Heat transfer will take place when materials at different temperatures come into contact with one another. It may initially appear that the surface of each material is entirely in direct contact. However, upon closer inspection, you’ll find that many materials have a surface roughness measurable at the micron or nanometer scale.

When materials are in direct contact, thermal conductivity is determined by the properties of the two materials. However, surface roughness introduces gaps between contacting materials, which are usually filled with air. The thermal conductivity of gasses, such as air, is typically much lower than the conductivity of common solid materials. Therefore, the heat flux due to conduction is smaller in noncontacting regions, leading to increased thermal resistance at the interface.

Yet, if you increase the structural stress over the gap, you’re going to decrease the size and extent of the gaps and, therefore, influence the thermal resistance. Most of the time, there is also surface-to-surface radiation in the gap, however, it can be neglected in many common applications as the temperature difference between the materials is usually sufficiently small.

In the Model Gallery, you can find the pre-solved model “Thermal Contact Resistance Between an Electronic Package and a Heat Sink“, which can be used to investigate the effect of thermal contact resistance on heat transfer in an electronic package.

The model is based off of a study by M. Grujicic, C.L. Zhao, and E.C. Dusel of Clemson University titled “The effect of thermal contact resistance on heat management in the electronic packaging“. In their paper, the authors use finite element analysis (FEA) to investigate the effect that thermal contact resistance has on heat management in a simple central processing unit (CPU) and heat sink design. They explore the effect of surface roughness, the mechanical and thermal properties of the contacting materials, the contact pressure, and the effect of the materials on the maximum temperature experienced by the CPU in detail in their paper.

In the COMSOL Multiphysics model, part of the Grujicic et al. study is reproduced, where we take a look at the influence of four main parameters on thermal contact resistance, and, thereby, the heat transfer:

- Contact pressure
- Microhardness
- Surface roughness
- Surface roughness slope

The model geometry is composed of a cylindrical electronic package that is located inside a heat sink constructed of eight cooling fins. Device efficiency is dependent on the eight cooling fins of the heat sink, as well as on the efficiency of heat transfer between the electronic package and heat sink. The device geometry is shown below, where radial symmetry has been used to reduce the geometry to one sixteenth of its original size.

*Left: Heat sink and electronic package geometry, showing the eight cooling fins around the cylindrical package. Middle and right: Radial symmetry and simplification of the geometry.*

The electronic package is modeled as a cylinder with a radius of 1 centimeter and a height of 5 centimeters and is made of silicon. The heat sink is made of aluminum with fins reaching a distance of 2 centimeters from the cylinder axis. The electronic package produces a total heat source of 5 W. In order to dissipate this heat, a cooling fan blows room-temperature air at 8.5 m/s across the heat sink.

To define the cooling due to the air flow, we use the built-in heat transfer coefficient in COMSOL Multiphysics. The four parameters — contact pressure, microhardness, surface roughness, and surface slope — can all be modeled using parametric sweeps set up in the *Thermal Contact* interface. Both a free triangular mesh and a swept mesh are used in the model.

Tip: You can find more information about the mesh in the model documentation.

The figure below shows the temperature profile obtained using reference values:

*Temperature profile with reference values for the parameters.*

Closer to the fan (on the left side of the model), the temperature of the fins reaches about 483 K. The temperature increases with greater distance from the fan, reaching 490 K at the other extremity.

Next, we further analyze the model to determine the effect of contact pressure, microhardness, surface roughness, and surface slope on constriction and gap resistance within the model. The amount that each of these four parameters affect both the constriction resistance and gap resistance directly influences the material characteristics at the surface of the heat sink and electronic packaging. Thus, the heat dissipation from the electronic device is altered.

Below are the results from this analysis:

*Left: Constriction resistance depending on contact pressure (*x*-axis) and microhardness (*y*-axis). Right: Constriction resistance depending on roughness (*x*-axis) and roughness slope (*y*-axis).*

*Left: Gap resistance depending on contact pressure (*x*-axis) and microhardness (*y*-axis). Right: Gap resistance depending on roughness (*x*-axis) and roughness slope (*y*-axis).*

Contact pressure, roughness, roughness slope, and microhardness all affect the constriction resistance within the model. However, roughness slope has little to no effect on the gap resistance. We can see this in the bottom-right image, where the plot shows constant values in the vertical direction.

In their paper, Grujicic et al. make the conclusion that surface roughness and mechanical and thermal properties can have a significant effect on thermal contact resistance, and, therefore, on thermal management. According to Grujicic et al., thermal contact resistance, and the parameters that influence it, can play a major role in the heat management of electronic devices. Therefore, it may significantly affect device performance, reliability, and life cycle.

- Read the paper: “The Effect of Thermal Resistance on Heat Management in the Electronic Packaging” by M. Grujicic, C.L. Zhao, and E.C. Dusel
- Download the model: Thermal Contact Resistance Between an Electronic Package and a Heat Sink

The coil heat exchanger we’ll consider is shown in the figure below.

*A copper coil carries hot water through a duct carrying cold air.*

Copper tubing is helically wound so that it can be inserted along the axis of a circular air duct. Cold air is moving through the duct, and hot air is pumped through the tubing. The air flow pattern and the temperature of the air and copper pipes will be computed using the *Conjugate Heat Transfer* interface. Since the geometry is almost axisymmetric, we can simplify our modeling by assuming that the geometry and the air flow are entirely axisymmetric. Thus, we can use the *2D axisymmetric Conjugate Heat Transfer* interface. Since the airspeed is high, a turbulent flow model is used; in this case, it is the k-epsilon model.

We can assume that the water flowing inside of the pipe is a fully developed flow. We can also assume that the temperature variation of the water is small enough that the density does not change, hence the average velocity will be constant. Therefore, we do not need to model the flow of the water at all and can instead model the heat transfer between the fluid and the pipe walls via a forced convective heat transfer correlation.

