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.

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.

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 10^{9} N/m^{3} 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.

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.

- Read the full COMSOL Conference paper: “Feasibility Study of Thermal Actuators for MEMS Variable Emittance Radiators“
- Browse additional blog topics relating to space exploration:
- Download the related tutorial: Joule Heating of a Microactuator

A few years ago, we talked on the blog about how selective laser sintering was taking the 3D print world by storm. Since then, the popularity of this rapid prototyping technique has continued to grow throughout various industries. The same can be said of a closely related technique, selective laser melting, which uses a laser beam to melt powdered material in order to produce a 3D part.

*A schematic describing the selective laser melting process. Image by Materialgeeza — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.*

Copper, aluminum, and stainless steel: these are just some metals that are already used in SLM. In recent years, researchers have experimented with adding high-melting materials into the mix. Molybdenum, shown below, is one example.

*Molybdenum is a high-melting material with potential use in SLM. Image by Alchemist-hp — Own work. Licensed under Free Art License 1.3, via Wikimedia Commons.*

With these new materials comes a new challenge: The processing window for refractory metals is significantly narrower. This means that further analysis is needed to gain an understanding of how high-melting materials behave during SLM. To address this multiphysics problem, researchers at Plansee SE in Austria turned to the COMSOL® software.

For their analysis, the researchers created a COMSOL Multiphysics model to analyze laser beam-matter interaction in SLM. The model geometry consists of a simple cubic metal powder layer, resting on top of a large base plate that is exposed to a Gaussian laser beam. Note that the model takes advantage of the symmetry in the direction that the laser moves.

*The meshed model geometry. Image by K.-H. Leitz, P. Singer, A. Plankensteiner, B. Tabernig, H. Kestler, and L.S. Sigl and taken from their COMSOL Conference 2016 Munich paper.*

To accurately model laser beam-matter interaction, there are several factors to account for:

- Laser radiation absorption
- Conductive and convective heat transfer
- Phase changes (melting and solidification as well as evaporation and condensation)
- Surface tension effects

Coupling thermal and fluid dynamics via the Heat Transfer Module and CFD Module enabled the researchers to investigate these factors. For this specific case, the angle dependency of absorption, shadowing effects, and various reflections are neglected.

In the analysis, the metal powder is represented by two different materials: stainless steel and molybdenum. The researchers compared the volume buildup at multiple stages of the SLM process for each material.

From the figures below, we can see a clear difference between the process dynamics for steel and molybdenum. In the case of steel, there is a long melt pool and significant effects via evaporation. In the case of molybdenum, the melt pool is confined to the size of the focal spot area and the temperatures are much lower than those causing evaporation. This difference can be traced back to the phase-transition temperatures and thermal conductivities for each material. Because of its high thermal conductivity, molybdenum experiences greater heat losses in SLM, which then restricts the melt pool size. These heat losses, in combination with molybdenum’s high evaporation temperature, prevent evaporation from occurring.

*The volume buildup during the selective laser melting of steel (left) and molybdenum (right). Images by K.-H. Leitz, P. Singer, A. Plankensteiner, B. Tabernig, H. Kestler, and L.S. Sigl and taken from their COMSOL Conference 2016 Munich paper.*

The above results provide a better understanding of the dynamics of SLM as well as the characteristics of the process that are specific to the material used. Since the core of the model describes laser beam-matter interaction, it can be used to study other manufacturing processes that involve lasers.

- Read the full COMSOL Conference paper: “Thermo-Fluiddynamical Modelling of Laser Beam-Matter Interaction in Selective Laser Melting“
- See how researchers used simulation to reduce laser-induced damage in internal optics
- For a more detailed guide, check out this blog post on how to model laser-material interactions in COMSOL Multiphysics

Building physics engineers aim to improve the energy performance and sustainability of building envelopes. Although their practices are based on past experience, new materials and building techniques are constantly being developed that offer a wide set of options in building design and thermal management. Let’s see how to model heat and moisture transport in building materials to help reduce energy costs and preserve buildings.

*Building envelopes can be analyzed by modeling heat and moisture transport.*

Controlling moisture is necessary to optimize the thermal performance of building envelopes and reduce energy costs. The thermal properties of insulation or isolation materials usually depend on both temperature and moisture content. Therefore, a coupled heat and moisture model helps us fully analyze the thermal performance of a building component. One example is the dependence of a lime silica brick’s thermal conductivity on relative humidity.

*The moisture dependence of thermal conductivity for lime silica brick.*

The figure above shows that lime silica brick becomes two times less thermally isolating for high relative humidity values.

In addition, we must consider moisture control in the building design process to choose building components that can reduce the risk of condensation. The coupled modeling of heat and moisture transport enables us to analyze different moisture variations and phenomena in building components, such as:

- Drying of moisture resulting from the initial construction
- Condensation due to the migration of moisture from outside to inside during warmer periods
- Moisture accumulation by interstitial condensation due to vapor diffusion during colder periods

Let’s consider a wood-frame wall between a warm indoor environment and a cold outdoor environment. Vapor diffuses through the wall from the high-moisture environment inside to the low-moisture environment outside. This creates high relative humidity values associated with high temperature values close to the exterior panel, with the risk of condensation as a direct consequence.

