell that is cooled by water on its back side. For details on setting up this model — including how to add conduction, surface-to-surface radiation, and convective cooling — take a look at the TPV cell model documentation.

The TPV system model geometry.

To minimize the computational costs of the simulation, we use sector symmetry and reflection to reduce the computational domain to one sixteenth of the original geometry. When modeling the surface-to-surface radiation, we expand this view to account for the presence of all of the surfaces in the full geometry.

Analyzing the Efficiency and Temperature of a TPV System with Simulation

First, let’s check the voltaic efficiency of the PV cell for a range of cell temperatures. In doing so, we see that the efficiency decreases as the temperature increases. When the temperature of the cell exceeds 1600 K, the efficiency is 0. As such, the maximum operational temperature for the PV cell design is 1600 K.

Plotting PV cell voltaic efficiency versus temperature.

In the next plots, we see how the temperature of the emitter affects the temperature of the PV cell and the electric output power. The cell temperature plot (left image below) indicates that the emitter temperature must be under ~1800 K to keep the PV cell below its maximum operating temperature of 1600 K.

Keeping this in mind, let’s take a look at the electric power output results (right image below). From the results, we conclude that the maximum electric power is achieved when the emitter temperature is ~1600 K.

Plotting PV cell temperature (left) and electric output power (right) against operating temperature.

Moving on, let’s examine the temperature distribution in the PV cell for the optimal operating condition (left image below) and compare it to a temperature that exceeds this operating temperature (right image below). The two plots highlight how the device’s temperature distribution varies due to operating conditions.

The stationary temperature distribution in the full TPV system when the emitter temperature is 1600 K (left) and 2000 K (right).

Looking closer at the plot of the optimal emitter temperature of 1600 K, we see that the PV cells are heated to a sustainable temperature of slightly above 1200 K. It is important to note that the outside part of the insulation reaches a temperature of 800 K, indicating that a large amount of heat is transferred to the surrounding air. In addition, the irradiative flux significantly varies around the PV cell circumference and insulation jacket.

To determine the cause of this variation, we generate a plot of the irradiative flux for a single sector of symmetry at a temperature of 1600 K. The graph indicates that the variation is caused by shadowing and is related to the mirror positions. Using this plot, we could optimize the cell size and placement of the mirrors for a PV design.

The irradiation flux at the TPV cell, insulation inner surface, mirrors, and emitter.

Next Step

Using models like the one discussed here, engineers can efficiently find optimal operating conditions for TPV devices, minimizing prototype development and testing.

To try this TPV cell example yourself, download the model files above.

Part 1: Introducing the Basics and Fundamentals of Cable Modeling

The beginning is a very good place to start, as most would say. Part 1 of the tutorial series is where you meet the model — a three-core lead-sheathed cross-linked polyethylene, high-voltage alternating current (XLPE HVAC) submarine cable. You’ll also get details on what to expect in the other five parts of the series.

A submarine cable similar to the one modeled throughout this series. Image by Z22 — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.

The overview of the fundamentals of electromagnetism and numerical modeling is helpful if you are new to the electromagnetics field, simulation, or both. Feel free to skip ahead if these topics are old hat to you, but if not, this primer covers subjects such as:

• Drawing geometry
• Adding material properties
• Creating selections
• Meshing your model

The cross section (left) and mesh (right) for a model of a typical lead-sheathed XLPE HVAC submarine cable with three cores. The geometry has been parameterized to allow for quick modification; any cable with the same basic structure can be investigated with ease.

Part 2: Capacitive Effects

The second tutorial focuses on modeling the cable’s capacitive properties and validates an important assumption: An analytical approach is sufficient for the analysis of capacitance and charging effects. This will be useful throughout the series.

This tutorial is included for beginners, but the results also support the other parts of the series, as it demonstrates the significance of the material properties and cable length. In the cross section of the cable model, the large contrast in material properties enables you to consider the XLPE as a perfect insulator and lead and copper materials as perfect conductors. These results correspond to the analytical approximations.

Left: The electric potential distribution after 10 km of cable for single-point bonding (at phase φ = 0). Right: The in-plane displacement current density norm in the insulators (primarily the XLPE).

In terms of cable length, you will see that the analytical approximations are sufficient for a 10-km cable. This stays true even under the worst possible nominal conditions, which occur when single-point bonding is applied and all voltage-inducing effects are in-phase.

Part 3: Bonding Capacitive

Part 3 of the series builds on the previous tutorial, which showed that you may neglect the capacitive coupling between phases and consider one isolated phase. This reduces the model to an axisymmetric problem. In order to cover the full 10 kilometers of cable, we use a scaled 2D axisymmetric geometry in the model.

Left: The 2D axisymmetric geometry of an isolated phase with three separate bonding sections and a different scale for transverse and longitudinal directions. Right: The norm of the resulting charging current accumulated along the cable (for cross bonding).

The charging currents that leak into the screen build up along the cable and reach a maximum at the ground point, or intersection. The Bonding Capacitive tutorial analyzes the current buildup for different bonding types as well as the corresponding losses. The results are as follows:

Part 4: Inductive Effects

This part of the series builds on the previous two tutorials, which show that there is a weak coupling between the inductive and capacitive parts of the cable. The relatively small losses caused by in-plane displacement and eddy currents justify approximating the cable using a 2D inductive model with out-of-plane currents only.

Animation of the instantaneous magnetic flux density norm in the cable’s cross section, for solid bonding and with armor twisting included.

Animation of the current density induced in the cable’s armor and screens, for solid bonding and with armor twisting included.

This model focuses on the importance of wire twist with respect to both phase conductors and armor, and investigates the corresponding losses. For instance, when armor twist is applied to the cable, the armor currents are suppressed and the total losses decrease by ~11%.

In addition to this, we demonstrate two different ways of modeling the central conductors. The first example assumes the central conductors to consist of solid copper, resulting in a typical skin and proximity effect. The other shows a perfectly stranded Litz wire approach, resulting in a homogenized current distribution.

The simulation results found in this tutorial are validated using actual product data sheets following the official international standards. The comparison shows a good match, especially for the inductance.

Part 5: Bonding Inductive

The objective of Part 5 is to further examine the different bonding types that were suggested in Part 3 (and 4): single-point, solid, and cross bonding. (Cross bonding is especially of interest for terrestrial cable systems.) As opposed to Part 3, this part focuses on inductive effects.