The Convective Heat Flux boundary condition uses a Nusselt number correlation for forced internal convection to compute the heat transfer between the water and copper tubing. This boundary condition is applied at all inside boundaries of the copper piping. As inputs, it takes the pipe dimensions, fluid type, fluid velocity, and fluid temperature. With the exception of the fluid temperature, all of these quantities remain constant between the turns of the tubing.

As the hot water is being pumped through the copper coils, it cools down. However, since the model is axisymmetric, each turn of the coil is independent of the others, unless we explicitly pass information between them. That is, we must apply a separate Convective Heat Flux boundary condition at the inside boundaries of each coil turn.

This raises the question: How do we compute the temperature drop between each turn and incorporate this information into our model?

Consider the water passing through one turn of the copper coil. The heat lost by the water equals the heat transfer into the copper pipes. Under the assumption of constant material properties, and neglecting viscous losses, the temperature drop of the water passing through one turn of the pipe is:

\Delta T = \frac{Q}{\dot m C_p} = \frac{\int q'' dA}{\dot m C_p}

where \dot m is the mass flow rate, C_p is the specific heat of water, and Q is the total heat lost by the water, which is equal to the integral of the heat flux into the copper, integrated over the inside boundaries of the coil. This integral can be evaluated via the Integration Component Coupling, defined over the inside coil boundaries.

*The Integration Component Coupling defined over a boundary. Note: The integral is computed in the revolved geometry.*

Using these coupling operators, we can define a set of user-defined variables for the temperature drop:

`DT1 = intop1(-nitf.nteflux/mdot0/Cp0)`

This evaluates the temperature drop along the first turn of the pipe. We can define a different temperature drop variable for each turn of the pipe and use them sequentially for each turn.

*The water temperature in the sixth turn considers the temperature drop in the first five turns.*

*Flow field and temperature plot (left) and the temperature along a line through the center of the coils (right).*

Since this is a 2D axisymmetric model, it will solve very quickly. We can examine the temperature and the flow fields and plot the temperature drop along a line down the center of the coils. We can observe that the water cools down between each turn of the coil, and the air heats up.

This can be considered a parallel-flow heat exchanger, since the hot and cold fluids flow in the same overall direction. If we wanted to change this model to the counter-flow configuration, we could simply switch the air inlet and outlet conditions so that the fluids travel in opposite directions.

What other kinds of heat exchanger configurations do you think this technique can be applied to?

]]>

*Phase change* is a transformation of material from one state of matter to another due to a change in temperature. Phase change leads to a sudden variation in the material properties and involves the release or absorption of latent heat. We can use the Heat Transfer Module to model this type of phase change. Let’s start with an example.

In the continuous casting process, liquid metal is poured into a cooled mold and starts to solidify. As the metal leaves the mold, the outside is solidified completely, while the inside is still liquid. To further cool down the metal, spray cooling is used. When the metal is completely solidified, it can be cut into billets. This is a stationary, time-invariant, process. The rate at which the metal enters and leaves the modeling domain does not vary with time, and neither does the location of the solidification front.

Here is an illustration of the continuous casting process:

*Sketch of a continuous casting process.*

In order to optimize and improve this process, we can turn to simulation. With COMSOL software, we can predict the exact location of the phase interface.

COMSOL Multiphysics and the Heat Transfer Module together offer a tailored interface for modeling phase change with the *Apparent Heat Capacity method*. The method gets its name from the fact that the latent heat is included as an additional term in the heat capacity. This method is the most suitable for phase transitions from solid to solid, liquid to solid, or solid to liquid. Up to five transitions in phase per material are supported.

When implementing a phase transition function, \alpha(T), a smooth transition between phases takes place, within an interval of \Delta T_{1\rightarrow2} around the phase change temperature, T_{pc, 1\rightarrow 2}. Within this interval, there is a “mushy zone” with mixed material properties. The smaller the interval, the sharper the transition.

The below figure shows the phase change function for the continuous casting model:

*COMSOL Multiphysics settings for phase change. Keep in mind that phase 1 is below T_{pc, 1\rightarrow 2} and phase 2 is above.*

The material properties for the solid and liquid phase are specified separately. These values are combined with the phase transition function so that there is a smooth transition from solid to liquid. The heat capacity of the material is expressed as C_p=C_{p,solid}\cdot(1-\alpha(T))+C_{p,liquid}\cdot\alpha(T), and similarly for the thermal conductivity and density. For a pure solid, \alpha(T)=0, and for a pure liquid, \alpha(T)=1. Within the transition interval, the material properties vary continuously.

The latent heat is included by an additional term in the heat capacity. Let us take a look at the derivative of the phase transition function:

*Derivative of the phase transition function.*

Integrating this function over \Delta T_{1\rightarrow2} gives 1 and multiplying by the latent heat L_{1\rightarrow 2} gives the amount of latent heat that is released over \Delta T_{1\rightarrow2}.

Consider the stationary heat transfer equation with a convective term, of the form:

\rho C_p\cdot \nabla T=\nabla\cdot\left(k\nabla T\right)

The Apparent Heat Capacity method uses the following expression for the heat capacity:

C_p=C_{p,solid}\cdot(1-\alpha(T))+C_{p,liquid}\cdot\alpha(T)+L_{1\rightarrow 2}\frac{d\alpha}{dT}

The advantage of this method is that the location of the phase interface does not need to be known ahead of time.

With the help of the *Heat Transfer with Phase Change* interface, the implementation is straightforward. Axial symmetry is assumed, and the model is reduced to a 2D domain. The casting velocity is constant and uniform over the modeling domain.