*The relative humidity distribution in a wood-frame wall.*

Condensation leads to mold growth, which directly affects human health and building sustainability. The rate of mold growth is key data for the preservation of historical buildings, for example. To prevent the risk of interstitial condensation, it is common practice to add a vapor barrier between the interior gypsum panel and the cellulose isolation board. This reduces the moisture values where they are at a maximum. The figure below shows the relative humidity distribution across the wood-frame wall through a wood stud (red lines) and a cellulose board (blue lines), with and without the vapor barrier (dashed lines and solid lines, respectively).

*Effect of a vapor barrier on relative humidity distribution across the wood-frame wall in a wood stud and cellulose board.*

For this model, we consider the building materials to be specific unsaturated porous media in which the moisture exists in both liquid and vapor phases and only some transport processes are relevant. The norm EN 15026 standard addresses the transport moisture phenomena that is taken into account in building materials, following the theory expressed in Ref. 1.

The transport equation established as a standard by the norm accounts for liquid transport by capillary forces, vapor diffusion due to a vapor pressure gradient, and moisture storage.

\xi\frac{\partial \phi}{\partial t} + \nabla \cdot \left(- \xi D_\textrm{w} \nabla\phi -\delta_\textrm{p}\nabla\left(\phi p_\textrm{sat}\right)\right) = G

We model the latent heat effect due to vapor condensation by adding the following flux in the heat transfer equation:

\mathbf{q}= -L_\textrm{V}\delta_\textrm{p}\nabla\left(\phi p_\textrm{sat}\right)

In addition, the moisture dependence of the thermal properties is assessed.

Find details about the moisture transport equation in building materials in the

Heat Transfer Module User’s Guide.

When using the Heat Transfer Module, the *Heat and Moisture Transport* interface adds a:

*Heat and Moisture*coupling node*Heat Transfer in Building Materials*interface*Moisture Transport in Building Materials*interface*Building Material*feature for heat transfer*Building Material*feature for moisture transport*Thin Moisture Barrier*feature for modeling the vapor barrier

Finally, the latent heat source due to evaporation is added to the heat transfer equation by the *Building Material* feature of the *Heat Transfer* interface.

*The model tree and subsequent subnodes when choosing the* Heat Transfer in Building Materials *interface, along with the Settings window of the* Building Material *feature.*

Modeling heat and moisture transport in an unsaturated porous medium is important for analyzing polymer materials for the pharmaceutical industry, protective layers on electrical cables, and food-drying processes, to name a few examples.

For these applications, phenomenological models, such as the one presented above for building materials, may not be available. However, by considering the conservation of heat and moisture in each phase (solid, liquid, and gas), and volume averaging over the different phases, we can derive a mechanistic model.

To compute the moisture distribution, we solve a two-phase flow problem in the porous medium. Two equations of transport are solved: one for the vapor and one for the liquid water. The coupling between the vapor and liquid water operates through the definition of saturation variables, *S*_{vapor} + *S*_{liquid} = 1. The changing water saturation is taken into consideration for the definition of the effective vapor diffusivity and liquid permeability.

For quick processes, with a time scale comparable to the time it takes to reach equilibrium between the liquid and gas phases inside the pores of the medium, a nonequilibrium formulation can be defined through the following evaporation flux:

g_\textrm{evap} =M_\textrm{v}K(a_\textrm{w}c_\textrm{sat}-c_\textrm{v})

In this definition, the equilibrium vapor concentration, defined as the product of the saturation concentration *c*_{sat} and the water activity *a*_{w}, is used to account for the porous medium structure. Indeed, due to capillary forces, equilibrium is reached for concentrations that are lower than in a free medium.

By letting the evaporation rate *K* go to infinity, an equilibrium formulation is obtained with the vapor concentration equal to the equilibrium concentration.

Let’s consider a food-drying process. A piece of potato, initially saturated with liquid water, is placed in an airflow to be dried. Inside the potato, the vapor is transported by binary diffusion in air. We use a Brinkman formulation to model the flow induced by the moist air pressure gradient in the pores. As the liquid phase velocity is small compared to the moist air velocity, Darcy’s law is used for the liquid water flow due to the pressure gradient. The capillary flow, due to the difference between the relative attraction of the water molecules for each other and the potato, is also considered in the liquid water transport.

The vapor and liquid water distributions over time for this model are shown in the following two animations. Note that water can leave the potato as vapor only.

*The liquid water concentration over time.*

The vapor is transported away by the airflow, as shown in this animation:

*The water vapor concentration over time.*

The evaporation causes a reduction of the temperature in the potato. The temperature distribution over time is shown below.

*Temperature distribution over time.*

You can implement the equations in the *Heat Transfer in Porous Media* interface within the Heat Transfer Module and the *Transport of Diluted Species* interface within the Chemical Reaction Engineering Module. This process requires some steps in order to couple the multiphase flow in a porous medium together with the evaporation process.

Read the article “Engineering Perfect Puffed Snacks” on pages 7–9 of

COMSOL News2017 to see how Cornell University researchers used COMSOL Multiphysics to model rice puffing. In this numerically challenging process, the rapid evaporation of liquid water results in a large gas pressure buildup and phase transformation in the grain.