You will learn how to individually consider three different cable sections by coupling three separate magnetic fields physics interfaces to a circuit. The resulting model allows for investigating debalanced cables and cables with dissimilar section lengths.

In addition to this, the tutorial demonstrates the effects of using a simplified geometry. Simplification is an overarching theme in this tutorial series: It is often justified to use a much simpler geometry than you think. It isn’t the quantity of details, but the quality that optimizes a model.

Part 6: Thermal Effects

In the final installment of the series, electromagnetic heating and temperature-dependent conductivity are added to the cable model. Building on Part 4, you’ll learn how to set up a two-way coupling between the electromagnetic field and heat transfer part by implementing a frequency-stationary study.

Left: An example of a preset resistance curve R_ac (T). Right: The resulting temperature distribution when using a temperature-dependent conductivity such that R_ac (T) is matched.

Results show the effect of temperature on losses for the cable’s phases and screens. When electromagnetic heating is added (without temperature-dependent conductivity) the cable heats up, but the electromagnetic properties are still identical to those reported in Part 4. When adding linearized resistivity to the phases specifically, phase losses increase but not the screen losses. The temperature reaches a maximum. If linearized resistivity is applied to the screens as well, the temperature lowers and losses decrease for both the phases and the screens.

In this case still, the material properties are provided and the numerical model determines the corresponding AC resistance. However, for thermal cable models, it’s common practice to use the temperature-dependent AC resistance as an input (as provided by the IEC 60287 series of standards). The final part of the tutorial demonstrates how to use any temperature-dependent resistance curve as an input and let the model determine the corresponding material properties.

Next Steps

Check out the Cable Tutorial Series if you’re looking for a self-paced electromagnetics modeling resource, whether you want to examine each section in detail or skip ahead depending on what interests you.

You can access the materials, which include step-by-step PDF instructions and MPH-file downloads, via the button below:

Model documentation is available with a COMSOL Access account. To download the MPH-files, you also need a software license.

You can also learn more about modeling cable systems by watching this archived webinar.

Electromagnetic Heating Around Sharp Corners

One of the most common problems solved with the COMSOL Multiphysics® software is that of electromagnetic heating, combining the solution to Maxwell’s equations, which solves for the current flow and resultant losses, and the solution to the heat transfer equation, which solves for the temperature distribution.

As mentioned in a previous blog post, when solving for the electromagnetic fields, sharp reentrant corners lead to locally nonconvergent electric fields and current density. Electromagnetic losses are the product of the electric field and current density, so the peak losses at a sharp corner will similarly go to infinity with mesh refinement.

However, the integral over the losses around the sharp corner will be convergent with respect to mesh refinement. This is one of the strengths of the finite element method, which solves the governing equation in the so-called “weak form”, which satisfies the governing partial differential equations in an integral sense: minimizing the total error in the model, but allowing (possibly infinite!) local errors.

A schematic of a simple electromagnetic heating problem with a singularity at the inside sharp corner.

Let’s review this concept with a simple example, as shown in the image above. A rectangular domain with a sharp notch has an electric potential difference applied, leading to current flow and resistive losses in the material.

Below, we see a color plot of the resistive losses and the mesh used for different levels of mesh refinement at the sharp inside corner. At the highest level of mesh refinement, the losses appear very localized around the sharp corner.

The electromagnetic losses on several different meshes.

At this sharp inside corner, the electric fields are actually theoretically infinitely large, since this geometry and the boundary conditions imply that the current must instantaneously change direction at a point. Also note that the sharp outside corners do not lead to singularities. As a consequence of the geometry and boundary conditions, the electric currents are not forced to change direction instantaneously at these points.

The resistive losses in log scale, plotted along a cut line, and a table of the integral of the losses for different meshes.

If we plot the losses at a cross-sectional line, as shown above, we can observe that the losses at the sharp point get larger and larger as the mesh is refined. However, the integral of the losses over the domain (roughly speaking, the area underneath the plotted curves) converges very quickly with mesh refinement.

Now let’s make this a multiphysics problem by additionally solving the heat transfer equation for the temperature distribution under steady-state conditions. These temperature fields are plotted below for several levels of mesh refinement, as well as the temperature at the sharp point.

The temperature field and a table of the temperature evaluated at the sharp corner for different levels of mesh refinement.

We can see from these results that the temperature at the singular point — and, of course, everywhere else — is also very insensitive to the mesh refinement. This is for two reasons. First, as we just saw, the total resistive losses are quite insensitive to the mesh. Second, the diffusive nature of the steady-state heat transfer governing equation will return very similar temperature solutions as long as the total heat loads are similar. The transient temperature solution, on the other hand, can predict very high local temperatures if the heat load is very high, but this is also a local and relative effect, albeit in time. That is, spikes in the heat load distribution in space will be smoothed out over time, and in the limit of very long simulation times, the transient solution will approach the steady state solution.

Closing Remarks

What can we conclude from all of this information? If you are solving an electromagnetic heating problem and are only interested in computing the total electromagnetic losses and temperature distribution, you can usually avoid adding fillets to your model.

The advantages here are twofold. You do not need to go through the CAD modeling effort of adding any fillets to your geometry and you do not need an overly refined mesh in the sharp corners, which can save you the most valuable resource: time!

Circuit Breakers Increase the Safety of Electrical Systems

In residential buildings, power is often distributed between various electrical circuits connected to a panel. If too many devices draw power from a single electrical circuit, it can exceed the circuit’s limit, causing an overload. Other electrical events, like overcurrents, can also harm critical parts of the system, requiring expensive repairs and possibly creating a fire hazard.

Circuit breakers and fuses help prevent these issues. Let’s focus on electric switch circuit breakers, which “trip” and interrupt the current flow by moving a plunger as soon as a defect is detected. Unlike fuses, circuit breakers do not need to be replaced after activation; they can simply be reset. As anyone who has used both options knows, resetting a circuit breaker is much easier.

A circuit breaker panel.

Circuit breakers are classified by features such as:

• Construction type
• Voltage rating
• Structure
• Interruption methods

So far, we’ve only described circuit breakers for household use. Breakers based on very different designs are used to protect systems with higher currents than you would deal with on a residential scale, such as power lines and factories. Here, we analyze one of these more heavy-duty types of circuit breakers, a magnetic power switch type of circuit breaker. This electromechanical device uses the magnetic attraction from a current flowing through coils to move iron plungers. The switch resets to its initial state when the driving current is turned off.