To get a sharp transition and thereby the exact location between the solid and liquid phase, we need a small transition interval, \Delta T_{1\rightarrow2}. Resolving such a small interval properly requires a fine mesh. However, we do not know the location of the solidification front in advance, so we first solve the model with a gradual transition interval, and then use adaptive mesh refinement to get better resolution of the solidification interface. The transition interval can then be made even smaller.

The results are compared below for two different transition intervals. As the transition interface is made smaller, the model better resolves the transition between liquid and solid. This information can be used to improve the continuous casting process, and this same approach can be used for similar applications involving phase change.

- Download the model: Cooling and Solidification of Metal
- Read a user story: Optimizing the Continuous Casting Process with Simulation

Continuous casting, also known as *concasting*, is a process by which a continuous strand of steel is produced during one casting sequence that is subsequently cut into pieces for rolling. Unlike the batch process of ingot casting, which entails the casting of a single ingot at a time, continuous casting allows for the production of waste metal to be reduced, is more energy efficient than ingot casting, and produces products of a superior quality.

During ingot casting, the head of each beam must be cropped after each casting process, producing waste metal. By producing a long continuous strand of steel, on the other hand, cropping is only required at the very beginning and very end of the casting sequence (during which hundreds of ingots or “blooms” are produced), thereby reducing waste.

During continuous casting, molten steel poured from a ladle into a tundish from where it is drawn through a copper mold by a series of rollers, as shown in the diagram below.

*Diagram of the continuous casting process.*

After solidifying, these metal strands are cut into 3- to 15-meter long pieces and left to cool. In continuous casting, the shape of the strand can be cast closer in shape to the final product than what would be produced during ingot casting, greatly reducing the cost of further processing the strands. In the image below, billets are exiting the concast machine onto a discharge table, where the strands are being cut.

*There are four types of semi-finished casting products: ingots, billets, blooms, and slabs. Depicted here are billets exiting a continuous casting machine onto a discharge table.*

Nicolas Grundy, head of Metallurgy & Process Continuous Casting at SMS Concast, has found that simulation can be a valuable tool for better understanding and optimizing the continuous casting process. In a recent article that appeared in *COMSOL News*, Grundy explained how COMSOL Multiphysics was used during his research. “We are constantly pushing the limits and the only way to understand something that we have never done before is to simulate it,” Grundy describes in the article.

In his research, Grundy and his team used simulation to analyze every step of the process. One of the main goals of their study was to learn more about the solidification process and the mechanical deformations that can take place during both quenching and slow cooling of the metal slabs. The team found that by minimizing the segregation of alloying elements in the center of the strand, removing any non-metallic substances, and improving the microstructure of the solidifying steel, they could improve the quality of the final product. They achieved these goals by implementing a stirring process, during which the molten steel is mixed through the use of electromagnetic stirrers and by optimizing the design of the tundish.

In the figures below, you can see the COMSOL Multiphysics simulation the team used to help ensure that the liquid steel flow in the tundish is designed correctly to achieve the best quality steel.

*Model of the tundish created with the CFD Module.*

In addition to analyzing the effect of using electromagnetic stirrers, Grundy also simulated hot charging. This is a recent trend in steelmaking where steel strands are charged to the rolling mill while still hot, instead of leaving them to cool and then reheating them in a reheating furnace. Using their COMSOL Multiphysics model, the team explored the heat exchange process that takes place during the first solidification of the molten steel. Their results were used in the design of a new type of mold that forms billets with large, rounded corners that stay warm after casting, resulting in a more even distribution of temperature on the billet’s surface. This technique was employed at Tung Ho Steel in Taiwan, allowing them to completely forgo the use of a reheating furnace.

The subsequent reduction in yearly emissions by the company is equal to the exhaust of about *20,000 cars* (that’s 40,000 tons of CO_{2})!

Interested in learning more about how simulation can be used to explore and optimize concasting?

The Continuous Casting model, available in the Model Gallery, is an already-solved model that you can download and run using COMSOL Multiphysics and the Heat Transfer Module. The model analyzes the casting of a metal rod from molten metal through both the thermal and fluid dynamic aspects of the process, including heat transfer, the melt flow field, and phase change. Using the model, you can explore how the casting process can be optimized by altering the casting and cooling rates.

*Model of the continuous casting process. Top: Velocity field with streamlines. Bottom: Temperature distribution. Download the model from the Model Gallery.*

With the use of COMSOL Multiphysics, Tingcheng Wu, Guillaume Escamez, Clement Lorin, and Philippe J. Mason from the Department of Mechanical Engineering at the University of Houston were able to perform simulations to analyze how individual components of the structure impacted its overall performance. By applying various parameters to the structural design, they were able to conclude which factors had the greatest effect on the machine, both structurally and thermally. Thus, they could determine how to achieve a balance between the two.

The team of researchers modeled the structure as a rotor shaft separated into five individual parts connected by bolts. As a means to provide thermal insulation, different materials were used on particular areas of the shaft, as you can see below.

*Bolts were used to connect stainless steel with G10, a glass fiber material characterized by low thermal conductivity and a high yield stress. Image by T. Wu, G. Escamez, C. Lorin, and P. Mason, and taken from their poster submission.*

Seeking to analyze the issues of heat transfer and solid mechanics in these machines, the researchers used the Heat Transfer Module and Structural Mechanics Module to create their simulations. The figure below highlights the team’s findings regarding temperature, depicting that the G10 components take on the greatest temperature gradient.

In the next figure, the connection bolts underwent the most stress within the structure, a factor that was found to decrease as the cross-section area was decreased.

*Simulations highlight the thermal and structural pressure that the shaft endured, especially along its connection bolts. Image by T. Wu, G. Escamez, C. Lorin, and P. Mason, and taken from their presentation.*

With the continued funding and efforts of NASA and other research teams, progress continues to be made in the design of aircraft. As torque transfer components within fully superconducting rotating machines continue to be optimized, researchers gain momentum in their quest for developing structures with greater power densities and the potential for electric propulsion. In addition to making air travel a quieter and more energy efficient process, implementing this technology paves the way for its potential use within modes of ground transportation as well.