In this blog post, we discussed COMSOL® software features for modeling heat and moisture transport in porous media. COMSOL Multiphysics (along with the Chemical Reaction Engineering Module and Heat Transfer Module) provides you with tools to define the corresponding phenomenological and mechanistic models for a large range of applications. Depending on the dominant transport processes, you can use predefined interfaces or define your own model.

Künzel, H. 1995. *Simultaneous Heat and Moisture Transport in Building Components. One and two-dimensional calculation using simple parameters.* PhD Thesis. Fraunhofer Institute of Building Physics.

- Check out the tutorial models featured in this blog post:

Humidified air not only affects human comfort, but also conditions the sustainability of buildings and the operation of electronic devices. This makes accounting for the presence of moisture critical when modeling heat transfer and phase change in ambient air surrounding equipment and in structures.

A standard variable used to quantify the amount of moisture in air is the relative humidity, *φ*. It expresses a relative state toward saturation and is the ratio between the vapor’s partial pressure in air, *p*_{v}, and the saturation pressure at a given (usually standard) temperature, *p*_{sat}(*T*):

\phi=\frac{p_\textrm{v}}{p_\textrm{sat}(T)}

As a first approximation, we can assume the vapor’s partial pressure *p*_{v} to be homogeneous. Yet, because of the dependence of saturation pressure on temperature, we should note that the relative humidity is not actually homogeneous, as temperature gradients are present.

Typical ambient moisture conditions can be defined from tabulated data such as weather records. These can be used to define, for example, the air’s thermodynamic properties when solving the heat transfer equation:

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

The moisture dependence on the density, thermal conductivity, and heat capacity at constant pressure are set through a mixture formula based on dry air and pure steam properties.

In a previous blog post, we detail how to use typical weather data for temperature and relative humidity in COMSOL Multiphysics®.

By exclusively solving the equation above for the temperature while knowing the (homogeneous) vapor’s partial pressure, we can already identify zones where condensation is likely to happen. Indeed, condensation happens at the saturation state, corresponding to *φ* = 1, where the detection of condensation relies on the relationship between both temperature and moisture.

As an example, let’s consider an electronic device within a box that produces 1 W of heat. Moist air flows through the box via 2 small slits located at the left and right sides of the box. From the computed temperature and relative humidity distributions, the risk of condensation inside the box is evaluated. Note that in this computation, the latent heat associated with the condensation is not accounted for in the heat transfer model. As shown in the figures below, condensation forms on the walls close to the slits after around 3 hours, for about 30 minutes, as well as after 4 hours and 30 minutes. These times correspond to when the ambient temperature is low and the relative humidity is high, at different points in the box.

*Temperature distribution after 3 hours (left); relative humidity distribution after 3 hours (center); and the evolution of the condensation indicator variable, ht.condInd, over time (right).*

You can find more information in a previous blog post on modeling convective heat transfer.

When using the Heat Transfer Module, the *Moist Air* option in the *Fluid* Settings window from the *Heat Transfer in Fluids* interface defines the moisture-dependent thermodynamic properties of the modeling domain. This option also provides the `ht.condInd`

variable to be used when postprocessing results to identify condensation detection.

*The model tree and Settings window of the* Fluid *feature with the* Moist air *option selected.*

In some situations, we need to describe the moisture distribution more precisely. This includes cases where the amount of moisture is locally high due to evaporation and when the diffusion and convection of vapor can’t be neglected.

Compared to the previous approach, we need to compute the moisture distribution by solving an additional transport equation for the convection and diffusion of the vapor concentration *c*_{v} in air:

M_\textrm{v}\left(\frac{\partial c_\textrm{v}}{\partial t} + \nabla \cdot \left(- D \nabla c_\textrm{v} \right) +\textbf{u}\cdot\nabla c_\textrm{v}\right) = G

Note that in this equation, the temperature dependence is still accounted for through the vapor concentration *c*_{v} = *φc*_{sat}(*T*), with *c*_{sat}(*T*) as the saturation concentration of vapor.

Let’s consider a beaker filled with hot water (80°C) and placed in an air flow with a velocity of 2 m/s. Evaporation occurs from the water surface due to the flow of the air. Evaporation creates a state of saturated vapor (dependent on the temperature) at the air-liquid water interface, where this is transported away, and replenished by unsaturated air through convection and diffusion (see the figure below).

*Vapor concentration distribution after 20 minutes, with contour lines for the relative humidity.*

The energy required to sustain the evaporation is primarily extracted from the internal energy of the liquid water, which cools down as a result, as shown in the animation below. This process is known as evaporative cooling. It is the main process used in evaporative coolers and cooling towers, taking advantage of water’s relatively large heat capacity and latent heat when heating and vaporizing water for air cooling.

*Temperature distribution over time and streamlines indicating the flow field.*

In the model, evaporation occurs when the vapor concentration stays below the saturation state and just above the liquid surface. The evaporation flux is expressed as:

g_\textrm{evap} =M_\textrm{v}K(c_\textrm{sat}-c_\textrm{v})

where *K* is an evaporation rate depending on the application.