Accurately Simulating a Magnetic Power Switch with the AC/DC Module

The tutorial model of the magnetic power switch has two main purposes:

1. Determining possible solutions to modeling a magnetic power switch
2. Investigating the working principles of the switch

The model geometry can be created using the built-in CAD tools in the COMSOL Multiphysics® software as well as parameterized Geometry Parts to more finely control the geometry. You can also use symmetry to reduce the geometry to 1/4 of its original size, and embed the model in an air domain to compute the electromagnetic fields.

The magnetic power switch geometry.

The geometry consists of two bulk E-shaped iron cores that are separated by an air gap. The lower E-core is fixed, while the upper E-core, also called the moving plunger, is held in place by a prestressed string. As a current flows through the copper coil on the central leg of the lower E-core, it exerts an attractive force onto the plunger. You simulate the attractive force with a Force Calculation feature and use the resulting value in an ordinary differential equation (ODE) describing the plunger dynamics based on Newtonian mechanics.

Eventually, the force reaches a threshold value, causing the plunger to close the air gap by moving down toward the lower E-core. The closing time depends on the spring stiffness. As such, the model takes into account the spring and constraint arrangements that keep the plunger at an equilibrium position.

You can model the closure by coupling the Magnetic Fields interface to the Moving Mesh interface. Together, these interfaces calculate the magnetic fields in a geometry that changes over time.

Magnetic power switch model in motion.

In the model, you can simulate the movement and time it takes for the plunger to close the air gap and account for the influence of magnetic forces and induced currents.

Next, let’s discuss the rigid body dynamics of the magnetic power switch.

Transient Analysis of a Circuit Breaker Device

The main time-dependent study step, which analyzes a time period from t = 0 s to t = 1 s, can be broken into different stages. The current grows in the first 45 ms of the simulation. During this time, the air gap remains open, since the electromagnetic attractive force isn’t enough to overcome the force of the opposing spring. It isn’t until 45–85 ms after the start of the simulation that the electromagnetic attractive force becomes enough to move the plunger down toward the iron core.

While this movement occurs, the current begins to decrease because of the inductance changing. The inductance reaches its minimum value when the plunger has closed the air gap and stopped at its new position. Then, since the contact between the plunger and core creates a new stationary RL circuit, the current increases again. The slope of this increase is dependent on the new characteristic time of the device.

Magnetic flux density norm results showing an open air gap at t = 0.05 s (left) and a closed air gap at t = 0.1 s (right). The induced eddy currents seen here appear to screen the interior of the core from the magnetic field. You resolve the currents in these plots by using a boundary layer mesh and restricting them to a region as large as the skin depth.

The simulation results show the evolution of both current density and magnetic flux density over time. As seen below, the spring reaches its maximum compression at t = 0.1 s (bottom left). Note that the induced currents in the core decay by t = 0.5 s (bottom right), well before the simulation time ends.

Plots showing the current density (surface) and magnetic flux density (streamlines) of the magnetic power switch. The plots are used for studying instances when the spring is prestressed (top left), begins to compress (top right), reaches a maximum compression (bottom left), and is completely compressed with the induced currents in the core decayed (bottom right).

You can also analyze the core losses caused by the induced current activity in the switch. This is useful to investigate potential overheating in the switch, which is a key concern in circuit breaker design.

Core losses caused by induced currents at 50, 100, and 200 ms.

Moving on, the 1D plots can be used to study the dynamics of the switch during the process. Start by looking at the beginning of the simulation, before switching (plunger movement). At this time, the spring isn’t compressed and the gap size remains the same (as indicated by the straight green line). Here, you can see that the normalized current (blue line) is similar to the response of the ideal system (red line).

Plot showing the magnetic flux density norm before the plunger motion.

By expanding the time scale, you can look at the dynamics of the system during and after the plunger moves. The plot below indicates that the mechanical power (red line) only rises above zero when the plunger is in motion, as expected. When the gap closes, the mechanical power returns to zero.

Plot showing the magnetic flux density norm during plunger motion.

Take a look at the induction losses in the core (the red line in the graph below) over a longer time period. These losses are significant when the plunger is in motion — a factor that may need to be considered when designing magnetic power switches, depending on the device details and its intended performance. When the movement ceases, the normalized current increases again. This is expected for a nonlinear RL circuit.

Plot showing the magnetic flux density norm throughout the entire simulation time.

Simulation can be a helpful tool when studying circuit breakers such as magnetic power switches. Using models like the one discussed here, you can study key magnetic power switch design factors. Give the tutorial model a try yourself for more information on this process.

The Evolution of Space Exploration Technology

The trend of miniaturization is one that we can see in a variety of applications, including mobile phones and computers. The same can be said for the design of satellites used in space missions. The devices used in NASA’s Space Technology 5 (ST5) mission are just one example.

Microsatellites mounted on a payload structure for the ST5 mission. Image by NASA. Licensed under the public domain, via Wikimedia Commons.

Due to the payload complexity of microsatellites — and the desire to extend their reach outside of Earth’s orbit — active thermal control is very important. Such control demands more power and also increases the mass of the satellite with added parts. The challenge is to design a thermal control system that can meet these power and mass demands while still removing excess heat in a controlled manner.

With this in mind, NASA used electrostatic comb drives for actuation in their ST5 mission. These actuation systems were paired with two different radiator designs: a louvre and a shutter configuration. The mission helped to validate the use of high-voltage MEMS technology in thermal subsystems.

Left: The shutter concept. Right: An optical microscope image of the shutter radiator design. Images by L. Pasqualetto Cassinis and taken from his COMSOL Conference 2016 Munich paper.

Looking to build upon these initial findings, a researcher from TU Delft considered an alternative to using electrostatic comb drives: thermal actuators. These devices provide relatively high displacement with little applied voltage and are less sensitive to radiation than their electrostatic counterparts. To validate their potential in such applications and further optimize their design, the researcher turned to the COMSOL Multiphysics® software.

Verifying the Potential of Using Thermal Actuators in Microsatellites

For this analysis, two models were built in COMSOL Multiphysics. The first is a 3D structural model of the shutter array, a configuration chosen based on its robustness.