- Access the paper, presentation, and poster: “FEA Mechanical Modeling of Torque Transfer Components for Fully Superconducting Rotating Machines“

GREENKITCHEN project, a European initiative that supports the development of energy-efficient home appliances with reduced environmental impact, was founded to help develop and share knowledge about home appliance manufacturing in order to advance eco-friendly strategies for improving energy efficiency in households across Europe.

*COMSOL Multiphysics model of Whirlpool’s Minerva oven showing the predicted temperature distribution of the oven’s surfaces.*

One of the main goals of the project is to improve heat exchange in domestic ovens. Since ovens are one of the least energy-efficient appliances in a household ( only about 10-12% of the input power is used to heat the food being prepared ), they offer one of the best areas for improvement in regards to energy efficiency.

Using Whirlpool’s Minerva oven as a model, researchers at Whirlpool R&D working on the GREENKITCHEN project modeled the heat transfer processes of conduction, convection, and radiation in the oven. “Our goal for the project is to reduce energy consumption of Whirlpool’s ovens by 20 percent,” described Nelson Garcia-Polanco, Research and Thermal Engineer at Whirlpool, in an article from the 2014 edition of * Multiphysics Simulation*. This improvement in efficiency would result in a reduction of 50 million tons in CO

The first step in the project was to create a COMSOL Multiphysics model of the oven and verify its accuracy by performing experimental tests. One of the tests used by the team was the “chilled wet brick test” or “brick test”, in which a wet brick representing a food matrix is heated from 5°C to 60°C over a set period of time. The temperature distribution and evaporation of the water from the brick is then measured to determine the oven’s performance and energy usage. Below are images showing both the model of the oven and brick (right), as well as a photo of the oven during experimental testing (left).

Using the model, Garcia-Polanco measured the temperature profile and water concentration in the brick over the course of a 50-minute heating cycle at 200°C. This information was then compared to the results from the experimental test in order to verify that the oven simulation could reasonably predict the temperature changes throughout the test’s duration.

Water evaporation from the brick was also taken into account. The model predicted that an average of 166 grams of water would have evaporated after the 50-minute heating cycle, while the actual value was found to be only slightly higher, at 171 grams. With this information about water concentration and heating profiles, Garcia-Polanco and the GREENKITCHEN team will be able to optimize the oven to achieve a high energy efficiency, while being able to maintain the quality of the final product — ensuring that the energy-efficient oven distributes heat throughout the oven and provides optimal performance in the cooking process.

Currently, Garcia-Polanco and the team are using the validated model to test design ideas for both efficiency and performance.

By the way, Garcia-Polanco will discuss their project and designs further in his upcoming keynote speech on “Multiphysics Approach to Improving the Performance of Domestic Ovens” at the COMSOL Conference 2014 in Cambridge, UK.

To learn more about the research by Nelson Garcia-Polanco and the team at GREENKITCHEN, check out:

- The full-length article: Simulation Turns Up The Heat and Energy Efficiency at Whirlpool Corporation
- Read the Garcia-Polanco et. al paper from the COMSOL Conference 2013, “Multiphysics Approach of the Performance of a Domestic Oven“

To optimize the Passive Vaccine Storage Device (PVSD), engineers at Intellectual Ventures, as part of the Global Good Program, turned to thermal and vacuum system modeling with COMSOL Multiphysics together with experimentation.

In the early development stages, they began with a design similar to a cryogenic dewar — a specialized vacuum container commonly used in the field. Typical dewars are able to store ice for a few days before it melts, which is not nearly enough time for long trips to remote destinations. Traveling from a source point to areas where people need vaccinations could take weeks depending on their locations. Long travel times in combination with extreme climates present major challenges for experts working in the medical community. The PVSDs need high-performance insulation to create the temperature-controlled environment required for vaccine storage. Each layer of the device can impact overall performance and is designed to add to its insulative strength.

*The Passive Vaccine Storage Device can hold vaccines in a temperature-controlled, easy-to-transport compartment for longer durations than ever before.*

The shell of the PVSD is made of multilayer insulation, which is similar to the materials used for temperature regulation in spacecraft. This design is especially necessary for areas of the world that get incredibly hot because the vaccines need to stay in a cool and narrow temperature range (between 0°C and 10°C). The multilayer insulation consists of several layers of reflective aluminum, a low conductivity spacer, and nonconducting vacuum space.

*When modeling the PVSD, the Intellectual Ventures team considered physics phenomena and design variables including heat transfer, outgassing, and hold time.*

Vaccines require cold chain storage, which entails proper handling from the moment they are manufactured up until they are administered to a patient. Live virus vaccines can quickly deteriorate as soon as they leave their temperature-controlled space, and inactivated vaccines can lose potency from very short temperature fluctuations. Each year, countless doses of vaccines are thrown away or rendered useless because they were not stored and handled correctly.

For experimental tests, the researchers used an environmental chamber to recreate extreme outdoor conditions. In addition to experimental evaluation, multiphysics models were implemented using the Molecular Flow Module and Heat Transfer Module with COMSOL Multiphysics to optimize the PVSD design with regard to thermal performance and hold time. The outside of the device, composed of metal, prevents air inflow and helps maintain the cool temperature within. Added rubber absorbs shock to protect the contents during bumpy travels. Inside the PVSD is a small insulating shell (pictured below) that contains several compartments where the life-saving vaccines are stored.

*The inner shell of the PVSD holds individual vaccine vials that aid workers can easily access without disrupting the vacuum space or controlled environment.
*

By breaking down geographical barriers for aid workers, many lives will be impacted by the PVSDs and their important cargo.