The latent heat variation in the liquid is taken into account by adding the following heat source in the heat transfer equation:

Q_\textrm{evap} =-L_\textrm{v}g_\textrm{evap}

where *L*_{v} is the latent heat of the evaporation of water.

When using the Heat Transfer Module, the *Heat and Moisture Transport* interface adds the subnodes shown in the screenshot below, including the:

*Heat and Moisture*coupling node*Heat Transfer in Moist Air*interface*Moisture Transport in Air*interface*Moist Air*feature for the heat transport in air*Moist Air*feature for the vapor transport in air*Wet Surface*feature for the evaporation from the liquid surface*Boundary Heat Source*feature, which adds the latent heat source due to evaporation to the heat transfer equation

*The model tree and subsequent subnodes when choosing the* Heat Transfer in Moist Air *interface, along with the Settings window of the* Moist Air *feature.*

When defining a fully coupled simulation of evaporative cooling, the *Heat Transfer in Moist Air* and *Moisture Transport in Air* interfaces are included together with the *Heat and Moisture* multiphysics interface. This also sets up the situation by including the first three subnodes under both interfaces by default. Further subnodes (e.g., the *Boundary Heat Source* and *Wet Surface* subnodes) can be included depending on the participating conditions of the process being simulated.

We have now reviewed the COMSOL® software features dedicated to the modeling of heat and moisture transport in moist air. Depending on the application, you may want to solve only for heat transfer and use the temperature prediction to detect condensation, or you may need to go further by computing the temperature and moisture distributions in a coupled way. In addition, you can account for the latent heat effects or disregard them. COMSOL Multiphysics (along with the Heat Transfer Module) provides the tools to define the corresponding models for a large range of applications.

Stay tuned for an upcoming blog post covering how to model heat and moisture transport in building materials and porous media.

*Editor’s note: You can read the follow-up post in this blog series here: “How to Model Heat and Moisture Transport in Porous Media with COMSOL®“.*

- Check out the tutorial models featured in this blog post:

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

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

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

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:

- Magnetron core, which generates electromagnetic microwaves
- Waveguide, which transmits the microwaves into the resonator
- Resonator (also called the reaction chamber), which includes a crucible to hold the silicon sample
- Tuner, which improves the absorption of the microwaves
- 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.

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.

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.

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.*

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.

- Explore other examples of solar simulation applications in these blog posts:

Artificial ground freezing is a construction technology that involves running an artificial refrigerant through pipes buried underground. As the refrigerant circulates through the pipe network, heat is removed from the ground and ice begins to form around the pipes. This in turn causes the soil to freeze. In other words, the process converts soil moisture into ice. Once the soil is frozen, it is both stronger (sometimes as hard as concrete) and has a greater resistance to water. This allows the soil to provide effective support to the relative infrastructures, particularly those that are larger and more complex.

*Once frozen, soil becomes stronger and more resistant to water.*

For the AGF method to be effective, we need to know the temperature distribution inside the system. Of the physical processes that occur in AGF, the most prominent is the phenomenon of transient heat conduction with phase change. Further, it is also important to consider the relationship between this phase change and the groundwater flow — particularly when there is a higher flow velocity. These elements can impact the development of the freezing wall and thus the strength and reliability of the AGF method.

To study the AGF method, a team of researchers from Hohai University turned to the COMSOL Multiphysics® software. Their case study involves using the method to strengthen soil at a metro tunnel entrance in Guangzhou, China.

For this specific example, the refrigerant that circulates throughout the pipe system is -30ºC brine. The subsurface temperature is reduced until the pore water is frozen and the freezing wall forms. The formation within the frozen area is made up of muddy sand, and the direction of the groundwater flow is primarily horizontal and normal in relation to the axial direction of the tunnel.

To simplify modeling heat transport in a saturated aquifer, the researchers used a 2D model based on a coupling of temperature and flow fields. The model, shown below, is 20 m in both length and height. Note that five monitoring points are included. These points are used to verify the accuracy of the model by comparing the calculated temperature results with *in situ* measurements.

*The AGF model’s geometry, with the monitoring points highlighted (left), and the model grid’s mesh (right). Images by Rui Hu and Quan Liu and taken from their COMSOL Conference 2016 Munich paper.*

In this analysis, the following assumptions are made:

- The ice is incapable of moving and the medium cannot deform
- The aquifer is fully saturated, with a total porosity that remains constant
- The freezing point depression caused by solute concentrations is negligible

According to previous temperature monitoring data from the frozen area, there is an initial ground temperature of 15°C. The figure below shows the initial temperatures in various holes of thermal observation.

*The initial temperatures in different holes for thermal observation. Image by Rui Hu and Quan Liu and taken from their COMSOL Conference 2016 Munich paper.*

The cooling source of the freezing system is the lateral wall of the freezing pipe. Changes in the temperature of the lateral wall have the greatest impact on the temperature distribution within the system. It is possible to use the values from the temperature monitoring of the main pipe as approximations for the estimated temperature of the lateral wall. The plot below shows the fitting function and curve for the lateral wall temperature of the main pipe after a monitoring period of 40 days.