3D shutter array model. Image by L. Pasqualetto Cassinis and taken from his COMSOL Conference 2016 Munich presentation.

The second is a 3D multiphysics model of a two-arm thermal actuator made of polysilicon — a model based on the Joule Heating of a Microactuator tutorial. An applied voltage generates electric current through the two hot arms, raising the temperature of the actuator. This temperature increase leads to thermal expansion, which then causes the actuator to bend. In addition to these hot arms, the thermal actuator includes a cold arm, with a gap that separates the two types. Note that the hot arms have more electrical resistance than the cold arm, thus greater Joule heating.

Thermal actuator model geometry. This image is taken from the documentation for the Joule Heating of a Microactuator tutorial.

To validate the thermal actuator model, the researcher compared the simulation results with analytical results and checked if the output displacement was close to the requirement of 3 µm. In the model, the displacement is 2.54 µm — a value comparable to that of analytical results (2.11 µm) and also near the required displacement. Note that the theoretical model only includes one hot arm, which can account for some of the differences in displacement values. Further, the simulation shows agreement in regard to temperature distribution, with the highest temperature at the center of the actuator.

A spring-like force is added to the shutter model to account for stiffness. With varying forces applied to the device, the shutter exhibits elastic behavior. The estimated stiffness obtained via the study is incorporated into the thermal actuator model. When varying the voltage to evaluate tip displacement via actuation, high voltages are needed to produce reasonable displacement. Additionally, as expected, the maximum displacement occurs at the center of the actuator instead of the tip.

Optimizing the Thermal Actuator Design

After verifying the thermal actuator model, the researcher sought to optimize its configuration. In this optimization study, the length of the actuator is varied along with the gap between the hot arms and the cold arm. Per analytical results, both variables are assumed to have a strong impact on tip displacement.

In the initial optimization study, an applied voltage of 2.7 V produces a shutter stiffness of 10⁹ N/m³ and a displacement of 2.98 µm. Additionally, the maximum temperature that the device reaches is significantly lower than the melting temperature of silicon.

The displacement (left) and temperature (right) of the thermal actuator with an applied voltage of 2.7 V. Images by L. Pasqualetto Cassinis and taken from his COMSOL Conference 2016 Munich presentation.

Reducing the required applied voltage was the focus of a later optimization study. Just a few volts can be crucial in, for instance, applications of CubeSats — a type of miniaturized satellite used for space research — where power demand is limited. For this study, multiple objective variables are considered and the gap between arms is included as a control variable. With this approach, the displacement comes closer to 3 µm and the applied voltage is reduced to about 2.5 V.

Multiphysics Simulation Helps Take Microsatellites to New Heights

Advancing the design of miniaturized satellites is key to extending their use in space exploration. As we’ve highlighted with this thermal actuator example, simulation is a useful tool for testing active thermal control techniques in these systems, improving their safety and reach. We look forward to seeing how this technology will continue to advance in the future and the potential role that simulation will play.

Learn More About Advancing Space Exploration with Simulation

• Read the full COMSOL Conference paper: “Feasibility Study of Thermal Actuators for MEMS Variable Emittance Radiators“
• Browse additional blog topics relating to space exploration:
  • Improving Atmosphere Revitalization for Manned Spacecraft
  • Rosetta and Philae: A Historic Landing on a Comet
  • Designing Reliable and Efficient CDRA Systems with Simulation
• Download the related tutorial: Joule Heating of a Microactuator

What Is Solar-Grade Silicon?

Solar-grade silicon is one of three grades of high-purity silicon. Each grade has different applications and specific purity percentage requirements:

1. Metallurgical-grade silicon is 98% pure
2. Solar-grade silicon is 99.9999% pure (6N or “six nines”)
3. Electronic-grade silicon is 99.9999999% pure (9N)

The structure of monocrystalline silicon. Solar-grade silicon is almost pure silicon.

Comparing Solar-Grade Silicon Production Methods

Traditionally, solar-grade silicon is produced using high temperatures (2000°C) to reduce silicon quartz and carbon, resulting in silicon with a 98.5% purity. This isn’t quite pure enough to be considered solar grade, so the silicon must be refined further through a gas phase. With multiple steps and different processes, this method isn’t efficient. It is also energy intensive, expensive, and requires experienced operators.

The method that JPM analyzed starts with raw materials that are highly pure. The silicon is placed into a contaminant-free microwave oven that performs both the heating and gas phase stages of the traditional production process. Since there’s no consecutive refinement processes, this approach is more efficient and cost effective.

The setup for the microwave furnace. Image by N. Rezaii and J.P. Mai and taken from their COMSOL Conference 2016 Munich paper.

The microwave furnace consists of five parts:

1. Magnetron core, which generates electromagnetic microwaves
2. Waveguide, which transmits the microwaves into the resonator
3. Resonator (also called the reaction chamber), which includes a crucible to hold the silicon sample
4. Tuner, which improves the absorption of the microwaves
5. Circulator, which keeps the magnetron from overheating by using a water bath to dissipate the reflective microwave energy

One advantage of an optimized microwave furnace design is that there is reduced heat loss. This is partially due to the selective heating, which heats materials on a volumetric heat input, leading to a temperature drop from the inside out. In addition, there’s less diffusion of the silicon’s impurities because the furnace has a faster warming time and shorter residence time.

To optimize the microwave furnace for solar-grade silicon production, JPM Silicon GmbH studied its internal processes with the COMSOL Multiphysics® software.

Simulating the Production of Solar-Grade Silicon in a Microwave Furnace

The research team set up their model to include the electromagnetic, chemical, and physical phenomena occurring within the microwave furnace. Since some materials have electromagnetic properties that are strongly temperature dependent, the model couples the electromagnetic field distribution and temperature field.

You can learn more about the model setup by reading the full conference paper “Multiphysics Modelling of a Microwave Furnace for Efficient Solar Silicon Production“.

It’s important to use chemically stable structural materials and an inert gas in the microwave furnace to avoid unwanted reactions. Also, the insulation materials must be effective in minimizing heat losses.

Electromagnetic Intensity and Distribution

The research team used the RF Module to simulate the electromagnetic intensity and distribution in the resonator and silicon sample. They used Maxwell’s equations to determine the propagation of the microwave radiation.