- Molecular Flow Module
- Heat Transfer Module
- Read the full-length article “Innovative Thermal Insulation Techniques Bring Vaccines to the Developing World” in the 2014 edition of
*COMSOL News*

Our test set-up consists of a copper coil wound around a laminated iron core with some plastic and aluminum parts for stability. A conventional computer fan is placed one meter away from it. The occurring electromagnetic losses have to be calculated as well as the turbulent non-isothermal fluid flow around the device. The iron core has an air gap, which is intentionally included in order to analyze the influence it has on currents inside the coil and aluminum parts.

*The inductor device.*

*Schematic of the test set-up.*

Engineers — especially those working within project deadlines — are always looking for the right balance between computational (and modeling) efforts and accuracy. Therefore, it is a good idea to start with thinking of a suitable simplification, since the aspect ratio of the model geometry is quite challenging.

The distance between the fan and device is roughly one meter, while the interior gaps between the copper winding are about 0.1 millimeters, resulting in an aspect ratio of 10,000. In order to keep the processing time as low as possible, we choose a submodeling approach. A first model with a simplified transformer geometry is used to calculate the large-scale flow field around the device. Due to symmetry, only half of the geometry is modeled. The results of this model are exported and used as an inlet condition for the following step.

*Streamline plot of the velocity field. This field was used as an inlet boundary condition in the detailed model (at the position of the slice plot).*

The geometry of the detailed electrical device is built in SolidWorks® software and imported into COMSOL Multiphysics® via the CAD Import Module. Only a small part is used for the non-isothermal flow calculation in the detailed submodel (about 400 mm by 900 mm). The electromagnetic part needs to be solved for an even smaller domain (200 mm by 200 mm).

The iron core is laminated in order to reduce eddy currents. We’ll use the same approach as described by TU Dresden & ABB. The material is homogenized and defined with an orthotropic electrical conductivity. This allows us to keep a single domain and a coarser mesh rather than resolving the lamination geometrically with all small plates.

Due to the alternating current at 500 Hz, inductive effects in the coil (skin and proximity effect) have to be resolved. Additionally, eddy currents in the aluminum plates and iron core will heat up the device.

Due to hysteresis, there are also some magnetization losses. These are quite small in comparison to the eddy current losses and are not explicitly solved for. The table below shows the magnetization losses as functions of the magnetic flux density Q_{mag} = f(B). We could use an interpolation function instead of solving hysteresis time-dependently.

Part |
Electromagnetic losses |
---|---|

Copper coil |
37.2 W |

Aluminum, eddy currents |
36.2 W |

Laminated core, eddy currents |
0.02 W |

Laminated core, magnetic losses |
0.004 W |

The device reaches a maximum temperature of 125°C on the backside of the coil.

Today, our task was to find the best solution for computing thermal designs of transformers. In the case of BLOCK Transformatoren, they decided that COMSOL Multiphysics was the most suitable for their application after comparing the handling and results of several simulation tools.

In the end, this model involved simultaneous solving for a maximum of 8 million degrees of freedom (DOFs), using a robust combination of direct and iterative solvers. Memory (RAM) usage peaked at 89 GB of memory.

In order to be able to solve highly complex models, they chose the Ready-to-Go+ (RTG+) package with a benchmarked cluster for optimal performance. With everything being set for advanced simulations at BLOCK, we can expect their products to be pushed even closer to the limit in the future.

*SolidWorks is a registered trademark of Dassault Systèmes SolidWorks Corp.*

Power transmission systems need to deliver a required level of current without overheating the cables. Cable structure, internal electric losses, installation system geometry, and environmental conditions — such as the ambient temperature or the properties of the surrounding material — all contribute to system’s thermal response. Even external loads, such as those caused by solar radiation (in the case of an aboveground cable) or in proximity to other systems (e.g., other cables crossing the route), must be accounted for in the design.

While simple mathematical models and calculations suffice for many cases, cable systems are more commonly being installed in environments with many external influences that make it hard to predict performance and side effects.

That’s where multiphysics simulation enters the scene.

*COMSOL Multiphysics simulation showing temperature distribution in the cross section of a double-armored umbilical cable.*

Engineers at Prysmian use the COMSOL software for optimizing the geometry and arrangements of components in their high-tech cable solutions. Such systems may require a careful combination of geometry, materials, and positioning of individual parts. For example, composite cables may contain power conductors, hoses for fluid delivery, and cables for signal transmission.

The team has carried out simulations that analyze the physical effects in individual cables and entire power transmission systems by accounting for the relevant loading and environmental conditions. This allowed them to test, in a virtual sense, their designs before building and running prototypes. In one case, they had to take mechanical loads into account and simulated impact testing in medium-voltage cables. They used their results to optimize the thickness and material choices used in the external layers.

The simulations proved particularly useful for understanding the coupled structural, thermal, and electromagnetic problems that are challenging to solve. Massimo Bechis, a modeling and simulation specialist at Prysmian, comments that they’ve used COMSOL Multiphysics to “do extensive transient analyses to account for daily variations in solar irradiation and ambient temperature conditions […] multiphysics simulation really solves these kinds of problems that were very difficult or even impossible to do before.” They have simulated a range of events and phenomena, including impact testing, temperature distribution in double-armored umbilical cables, and computational fluid dynamics (CFD) analysis in high-voltage systems with limited ventilation.

*Simulation results showing a thermal and fluid flow coupled analysis of a high-voltage cable system installed inside a tunnel with only natural ventilation.*

Bechis explains that simulation has improved the way products are developed, and that using the COMSOL software has helped them take a big step forward in the level of services they are able to offer customers and designers. “COMSOL is able to solve these kinds of problems because we can build a parametric model to optimize the geometry, the laying of the cables, and we can include the physics needed to account for the convection in the air […] we can account for current load changes instead of considering constant operating conditions. This allows us to satisfy requests to consider transient conditions due to load changes.”