*The fitting function and curve for the lateral wall temperature. Image by Rui Hu and Quan Liu and taken from their COMSOL Conference 2016 Munich paper.*

With regards to groundwater flow, a flow velocity of 0.2 m/d is obtained via field tests. Between upstream and downstream, the head difference is calculated as 0.8 m.

Now onto the results. Let’s consider the temperature distribution and permeability coefficient for a range of times. In terms of temperature, when the freezing time increases, the cold temperature from the freezing pipes is primarily led downstream — with less of an influence upstream. The permeability coefficient results, which illustrate the formation of the freezing wall, indicate that the top and bottom walls form at a faster rate than those walls at upstream and downstream. Note that the freezing wall is entirely closed after 35 days.

*The temperature distribution (left) and permeability coefficient results (right) at various points in time. Images by Rui Hu and Quan Liu and taken from their COMSOL Conference 2016 Munich paper.*

When comparing the closure of the freezing wall and flow velocity, the closing time increases nonlinearly as the flow velocity increases. The time of closure dramatically increases when the velocity is greater than 1.5 m/d. As for the average wall thickness in all directions and relative flow velocity, the influence of the flow velocity on the thickness of the upstream wall is most prominent.

The successful validation of this model offers guidance for the metro tunnel project in Guangzhou, China. With plans to further develop this model, the researchers hope to use it as a resource for improving applications of the AGF method.

- Read the full COMSOL Conference paper: “Simulation of Heat Transfer during Artificial Ground Freezing Combined with Groundwater Flow“
- Browse some additional examples of modeling subsurface flow:

Let’s start by considering a model of the electrical heating of a busbar, shown below. You may recognize this as an introductory example to COMSOL Multiphysics, but if you haven’t already modeled it, we encourage you to review this model by going through the *Introduction to COMSOL Multiphysics* PDF booklet.

*Electric currents (arrow plot) flowing through a metal busbar lead to resistive heating that raises the temperature (color surface plot).*

In this example, we model electric current flowing through a busbar. This leads to resistive heating, which in turn causes the temperature of the busbar to rise. We assume that there is only heat transfer to the surrounding air, neglecting any conductive heat transfer through the bolts and radiative heat transfer. The example also initially assumes that there isn’t any fan forcing air over the busbar. Thus, the transfer of heat to the air is via natural, or free, convection.

As the part heats the surrounding air, the air gets hotter. As the air gets hotter, its density decreases, causing the hot air to rise relative to the cooler surrounding air. These free convective air currents increase the rate of heat transfer from the part to the surrounding air. The air currents depend on the temperature variations as well as the geometry of the part and its surroundings. Convection can, of course, also happen in any other gas or liquid, such as water or transformer oil, but we will center this discussion primarily around convection in air.

We can classify the surrounding airspace into one of two categories: *Internal* or *External*. Internal means that there is a finite-sized cavity (such as an electrical junction box) around the part within which the air is reasonably well contained, although it might have known air inlets and outlets to an external space. We then assume that the thermal boundary conditions on the outside of the cavity and at the inlets and outlets are known. On the other hand, External implies that the object is surrounded by what is essentially an infinitely large volume of air. We then assume that the air temperature far away from the object is a constant, known value.

*The settings for a constant heat transfer coefficient.*

The introductory busbar example assumes free convective heat transfer to an external airspace. This is modeled using the following boundary condition for the heat flux:

q=h \left(T_{ext}-T \right)

where the external air temperature is *T _{ext}* = 25°C and is the heat transfer coefficient.

This single-valued heat transfer coefficient represents an approximate and average of all of the local variations in air currents. Even for this simple system, any value between could be an appropriate heat transfer coefficient, and it’s worth trying out the bounding cases and comparing results.

If we instead know that there is a fan blowing air over this structure, then due to the faster air currents, we use a heat transfer coefficient of to represent the enhanced heat transfer.

If the surrounding fluid is a liquid such as water, then the range of free and forced heat transfer coefficients are much wider. For free convection in a liquid, is the typical range. For forced convection, the range is even wider: .

Clearly, entering a single-valued heat transfer coefficient for free or forced convection is an oversimplification, so why do we do it? First, it is simple to implement and easy to compare the best and worst cases. Also, this boundary condition can be applied with the core COMSOL Multiphysics package. However, there are some more sophisticated approaches available within the Heat Transfer Module and CFD Module, so let’s look at those next.

A *convective correlation* is an empirical relationship that has been developed for common geometries. When using the Heat Transfer Module or CFD Module, these correlations are available within the Heat Flux boundary condition, shown in the screenshot below.

*The Heat Flux boundary condition with the external natural convection correlation for a vertical wall.*

Using these correlations requires that you enter the part’s characteristic dimensions. For example, with our busbar model, we use the *External natural convection, Vertical wall* correlation and choose a wall height of 10 cm to model the free convective heat flux off of the busbar’s vertical faces. We also need to specify the external air temperature and pressure. These values can be loaded from the ASHRAE database, a process we describe in a previous blog post.