The electric field is higher at the height of the waveguide ports than at any other part of the reaction chamber. The field enhancement in the crucible’s core indicates that this is the optimal location for the crucible to be heated, as shown in the results below.

The distribution of the electric field in the resonator and waveguide. Image by N. Rezaii and J.P. Mai and taken from their COMSOL Conference 2016 Munich paper.

The researchers also wanted to see how varying the height of the insulation plate affects the operation of the furnace. They tested three different heights for the plate (which the crucible sits on top of) and reexamined the electric field. The different insulation plate heights include:

• 30 mm
• 40 mm
• 50 mm

The distribution of the electric field when the height of the insulation plate is 30 mm (left), 40 mm (middle), and 50 mm (right). Images by N. Rezaii and J.P. Mai and taken from their COMSOL Conference 2016 Munich paper.

The simulation results show that the 40-mm insulation plate performs best. The electric field is focused at the center of the crucible, thus on the silicon sample.

Gas Flow Velocity Distribution

The CFD Module solves for the Navier-Stokes equations, allowing the researchers to find the gas flow velocity distribution. The gas flows from the inlet over the surface of the silicon sample, rather than having a homogeneous velocity. The wall then deflects the flow toward the outlet. The simulation shows that only a slight gas flow exists near the waveguide ports as well as near the top and bottom walls.

The distribution of gas velocity in the resonator. Image by N. Rezaii and J.P. Mai and taken from their COMSOL Conference 2016 Munich poster.

Heat Distribution

To analyze how well the electromagnetic waves heat the silicon sample, the research team examined the heat distribution in the resonator. Their model includes forced heat equations to calculate conduction, convection, and radiation from solids and liquids (Planck’s radiation law) as well as gases (Stefan-Boltzmann law). The dissipated heat, solved with the RF Module, is used as a volumetric heat source. The gas velocity profile, calculated with the CFD Module, helps find the convective thermal losses.

As expected from the electromagnetics study, the hottest point in the resonator is at the crucible’s core. Further, the surrounding insulation layers don’t heat up as much, thanks to their lower thermal conductivity.

The distribution of heat in the resonator. Image by N. Rezaii and J.P. Mai and taken from their COMSOL Conference 2016 Munich poster.

By gaining insight into the internal processes of a microwave furnace, researchers from JPM Silicon GmbH were able to optimize their design and pave the way for efficient solar-grade silicon production.

Read More on Simulating Solar Applications

• Explore other examples of solar simulation applications in these blog posts:
  • Optimizing the Production Process for Solar Energy Cells
  • Analyzing a Silicon Solar Cell Design with the Semiconductor Module

The Large Hadron Collider: The Most Powerful Particle Accelerator

Built by the European Organization for Nuclear Research (CERN), the Large Hadron Collider (LHC) is a structure that holds many records. It’s not only the most complex experimental facility that has ever been built and the largest single machine that exists, but it’s also currently the world’s largest and most powerful particle accelerator. The LHC has the potential to provide answers to various physics-related questions. (Take a virtual tour to see it for yourself.)

Part of the Large Hadron Collider’s tunnel. Image by Julian Herzog — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.

Behind the operation of the accelerator is a 27-kilometer ring of superconducting magnets and multiple accelerating structures that give particles an energy boost. These magnets, which are made of coils that can operate in the superconducting state, maintain a strong magnetic field that guides particle beams around the accelerator ring.

To create such strong fields, iron-yoked electromagnets, fully wound with rectangular cables, are used. Nb-Ti filaments are embedded inside a copper matrix to form a strand, which is then twisted and wrapped in a polyamide insulation layer. When the cable is cooled to 1.9 K, the filaments are able to reach the superconducting state, which in turn allows the cable to carry greater current densities.

Left: Cross section of the magnet. Right: Cable layout. Images by L. Bortot, M. Maciejewski, M. Prioli, A.M. Fernandez Navarro, S. Schöps, I. Cortes Garcia, B. Auchmann, and A.P. Verweij and taken from their COMSOL Conference 2016 Munich paper.

When designing superconducting magnets like those in the LHC, it’s important to consider potential scenarios that could cause disruptive effects. One such example is quenches.

Studying Quenches in Superconducting Magnets

A quench refers to the sudden transition of a magnet from the superconducting state to a normal state. This process occurs when the working point of a superconductor magnet moves out of what is called the critical space, which then causes energy stored in the magnetic field to be released as Ohmic losses.

Plot of the critical surface for the filaments, with the maximum current density shown as a function of the magnetic and temperature fields. Image by L. Bortot, M. Maciejewski, M. Prioli, A.M. Fernandez Navarro, S. Schöps, I. Cortes Garcia, B. Auchmann, and A.P. Verweij and taken from their COMSOL Conference 2016 Munich paper.

When a quench takes place, it forces the conduction current to travel from the filaments to the copper matrix in which they are embedded. Such movement can cause the coils inside the magnet to overheat. In an effort to prevent possible disruption effects, magnet designs typically include quench detection and protection systems. Ensuring the effectiveness of these systems, however, requires an understanding of the electrothermal transient phenomena that takes place within the magnet.

Recognizing this, a team of researchers from CERN simulated a quench event in a superconducting magnet, using a main dipole from the LHC as their point of analysis. Let’s see how the flexibility and functionality of the COMSOL Multiphysics® software helped them to simulate this complex design.

Building an Electrothermal Model of a Superconducting Magnet

When modeling a superconducting magnet, one of the challenges is accounting for the number of half turns. For the main dipole of the LHC, this number is 320. Each of these half turns must be set up with its respective variables and operators in order to compute relevant quantities. This process is not only time consuming, but it is also prone to error.

Cross section of a magnet’s coil with highlighted half turns. Image by L. Bortot, M. Maciejewski, M. Prioli, A.M. Fernandez Navarro, S. Schöps, I. Cortes Garcia, B. Auchmann, and A.P. Verweij and taken from their COMSOL Conference 2016 Munich paper.

To speed up the model creation process and reduce the chances of error, the researchers from CERN developed an automated Java® workflow that relies on the COMSOL API. The structure of this application is based on three main functional layers:

1. Top layer, which enables users to describe a desired model with text input files
2. Middle layer, which includes numerical methods needed to formulate input parameters for the API
3. Bottom layer, which offers classes to embed functionalities from the COMSOL API for use with Java®

With this workflow in place, the magnet’s half turns were implemented in the model design with the indexing feature. This feature is quite useful as it allows you to redefine variables that have a formulation that’s common to every turn. Therefore, only a single variable is needed to describe a group of domains that share the same property.