The superb results at Prysmian show strong promise for future developments. Simulation has opened up a whole world of possibilities for design, testing, and optimization prior to prototyping. It has reduced their costs, prototyping time, the number of lab tests, and has provided a straightforward way to optimize designs ahead of time, affirming their already-spectacular product record. The team expects that these methods will eventually lead to changes not only in the design phase of development, but also in manufacturing processes.

Simulation has also given them an avenue for communicating their design choices. As Bechis concluded, “Now we are able to optimize, among other things, the structure of our cables and still meet the specifications. We can also explain why we use a certain amount of material in a certain layer and show how we came to our decisions based on the modeling [...] we have improved procedures for designing our cables and power transmission systems. We have an additional and powerful way to respond to requests from clients.”

- Read the full story, “Simulation Software Brings Big Changes to Cable Industry” from the
*IEEE Spectrum*insert*Multiphysics Simulation*.

Before going into the physics, we should briefly recall the system of frames used in COMSOL Multiphysics. When geometric nonlinearities are considered, the *Solid Mechanics* interface makes the distinction between *material* and *spatial* frames. The material frame expresses the physical quantities in the coordinates of the initial state \mathbf{X} = (X, Y, Z), while the spatial frame uses the coordinates \mathbf{x} = (x, y, z) of the current state.

The two figures below present the example of a square submitted to compressive strain. The square is ten centimeters long and its bottom-left corner is initially located at (X, Y) = (1~\textrm{cm}, 1~\textrm{cm}). It is then compressed by boundary loads at its left and right sides. This deformation modifies the position of almost all points of the square. For instance, the bottom-left corner moved to a new location (x, y) = (1.54~\textrm{cm}, 0.82~\textrm{cm}).

*A deformed square represented in material coordinates, initial state on the left and final state on the right.*

*A deformed square represented in spatial coordinates, initial state on the left and final state on the right.*

The material coordinates always refer to the same particle in time, which was initially at a given point (X, Y, Z). The momentum equation of Solid Mechanics is formulated in this coordinate system. On the other hand, a point (x, y, z) in spatial coordinates refers to any particle that would be located there at the current state. The heat equation is formulated in this coordinate system.

In these two frames, volume-related physical quantities have different values. For instance, without any mass source, the density in material coordinates remains constant before and after transformation while the density in spatial coordinates changes according to the volume change. Hence, in order to couple an equation formulated on the material frame (structural mechanics) and another equation formulated on the spatial frame (heat transfer), these values need to be properly evaluated on each frame. The following table provides a list of conversions for some thermal physical quantities from spatial to material frame. These conversions involve the deformation gradient \mathbf{F} = {\partial \mathbf{x}} / {\partial \mathbf{X}} and its determinant, J. Both are evaluated using the displacement field computed by the *Solid Mechanics* interface.

Quantity | Material | Spatial |
---|---|---|

Temperature | T | T |

Density | \rho_0 | \rho = J^{-1} \rho_0 |

Thermal conductivity tensor | \bold{k}_0 | \bold{k} = J^{-1}\mathbf{F}^T \bold{k}_0 \mathbf{F} |

Pressure work | W_{\sigma, 0} = \boldsymbol{\alpha} T : \frac{\mathrm{d} \mathbf{S}}{\mathrm{d} t} | W_\sigma = J^{-1} W_{\sigma, 0} |

Heat source | Q_0 | Q = J^{-1} Q_0 |

*Conversion of thermal physical quantities from material to spatial frame.*

These conversions also reflect the fact that stress and strain affect the heat transfer by modifying the geometrical configuration (represented in the spatial frame). For example, a stretched boundary is more likely to receive a higher amount of heat by radiation (Q_\mathrm{r} > Q_{\mathrm{r}, 0}), as shown below.

*Radiative heat flux received at the top surface of a solid, initial state (left) and after stretching the top surface (right).*

Another example, the thermal conductivity expression in the spatial frame, usually using the initial state value \bold{k}_0, involves the quantities \mathbf{F} and J related to solid strain.

*Modification of the thermal conductivity on the spatial frame after deformation of a solid.*

The equations of Solid Mechanics are defined in the material frame. They relate the displacement, \mathbf{u}, the second Piola-Kirchhoff stress tensor, \mathbf{S}, and the elastic strain tensor, \mathbf{E}_\mathrm{el}, by a linear momentum balance equation and a stress-strain relation:

(1)

\rho_0 \frac{\mathrm{d}^2 \mathbf{u}}{\mathrm{d}t^2} = \nabla \cdot (\mathrm{\mathbf{FS}}) + \mathbf{f}_\mathrm{vol}

(2)

\mathbf{S} = \mathbf{C}:\mathbf{E}_\mathrm{el}

Here, \mathbf{C} is the elasticity tensor, which is often defined from the Young’s modulus and the Poisson coefficient. It may depend on the temperature as it is the case for Carbon Steel 1020.

*Young’s modulus of Carbon Steel 1020, depending on the temperature.*

Without any plastic effects, the elastic strain tensor, \mathbf{E}_\mathrm{el}, carries the temperature dependence via the thermal strain tensor, \mathbf{E}_\mathrm{th}, according to:

(3)

\mathbf{E}_\mathrm{el} = \mathbf{E}_\mathrm{tot}-\mathbf{E}_\mathrm{th}

(4)

\mathbf{E}_\mathrm{tot} = \frac{1}{2}(\mathbf{F}\mathbf{F}^\mathrm{T}-\mathbf{I})

(5)

\mathbf{E}_\mathrm{th} = \boldsymbol{\alpha} (T-T_\mathrm{ref})

The coefficient of thermal expansion, \boldsymbol{\alpha}, characterizes the ability of the material to contract and expand because of temperature variations. It is often scalar but may more generally take a tensor form. The table below shows a list of typical values of isotropic \boldsymbol{\alpha}.