The table below shows schematics for all of the available correlations. They take the information about the surface geometry and use a Nusselt number correlation to compute a heat transfer coefficient. For the horizontally aligned faces of the busbar, for example, we use the *Horizontal plate, Upside* and *Horizontal plate, Downside* correlations.

When using the Forced Convection correlations, you must also enter the air velocity. These convective correlations have the advantage of being a more accurate representation of reality, since they are based on well-established experimental data. These correlations lead to a nonlinear boundary condition, but this usually results in only slightly longer computation times than when using a constant heat transfer coefficient. The disadvantage is that they are only appropriate to use when there is an empirical relationship that is reasonable for the part geometry.

Free Convection | Forced Convection | |
---|---|---|

External | ||

Internal |

*The available* Convective Correlation *boundary conditions.*

Note that all of the above convective correlations, even those classified as Internal, assume the presence of an infinite external reservoir of fluid; e.g., the ambient airspace. The heat carried away from the surfaces goes into this ambient airspace without changing its temperature, and the ambient air coming in is at a known temperature. If, however, we are dealing with convection in a completely enclosed container, then none of these correlations are appropriate and we must move to a different modeling approach.

Let’s consider a rectangular air-filled cavity. If this cavity is heated on one of the vertical sides and cooled on the other, then there will be a regular circulation of the air. Similarly, there will be air circulation if the cavity is heated from below and cooled from above. These cases are shown in the images below, which were generated by solving for both the temperature distribution and the air flow.

*Free convective currents in vertically and horizontally aligned rectangular cavities.*

Solving for the free convective currents is fairly involved. See, for example, this blog post on modeling natural convection. Therefore, we might like to find a simpler alternative. Within the Heat Transfer Module, there is the option to use the *Equivalent conductivity for convection* feature. When using this feature, the effective thermal conductivity of the air is increased based upon correlations for the horizontal and vertical rectangular cavity cases, as shown in the screenshot below.

*The Equivalent conductivity for convection feature and settings.*

The air domain is still explicitly modeled using the *Fluid* domain feature within the *Heat Transfer* interface, but the air flow fields are not computed and the velocity term is simply neglected. The thermal conductivity is increased by an empirical correlation factor that depends on the cavity dimensions and the temperature variation across the cavity. The dimensions of the cavity must be entered, but the software can automatically determine and update the temperature difference across the cavity.

*Temperature distribution in vertically and horizontally aligned cavities using the Equivalent conductivity for convection feature. The free convective air currents are not computed. Instead, the thermal conductivity of the air is increased.*

This approach for approximating free convection in a completely closed cavity requires us to mesh the air domain and solve for the temperature field in the air, but this usually adds only a small computational cost. The disadvantage of this approach is that it is not very applicable for nonrectangular geometries.

Next, let’s consider a completely sealed enclosure, but with a fan or blower inside that actively mixes the air. We can reasonably assume that well-mixed air is at a constant temperature throughout the cavity. In this case, it is appropriate to use the *Isothermal Domain* feature, which is available with the Heat Transfer Module when the *Isothermal domain* option is selected in the Settings window.

*The settings associated with using the Isothermal Domain interface.*

A well-mixed air domain can be explicitly modeled using the Isothermal Domain feature. In the model, the temperature of the entire domain is a constant value. The temperature of the air is computed based upon the balance of heat entering and leaving the domain via the boundaries. The Isothermal Domain boundaries can be set as one of the following options:

*Thermally Insulated*: no heat transfer across the boundary*Continuity*: continuity of temperature across the boundary*Ventilation*: a known mass flow of fluid, of known temperature, into or out of the isothermal domain*Convective Heat Flux*: a user-specified heat transfer coefficient, as described earlier*Thermal Contact*: a specific thermal resistance

Of all of these boundary condition options, the *Convective Heat Flux* is the most appropriate for well-mixed air in an enclosed cavity.

*Representative results when using an Isothermal Domain feature. The well-mixed air domain is a constant temperature and there is heat transfer to the surrounding solid domains via a specified heat transfer coefficient.*

The most computationally expensive approach, but also the most general, is to explicitly model the airflow. We can model both forced and free convection as well as simulate an internal or external flow. This type of modeling can be done with either the Heat Transfer Module or CFD Module.

*An example of computing air flow and temperature within an enclosure.*

If you finished the *Introduction to COMSOL Multiphysics* booklet, you have already solved one example of an internal forced convection model. You can learn more about explicitly modeling airflow in the resources mentioned at the end of this post.

We will finish up this topic by addressing the question: When can free convection in air be ignored and how can we model these cases? When a cavity’s dimensions are very small, such as a thin gap between parts or a very thin tube, we run into the possibility that the viscous damping will exceed any buoyancy forces. This balance of viscous to buoyancy forces is characterized by the nondimensional Rayleigh number. The onset of free convection can be quite varied depending on boundary conditions and geometry. A good rule of thumb is that for dimensions less than 1mm, there will likely not be any free convection, but once the dimensions of the cavity get larger than 1cm, there likely will be free convective currents.

So how can we model heat transfer through these small gaps? If there is no air flow, then these air-filled regions can simply be modeled as either a solid or a fluid with no convective term. This is demonstrated in the Window and Glazing Thermal Performances tutorial. It is also appropriate to model the air as a solid within any microscale enclosed structure.