Further, through the Java® workflow, the team at CERN was able to define various geometrical primitives, from points to lines, to construct their 2D model. Including model symmetries in the application helped to simplify the modeling process.

To minimize the number of mesh nodes, and thus the computational time, a combination of unstructured and structured elements was included in the model’s mesh. In order to ensure the accuracy of the results, the researchers performed a mesh sensitivity analysis.

The physics implemented in the model accounted for nonlinear temperature and field-dependent material properties and eddy currents induced within the superconducting cable. The latter enabled the team to calculate the quench initiation and propagation.

Simulation Results for Two Time-Dependent Studies

The analysis includes two time-dependent studies that were performed consecutively:

1. The magnet’s current, linearly ramped up to the nominal value
2. The exponential current decay, simulated at a time constant of 0.1 s

Note that the second study uses the final state of the first study as its initial condition.

The researchers first looked at the behavior of the magnet during fast discharge. The plot on the left shows the magnetic field in the magnet at nominal conditions. During a linear ramp up of 100 A/s, the field’s variation produces eddy currents. The equivalent magnetization of these currents is shown in the plot on the right.

Left: Magnetic field in the magnet at nominal current. Right: Equivalent magnetization of eddy currents during linear ramp up of 100 A/s. Images by L. Bortot, M. Maciejewski, M. Prioli, A.M. Fernandez Navarro, S. Schöps, I. Cortes Garcia, B. Auchmann, and A.P. Verweij and taken from their COMSOL Conference 2016 Munich paper.

The losses that are generated impact the magnetic field, affecting the magnetic equivalent electrical impedance. They also deposit energy in the magnetic coil, dissipating some of the energy stored inside the magnetic field. If they are high enough, these losses can cause the temperature of the superconductor to rise beyond the critical surface. This can in turn cause the superconductor to transition to a normal state. At this stage, Ohmic losses are dominant in causing the magnet’s coil to heat up. The temperature of the coil is extracted after 0.5 s and visualized in a plot.

Left: Eddy current losses deposited in the coil. Center: Ohmic losses deposited in the coil. Right: Temperature distribution in the coil. Images by L. Bortot, M. Maciejewski, M. Prioli, A.M. Fernandez Navarro, S. Schöps, I. Cortes Garcia, B. Auchmann, and A.P. Verweij and taken from their COMSOL Conference 2016 Munich paper.

Along with the coil temperature, coil resistance and voltage are also extracted from the simulation results. These values can be used as input when designing protection systems for superconducting magnets.

The coil’s resistance (left) and resistive voltage (right) as a function of time. Images by L. Bortot, M. Maciejewski, M. Prioli, A.M. Fernandez Navarro, S. Schöps, I. Cortes Garcia, B. Auchmann, and A.P. Verweij and taken from their COMSOL Conference 2016 Munich paper.

To learn more about this simulation research, read the full COMSOL Conference paper: “Simulation of Electro-Thermal Transients in Superconducting Accelerator Magnets“. For more examples of using COMSOL Multiphysics to simulate superconductors, browse the resources listed below.

Additional Resources on Simulating Superconductors

• Learn about modeling superconductivity in a YBCO wire
• Explore the use of simulation in designing fully superconducting rotating machines

Oracle and Java are registered trademarks of Oracle and/or its affiliates.

Properties of Underground Power Cables

A typical underground three-phase electrical cable is made up of a bundle of three conductive cables. Each individual cable is stranded, meaning it is composed of many wires that are twisted together and compressed so that the strands are in good electrical contact. The cable can also have shielding such as a metal foil. A polymer material between the cable and shielding provides electrical insulation. Wound paper, fluids, and even pressurized gases are also used as electric insulators. The entire insulated cable bundle is then encapsulated within another dielectric and a metal sheath as well as an outer polymer coating, which protects the cable from the environment.

Left: An underground three-phase electrical cable. Right: Cross-sectional schematic of a buried three-phase cable.

The alternating current passing through the cable results in a time-varying magnetic field, which causes induced currents in the cable as well as in the surrounding metal sheaths and foil. The currents lead to a combination of Joule heating and induction heating. The cable bundle then begins to heat up, possibly causing it to fail, hence our interest in building a predictive computational model.

The electrical analysis of the cable is fairly straightforward. We usually know all of the relevant material properties (electric conductivity, permeability, and permittivity) in the cable bundle as well as exactly how much current flows through the cable and at what frequency. However, we only have a rough understanding of the electrical properties of the surrounding soil.

Thermally speaking, there are even more unknowns. The thermal properties of the surrounding soil vary based on its composition and moisture content. Even within the cable, although we know the material properties, there can be thin layers of material and small air gaps that significantly change the peak temperature.

Let’s find out how we can model these types of cables using COMSOL Multiphysics.

Modeling Electromagnetic Fields in an Underground Cable

We can reasonably assume that underground cables are long and the surrounding environment is relatively uniform. These assumptions allow us to simplify our model by considering a 2D cross-sectional slice, similar to the one shown in the schematic above. We know that the three-phase current in the cables varies at a fixed frequency. We also know the maximum current.

We assume that the stranded bundle of copper wires is compressed together with good electrical contact, so we treat each of the three copper cables as one uniform domain across which the current can redistribute itself. Thus, we use three different Coil features to excite the three copper cables, as shown in the screenshot below. The applied excitation is of the form: 1[kA]*exp(-i*120[deg]).

That is, a 1kA peak current flows through the three cables, but the relative phases are shifted by 120° between each.

The Coil feature sets up the current flowing through one of the cables. The other two cables carry the same current, but with a 120° phase shift.

Next, we consider the thin layer of metal shielding. If the thickness of this layer is small compared to the other dimensions, then we model this metal layer via the Transition boundary condition, as shown below. This boundary condition allows us to enter a thickness and specify a set of material properties at an internal boundary of the model. The advantage of this condition is that we don’t need to explicitly model the geometry and thus don’t need to mesh this thin layer of material.

The Transition boundary condition in which the layer thickness and material properties can be entered.