Material | Coefficient of Thermal Expansion (10^{-6} K^{-1}) |
---|---|

Acrylic plastic | 70 |

Aluminum | 23 |

Copper | 17 |

Nylon | 280 |

Silica glass | 0.55 |

Structural steel | 12.3 |

*Coefficients of thermal expansion for some materials.*

In addition, \boldsymbol{\alpha} can, itself, depend on the temperature as shown by the example below.

*Coefficient of thermal expansion of Carbon Steel 1020, depending on the temperature.*

As seen in these examples, the values of \boldsymbol{\alpha} are most often of the order of 10^{-5} K^{-1}. Hence, for \mathbf{E}_\mathrm{th} to become significant, a high temperature difference from the reference state is necessary. For instance, aluminum needs to reach about 500 K above the reference temperature to show a thermal elongation of only 1.2%.

*Example of thermal expansion of a constrained aluminum beam heated 500 K, using a deformation scale of 1:1.*

Note that in the formulation of Equations (3)-(5), the thermal strain is subtracted from the total strain. This is an appropriate approximation for small strains, which the thermal strains normally are, due to usually low values of \boldsymbol{\alpha}. The more accurate multiplicative formulation, valid for large thermal strains, is shown below but not discussed further. This formulation is used for the hyperelastic materials in COMSOL Multiphysics.

(6)

\mathbf{E}_\mathrm{el} = \frac{1}{2}(\mathbf{F}_\mathrm{el}\mathbf{F}_\mathrm{el}^\mathrm{T}-\mathbf{I})

(7)

\mathbf{F}_\mathrm{el} = J_\mathrm{th}^{-1/3} \mathbf{F}

(8)

J_\mathrm{th} = \left( 1 + \alpha (T-T_\mathrm{ref}) \right)^3

The heat equation is an energy balance equation deduced from the First Law of Thermodynamics. For solids, it takes the following form when formulated on the spatial frame:

(9)

\rho C_p \left( \frac{\partial T}{\partial t} + \mathbf{u} \cdot \nabla T \right) = \nabla \cdot (k \nabla T) + W_\sigma + Q

The coupling term W_\sigma is the heat source due to compression or expansion of the solid and is defined by:

(10)

W_\sigma = \mathrm{det}(\mathbf{F})^{-1} T \frac{\partial \mathbf{E}_\mathrm{tot}}{\partial T} : \frac{\mathrm{d}\mathbf{S}}{\mathrm{d}t}

which, in the case of \boldsymbol{\alpha} being independent from temperature, reduces to:

(11)

W_\sigma = \mathrm{det}(\mathbf{F})^{-1} \boldsymbol{\alpha} T : \frac{\mathrm{d}\mathbf{S}}{\mathrm{d}t}

Here, \boldsymbol{\alpha} is the same coefficient of thermal expansion as in \mathbf{E}_\mathrm{th}. The low value of \boldsymbol{\alpha}, as seen in the table above, has to be compensated for by high enough values of T {\mathrm{d} \mathbf{S}} / {\mathrm{d} t} to make W_\sigma a significant heat source, that is:

- by a high temperature
- by rapid and high variations of stress

We have now described four key contributions to the multiphysics coupling between Heat Transfer and Solid Mechanics:

- The influence of strain and stress on thermal quantities and boundary heat fluxes in the material or spatial frames
- The temperature dependence of the elasticity matrix
- The temperature dependence of the elastic strain tensor via the thermal strain tensor
- The heat source, W_\sigma, corresponding to pressure work in the solid

Next, we will illustrate the last two coupling contributions and show how to handle them in COMSOL Multiphysics with a couple of modeling examples.

My colleague Nicolas previously described in more detail how to model thermal stress in a turbine stator blade. Here, we display only the results in order to show the effects of J_\mathrm{th}. Because this is a steady-state model, the pressure work, W_\sigma, can be ignored.

*Temperature field on the blade surface, representation in the material frame.*

Due to a hot environment, the temperature field shows values between 870 K and 1100 K compared to the reference temperature of 300 K that the shape of the stator blade is initially. Such high temperatures make the material more prone to thermal deformations. The average coefficient of thermal expansion and temperature being around 1.2·10^{-5} K^{-1} and 1070 K, \mathbf{E}_\mathrm{th} is around 0.9%.

The volume expansion, due to thermal effects, for large deformations is \Delta V/V_0 = J_\mathrm{th}-1 (where J_\mathrm{th} was introduced in Equation (8)). It is still a good approximation for a small strain, giving an expansion of around 2.80%. In postprocessing, the actual volume expansion is found to be 2.76%.

*Temperature field and deformation of the stator blade, exaggerated plot with a scale factor of 3 for more visibility.*

The Bracket — Transient Analysis model is available both in the Structural Mechanics Module Model Library and the Model Gallery. In this model, the arms of the brackets move according to rapid time-dependent loads. Consequently, small variations of temperature should occur.

The existing model neglects these thermal effects, so we need to add a new *Heat Transfer in Solids* interface.

Then, we add the two multiphysics features below to couple the *Heat Transfer in Solids* and *Solid Mechanics* interfaces:

- Thermal Expansion
- This modifies the thermal strain tensor, \mathbf{E}_\mathrm{th}, applied on the whole bracket domain

- Temperature Coupling
- This couples the temperature variable computed by the
*Heat Transfer in Solids*interface with the*Solids Mechanics*interface

- This couples the temperature variable computed by the

Finally, we add the *Pressure Work* subfeature to handle the thermoelastic heat source, W_\sigma.