If these thin gaps are very small compared to the other dimensions of the system being analyzed, you can further simplify the gaps by modeling them via the Thin Layer boundary condition with a *Thermally thick approximation* layer type. This boundary condition introduces a jump in temperature across interior boundaries based on the specified thickness and thermal conductivity.

*The Thin Layer boundary condition can model a thin air gap between parts.*

We can use the previous two approaches within the core COMSOL Multiphysics package. In the Heat Transfer Module, there are additional options for the Thin Layer condition to consider more general and multilayer boundaries, which can be composed of several layers of materials.

Before closing out this discussion, we should also quickly address the question of radiative heat transfer. Although we haven’t discussed radiation here, an engineer must always take it into consideration. Surfaces exposed to ambient conditions will radiate heat to the surroundings and be heated by the sun. The magnitude of radiative heating from the sun is significant — about 1000 watts per square meter — and should not be neglected. For details on modeling radiative heat transfer to ambient conditions, read this previous blog post.

There will also be radiative heat transfer between interior surfaces. Radiative heat flux between surfaces is a function of the difference of temperature to the fourth power. Keep in mind that radiative heat transfer between two surfaces at 20°C and 50°C will be 200 watts per square meter at most, but rises to 1000 watts per square meter for surfaces at 20°C and 125°C. To correctly compute the radiative heat transfer between surfaces, it is also important to compute the view factors with the Heat Transfer Module.

Today we looked at several approaches for modeling convection, starting from the simplest approach of using a constant convective heat transfer coefficient. We then discussed using an Empirical Convective Correlation boundary condition before going over how to use an effective thermal conductivity within a domain and an isothermal domain feature, approaches with higher accuracy and only a slightly greater computational cost. The most computationally intensive approach — explicitly computing the flow field — is, of course, the most general. We also touched on when it is appropriate to neglect free convection entirely and how to model such situations. You should now have a greater understanding of the available options and trade-offs for modeling free and forced convection. Happy modeling!

- Learn about explicitly modeling air flow and heat transfer on the COMSOL Blog
- Get an introduction to simulating heat transfer in an archived webinar

Baking is designed to not only heat a product, such as a cake, but also to prompt the biochemical reactions of the recipe’s ingredients. The combination of dry and wet ingredients creates a mixture that gives the cake flexibility, allowing it to expand while still holding the mixture together.

*Cake batter baking in the oven (left) and the finished cake (right).*

Although ensuring the right amount of each ingredient is important to this end, it is also key to account for the heat and mass transfer phenomena that take place during the baking process. These underlying mechanisms can have a large impact on the temperature and moistness of the cake as well as the degree to which it swells. This in turn affects the overall quality and taste of the baked good.

In an effort to better understand and predict heat and mass transfer phenomena during cake baking, one team of researchers created a numerical model with COMSOL Multiphysics and ran a series of simulation studies. Here’s a taste of what they found.

For their analysis, the researchers created a 2D axisymmetric model. The medium was assumed to be deformable and porous, containing three phases:

- Solid (batter)
- Liquid (water)
- Gas (combination of vapor and CO
_{2})

To address this problem, a system of five coupled partial differential equations was solved. The five variables included in the analysis were:

- Temperature
- Moisture content
- Total gas pressure
- Porosity
- Displacement

To predict the swelling of the batter (a result of the increase in total gas pressure), the researchers used a viscoelastic model from the Structural Mechanics Module, an add-on product to COMSOL Multiphysics.

*The physical phenomena that occur during cake baking. Image by R. Cutté, P. Le Bideau, P. Glouannec, and J.F. Le Page and taken from their COMSOL Conference 2016 Munich presentation.*

As a means of validating the model, the researchers carried out a series of experimental tests. These experiments involved baking a cake for 18 minutes in an oven with a floor wall temperature of 175ºC and top wall temperature of 195ºC. Instruments used within the setup provided information about the batter’s thermal and moisture content as well as the boundary conditions. A camera was used to track the swelling of the cake.

In the simulation analysis, the researchers plotted the temperature and moisture content within the cake at three different intervals:

- Initial state
- Halfway through the baking process
- Final state

The plots below depict the results for each instance, also showing the swelling of the cake. These results indicate that the evaporation-condensation phenomenon causes the water content at the core of the cake (crumb) to increase. On the other hand, the water content decreases at the surface of the cake (crust). As observed in other baking processes, this physical phenomenon contributes to the formation of large moisture content gradients. These gradients create heterogeneity between porosity and thermal, hydric, and mechanical properties. Such heterogeneous characteristics are further driven by the heating mode.

*The temperature and moisture content inside the cake at various time intervals. Image by R. Cutté, P. Le Bideau, P. Glouannec, and J.F. Le Page and taken from their COMSOL Conference 2016 Munich abstract.*

When comparing these simulation results to the experimental findings, there is agreement with respect to temperatures, mass losses, and global deformation. Note that because of the model that is used, the expansion-reduction effect is not accounted for in this case. However, there are plans to improve this by testing other mechanical constitutive laws in the future. To make the model even more accurate, the researchers also plan to add a gas phase with three species (water, CO_{2}, and air) to the model; implement reaction kinetics; and predict the brownness of the cake.