The magnetic fields can extend some distance outside of the cable. Since we want to know how quickly the fields drop off, we model a region of soil around the cable. We choose this region’s size by studying progressively larger domains until the field solution shows minimal variation with an increasing domain size, a procedure described in an earlier blog post on choosing boundary conditions for coil modeling. The results of such an analysis, seen in the image below, show the magnetic field and cycle-averaged losses. It is these losses that lead to a rise in temperature.

The losses in the cable bundle and shielding with arrows representing the magnetic field. The arrow lengths are logarithmically scaled relative to the magnetic field strength.

Predicting the Temperature Rise in COMSOL Multiphysics®

Modeling the temperature rise of the cable seems relatively straightforward — we simply take the computed losses and include them in a thermal model. We add the Heat Transfer in Solids physics interface to our model and, within the Multiphysics branch, use the predefined features to set up a bidirectional coupling between the electromagnetic and thermal problem. Either the Frequency-Stationary or Frequency-Transient study type can be used to solve the electromagnetic problem in the frequency domain while solving the thermal problem in the steady state or time domain.

The Frequency-Stationary solver and multiphysics settings for creating the bidirectionally coupled electromagnetic heating model.

Using the Thin Layer Boundary Condition

Now, even though the thermal model appears straightforward, there are a number of things that we need to keep in mind as well as features of the software that we want to be aware of. For one, the cable bundle has several thin layers of material, such as the shielding and coatings, that we might not want to model explicitly. For these, we use the Thin Layer boundary condition, which has the option of modeling the thin layer as a Thermally thin approximation, Thermally thick approximation, or General layer, as shown in the screenshot below.

The Thermally thin approximation is appropriate when the material layers have a relatively much-higher thermal conductance than their surroundings, whereas using the Thermally thick approximation is better for material layers with a relatively much-lower conductance. The General type should be used for any intermediate cases, where there are significant thermal gradients both normal to and tangentially along the layer of material. All of these options allow you to specify the layer thickness and properties, and the General type additionally allows you to specify a composite of up to five different layers.

The Thin Layer boundary condition.

The Thin Layer boundary condition is appropriate for well-defined layers of material, with known thicknesses and properties. We also need to consider the thermal resistance that arises when two materials are in contact. Heat transfer between rough surfaces in contact occurs when there is:

• Conductive heat transfer due to the solid materials being pushed together
• Conductive heat transfer through the thin layer of air
• Radiative heat transfer between the exposed surfaces

These effects can all be modeled via the Thermal Contact feature, seen in the following screenshot.

The Thermal Contact feature and equations.

The conductive heat transfer through the solids is strongly affected by the contact pressure. This pressure can be computed from (and coupled with) a structural analysis, as exemplified by these tutorial models:

• Thermal Contact Resistance Between an Electronic Package and a Heat Sink
• Contact Switch

Modeling the Thermal Environment and Domain

Next, we need to consider the thermal environment, which has a great deal of variability that directly affects the cable temperature. The surrounding soil, concrete, and rocks have thermal conductivities that range from 0.1 to 5 W/m/K and densities that range from just over 1000 kg/m³ for very loosely packed soil to over 3000 kg/m³ for solid rocks. Their material-specific temperatures also vary from ~500 to 1500 J/kg/K. Furthermore, these values do not remain constant. For example, the thermal conductivity of dry and wet sand can differ by over an order of magnitude: from ~0.2 to 4 W/m/K. It is also helpful to introduce the thermal diffusivity, which is defined as and ranges from roughly for these materials.

In addition to the huge variability of the soil’s thermal properties, the thermal boundary conditions at the surface are rarely well defined. There is both convective cooling to the air and radiative cooling to the sky. The magnitude of this cooling is greatly affected by the local and temporary features at the surface. For example, dead leaves or loosely packed snow can act as a very good layer of thermal insulation that is difficult to quantify with any precision.

Lucky for us, the cables are buried deep enough that these temporary variations at the surface can often be neglected. Thus, it is reasonable to approximate the heat balance at the surface with a combination of three boundary conditions:

1. A Heat Flux boundary condition representing the solar heat load based on the latitude and time of year
2. Another Heat Flux boundary condition representing the convective cooling to the average ambient air temperature
3. A Diffuse Surface boundary condition representing the radiative cooling to the effective sky temperature

The solar heat load and ambient air temperature can be entered approximately or from the American Society of Heating, Refrigerating, and Air-Conditioning Engineers database of weather station data, as we describe in a previous blog post. The effective sky temperature ranges from about 230 K to 285 K (-45°C to 10°C), depending on the air temperature and cloud cover, with a typical ground surface emissivity of 0.8–0.95.

We must also consider the width and depth of our thermal domain. We need to model a sufficiently large domain of soil such that the boundary conditions don’t affect the results. For thermal loads that vary sinusoidally in time with cycle period , the distance D from the boundary at which the temperature oscillation is reduced by approximately 90% relative to the oscillation at the surface is given by: .

Assuming that the thermal boundary conditions vary sinusoidally over the year, and assuming a very high thermal diffusivity, a good rule of thumb is to model a domain that extends at least eight meters beneath the surface and is at least three times the burial depth on either side, with thermal insulation boundary conditions on the vertical boundaries and a fixed temperature boundary condition at the bottom boundary. A Temperature boundary condition is used to fix the temperature at the bottom to the average of the surface temperature over an entire year. This is a good approximation of the large thermal mass of the ground.

We can also investigate a larger soil domain to see if peak temperatures are noticeably affected. Of course, if there are known subsurface features, such as nearby water mains or building foundations, these should be included in the model.

Solving the Model

We can solve the model using either a Frequency-Stationary or Frequency-Transient study type. Both solve for the frequency-domain form of Maxwell’s equations, but solve the thermal model as either a steady-state or time-dependent problem. Solving for the steady-state temperature requires a bit of care in interpreting the results. A steady-state analysis will assume that all thermal transients have died out, a rather severe assumption. Such results must be interpreted with care. Solving the transient problem, on the other hand, can consider all of the changing environmental conditions and loads and will give not just the peak temperatures but also the duration at which different materials are at different temperatures.

The screenshot below shows a typical model setup and sample results.

The thermal model and representative results of the cable’s temperature. The magnetic fields are only solved for the smaller circular domain around the cable, since the magnetic fields drop off rapidly in intensity.