The study can also be extended to 30 milliseconds to observe more load periods.

Starting from an isothermal profile of 20°C everywhere, the small temperature variations lead to a negligible thermal strain tensor. The main contribution to thermal effects is now the thermoelastic heat source due to rapid stress variations.

*Temperature profile of the bracket over time, exaggerated plot with a scale factor of 10 for more visibility.*

Differences of about 0.8 K can be observed between the extreme temperatures in the bracket. The heating and cooling process is, as expected, located at corners where the stress is more important and its variations stronger.

The heat transfer in a deformed solid is numerically computed by solving the heat equation and the momentum balance equation. For practical reasons, we made the distinction between two systems of coordinates:

- The material frame where the equation of motion is formulated
- The spatial frame for the heat equation

Volume-related quantities in both frames have different values and need a conversion from each other, in particular for specific energies and density.

The two governing equations each contain coupling terms that makes the solid motion dependent on the temperature and the heat transfer dependent on the solid deformation. As shown in the previous two examples, COMSOL Multiphysics provides appropriate functionalities to conveniently account for them.

When temperatures remain near the reference state and without too rapid stress variations, these coupling effects are negligible. Otherwise, they shall be added to the formulation on the model.

To delve deeper into this topic, you can download the files related to the models mentioned here and read a couple of related blog posts via the links in the section below.

- Model downloads:
- Previous blog posts:

There are two different ways in which hyperthermia can be used to treat cancer. The first is called *local hyperthermia* or *thermal ablation*, and it is used to treat a small area of cells, such as a tumor. This treatment is more successful when used to treat tumors that are located on or near the skin’s surface. In these treatments, the area is generally heated to temperatures between 40 and 45°C (104 to 113°F). The second type is called *regional hyperthermia*, and it uses low heat to elevate the temperature of a region of the body, such as a limb or organ. Generally, the temperature isn’t hot enough to destroy tissues, but it is instead combined with chemotherapy or radiation to make these treatments more effective.

Currently, hyperthermic oncology treatments are still undergoing extensive experimental testing. The procedure is only offered in a few treatment centers around the world. Because measuring the temperature inside a tumor is difficult, it is hard to ensure that the area being treated is kept within the exact temperature range needed over a precise period of time, without affecting surrounding tissues. This is where simulation is proving to be a powerful tool. It can be used to study the responses of different tissue types to hyperthermia, to determine the temperatures reached within a tumor, and to help decide the optimal design and placement of temperature sensors.

Let’s take a look at a model of a local hypthermia treatment method that uses microwave coagulation to heat a tumor by inserting an antenna into it. This method is most commonly used to treat hepatocellular carcinoma (small size liver tumors). The microwaves heat up the tumor and create a coagulated area where the cancer cells are destroyed.

The model example takes into account the temperature distribution, radiation, and specific absorption rate (SAR) in liver tissue. The model geometry, which was based off of the paper by Saito et al., is shown below. The model is composed of liver tissue, the antenna from which the microwave coagulation is delivered, and a ring shaped slot that serves to increase the size of the coagulated region in the liver tissue.

*Antenna geometry for microwave coagulation therapy. A coaxial cable with a ring-shaped slot cut on the outer conductor is short-circuited at the tip. A plastic catheter surrounds the antenna.*

Since the model contains rotational symmetry, the computational domain can be represented in 2D using cylindrical coordinates. Because of the computation time saved by modeling in 2D, a fine mesh can be selected, which will provide very accurate results. In the model, the metallic parts of the antenna and slot have been modeled as boundaries. Heat transfer has been modeled in the liver domain and insulation is used where the domain ends. In addition to analyzing heat transfer, the simulation also gives an idea about the degree of tissue injury during the procedure, as computed by the Arrhenius equation. The model assumes that blood enters the liver at 37°C.

The figure below show the results of the temperature distribution in the liver tissue for a microwave input power of 10 W after 10 minutes. The temperature is highest near the antenna and decreases with distance, reaching 37°C near the outer regions of the domain (top image). It appears that the relatively cool temperature of the blood limits the region of tissue that is heated. The image on the bottom shows the fraction of necrotic tissue.

*Temperature in the liver tissue (top) and the fraction of necrotic tissue (bottom).*

The model can also compute the SAR value within the liver, where the graph below shows it computed along a line parallel to the antenna and at a distance 2.5 mm from the antenna’s axis.

*SAR value in W/kg along a line parallel to the antenna and at a distance 2.5 mm from the antenna axis. The tip of the antenna is located at 70 mm, and the slot is at 65 mm.*

Clinical trials have shown that hyperthermia treatment significantly increases the effectiveness of radiotherapy and chemotherapy. However, there are two major complications that make treatment with hypothermia a challenge: the ability to *precisely* determine the temperature within the tumor, and the ability to ensure that a uniform temperature is achieved within the desired area. However, advances in tools for both heat delivery and temperature monitoring are promising for the future of hyperthermia treatments. Advancements in computational techniques have also played a role in its development. In addition to providing more accurate information on how to design, optimize, and implement hyperthermia treatments, simulation is also being used to developed personalized hyperthermia treatments and planning. Current studies continue to rely on simulation to better understand, improve, and personalize hyperthermic oncology.

Check out these resources to learn more about hyperthermia treatments and microwave coagulation:

- Model download: Microwave Cancer Therapy
- COMSOL webinar: Multiphysics Simulation in Bioheating and BioTechnology
- National Cancer Institute: Hyperthermia in Cancer Treatment
- Estimation of SAR Distribution of a Tip-Split Array Applicator for Microwave Coagulation Therapy Using the Finite Element Method
- Hyperthermia in combined treatment of cancer
- Treating cancer with heat: hyperthermia as promising strategy to enhance apoptosis