Closing Remarks on Modeling the Electromagnetic Heating of Underground Cables

Here, we have shown the appropriate COMSOL® software features and modeling approaches for computing the temperature rise in underground power cables. When solving such problems, keep in mind the variability in the solution that can be introduced due to the changing thermal environment, imprecisely known soil properties, and even small air gaps or standoffs within the cable itself. Of course, COMSOL Multiphysics (along with the AC/DC and Heat Transfer modules) is a great tool for modeling these situations and for considering all of the variability in the model inputs.

Interested in using COMSOL Multiphysics to model electromagnetic heating?

Further Reading

• Learn more about electromagnetic simulation on the COMSOL Blog:
  • How to Choose Between Boundary Conditions for Coil Modeling
  • How to Model Electrodynamic Magnetic Levitation Devices
  • Analyzing a Component of the ITER Tokamak with Simulation
  • Streamlining Capacitive Touchscreen Design with Apps
• Download the Cable Tutorial Series, which features six tutorial models and related documentation

Induction Heating: A Viable Technique for Food Processing

Since ancient times, food processing has played an important role in delivering quality products to consumers. Techniques such as salt preservation, fermentation, and sun drying were used early on to remove harmful toxins and microorganisms from food, enhance its overall consistency, and enable its transportation over long distances. While some of those same techniques are still used today, the food industry has turned to more sophisticated approaches over time that foster greater efficiency and higher quality.

Sun drying is one type of food processing technique. Image by ArianeCCM — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.

Induction heating, for instance, has become a favorable option for food processing. Compared to methods where pipes are heated by hot water or steam, induction heating enables better control of temperature, which, in turn, helps prevent local high temperatures. This better preserves the quality of the food, while optimizing the efficiency of food processes like pasteurization and sterilization. What’s more: Pipes can easily be swapped out to treat different types of food, as they don’t need to be in direct contact with the coil. And, as a further advantage, the process itself requires little to no water as it is based on electricity.

When it comes to the reliability and efficiency of induction heating in food processing, the coil and pipe configurations have a significant effect. As we will highlight in the next section, numerical modeling apps serve as a useful tool for testing various design configurations, delivering fast and accurate results to its users.

Testing Various Design Configurations for an Inline Induction Heater with an App

To begin, let’s take a look at the underlying model of our Inline Induction Heater app. The model consists of a circular electromagnetic coil that is wound around a set of pipes through which a fluid (liquid food) flows. As an alternating current passes through the coil, it generates an alternating magnetic field. This field then penetrates the pipes, producing eddy currents within the pipes and heating them up.

Geometry of an inline induction heater model.

The app itself is designed to measure the inline induction heater’s efficiency in heating the fluid as it flows through the pipes. It investigates the use of both nonferritic and ferritic stainless steel pipes. Due to its relatively low and stable price, a result of the absence of nickel in its components, ferritic stainless steel has grown in popularity in the food industry, offering new potential in food processing.

Now that we’ve covered the foundations of the app, we can shift gears to its design. Keep in mind that when designing an app of your own, you have control over the parameters that are included and its particular layout and structure. The result: an easy-to-use tool that is tailored to your specific needs.

The user interface (UI) of the Inline Induction Heater app.

The Inline Induction Heater app features two main sections: the Settings section on the left-hand side and the Graphics section on the right-hand side. Under Settings, users can easily modify the dimensions of the pipes and coil. For further guidance, the validity of the Geometry status is shown after each geometric input field is updated. The following criteria must be met for the status to be valid:

• The coil’s length and position must fit with the length of the pipes
• The thickness of the pipes must be proportional to the internal radius

If these criteria are not met, the status will render as invalid until the necessary modifications are made to the geometry. This helps ensure that users generate realistic simulation results when running their tests.

Tabs for the Settings section.

The Materials and Operating Conditions section, meanwhile, features a series of input parameters that allow specification of the pipe material, operating conditions of the coil, flow conditions, and target temperatures. In the case of the pipe material, users have the ability to choose from two predefined options: Stainless Steel 410 (martensitic alloy) and Stainless Steel 410s (ferritic alloy). Fitting with the theme of customization, they can also choose the Stainless Steel (User defined) option in which it is possible to modify the electrical conductivity and relative permeability.

After making their respective modifications to the parameters, users can then shift their focus to the Graphics section. Several graphical plots are shown here, including the geometry, temperature and magnetic flux density, fluid temperature along the pipes, magnetic flux density, fluid temperature, and fluid velocity magnitude.

Modifying the geometry and materials and conditions of the inline induction heater, with numerical results shown. Note that no sounds are included with this recording.

The Numerical Results section displays the maximum temperature in the pipes and the minimum temperature at the outlet. If either of these values are above or below the critical value, users will be notified via a message under the Status sections. The average temperature elevation at the outlet and the thermal efficiency of the inline induction heater are also included under Numerical Results.

Once the users have obtained their results, they can generate a report by simply clicking the Create Report button in the ribbon. Doing so provides them with an easy way to share and communicate their findings with other people involved in the design workflow.

Optimizing the Design of an Inline Induction Heater for Food Processing

Apps are powerful tools that hide complex physics beneath an easy-to-use interface. By empowering more people to run their own simulation tests, these tools facilitate greater efficiency in design workflows, while ensuring accuracy in the results that are obtained.

Here, we have demonstrated how this approach can benefit the design of an inline induction heater for food processing, allowing users to easily test out different configurations until achieving the desired results. As this example illustrates, apps serve as a viable resource for helping the food industry accelerate the optimization of food processing techniques, pairing quality with efficiency.

Like the other demo apps featured in our Application Gallery, the Inline Induction Heater app is designed to serve as a source of inspiration and guidance in your own app-building processes. We encourage you to use it as a foundation for creating apps of your own, giving users the ability to change these same parameters or alternate ones that fit your modeling needs.

Further Resources on Building Apps and Modeling Food Processing

• Try it out yourself…Download the Inline Induction Heater app
• To get started on building apps of your own, explore our Intro to Application Builder Videos series
• Browse additional blog posts pertaining to food processing:
  • Analyze Solar Food Dryer Designs with Heat Transfer Modeling
  • Simulating the Freeze-Drying Process
  • Coupling Transport and Solid Mechanics Models for Better Puffed Rice