*The simulation results and experimental findings for the temperature (left) and moisture content (right) inside the cake. Images by R. Cutté, P. Le Bideau, P. Glouannec, and J.F. Le Page and taken from their COMSOL Conference 2016 Munich poster.*

*Left: The simulation results and experimental findings for the swelling of the cake. Right: Comparison between the final deformed meshing and the real geometry. Images by R. Cutté, P. Le Bideau, P. Glouannec, and J.F. Le Page and taken from their COMSOL Conference 2016 Munich presentation.*

Baking a cake is not just an art; it’s a science. With COMSOL Multiphysics, you can create a simple yet realistic model to describe this complex process, specifically the heat transfer and mass transfer phenomena that are involved. The results generated from this model provide a better understanding of the cake baking process as a whole.

- Take a look at the COMSOL Conference presentation: “Numerical Model for Predicting Heat and Mass Transfer Phenomena during Cake Baking“
- Browse additional topics relating to simulation and food science:

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.

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.

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:

- Top layer, which enables users to describe a desired model with text input files
- Middle layer, which includes numerical methods needed to formulate input parameters for the API
- 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.

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

- The magnet’s current, linearly ramped up to the nominal value
- 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.

- 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.*

The thermoelectric effect is the conversion of a temperature gradient into an electric voltage and vice versa. The ratio of how an applied temperature difference results in an induced voltage is described by the Seebeck coefficient S(V/K). In principle, all materials have a Seebeck coefficient , but a few of them have coefficients significant enough to be useful. If a material is suitable for thermoelectric applications, it is not determined by the Seebeck coefficient alone. A high thermal conductivity works against the thermoelectric effect and a high electrical conductivity amplifies it. The figure of merit, *Z*, determines the efficiency of a material:

Z=\frac{S^2\sigma}{\kappa}

Seebeck discovered in 1821 that temperature differences lead to electricity. In 1834, Peltier discovered that heating or cooling occurs when an electric current is present. Finally, Thomson determined that the Seebeck coefficient is temperature dependent and the heat produced is the product of current density and temperature gradient. These effects, known together as the Seebeck, Peltier, and Thomson effects, form the general formulation of the thermoelectric effect that is used in the COMSOL Multiphysics® software. Modeling the thermoelectric effect also means that Joule heating must be considered (this is done in COMSOL Multiphysics by default).

Thermoelectric coolers, also known as Peltier coolers or TECs, are mainly used for cooling purposes, but can also be used for heating or as temperature controllers. The main advantage of a TEC is that it is almost maintenance free because there are no moving parts or liquids. They are also useful to cool devices where other methods of cooling, such as air or coolant flow, are impossible to apply. On the other hand, they are made from relatively expensive materials and do not provide efficient cooling effects in relation to their manufacturing costs. Therefore, optimizing the performance of thermoelectric coolers is an important design task.

*The geometry (left) and temperature profile (right) of a thermoelectric cooler.*

The basic design and operating principle of a thermoelectric cooler is depicted in the figures above. The device is made up of an alternating array of p-type and n-type semiconductors (thermoelectric legs) that are electrically connected and sandwiched between thermally conductive plates.

Thermoelectric coolers are often used in products when there is limited space for cooling devices. When designing the app, we should keep in mind that the app user needs to be able to change the size of the thermoelectric cooler geometry. Just as the overall size is variable, the size of the legs and the thickness of ceramics and conductors should be variable too. Different thermoelectric materials exist and the app user should be able to select between different options to accurately represent the design. Depending on the application, the temperature of the hot side differs, which influences the material properties and hence the thermoelectric behavior. The app should also have inputs for these different variables.

There are common parameters that characterize the efficiency of a thermoelectric cooler, including:

- The figure of merit,
*Z*(mentioned above) - Maximum temperature difference between the hot and cold sides,
- Electric current required to achieve the maximum temperature difference,
- Corresponding voltage,
- Overall resistance,
- Maximum heat load,

Besides these values, several performance charts are of interest when analyzing a thermoelectric cooler’s performance. For instance, depends on the heat load and the applied current . The coefficient of performance (COP), defined as

\textrm{COP}=\frac{Q}{U I}

gives the most efficient TEC performance when the heat that is absorbed divided by input power is at its maximum. Curves describing how COP depends on help us choose thermoelectric cooler parameters that are suitable for a certain application.

After running the app, the computed results are displayed in the Graphics section of the user interface (UI). The app automatically shows the temperature plot together with the performance parameters.

Here is a screenshot of the app UI showing the performance parameters for the default thermoelectric cooler design:

Three separate tabs display the performance charts for the temperature difference depending on the heat source (), the temperature difference depending on the applied current , and the coefficient of performance.

*Performance charts for the thermoelectric cooler design with default inputs. Left: Dependence of the temperature on the applied current. Right: The coefficient of performance for three different temperature differences.*

The Thermoelectric Cooler app, built using the Application Builder in COMSOL Multiphysics, combines all of these inputs and outputs into a user-friendly interface.

*Video (without sound) demonstrating the Thermoelectric Cooler app in use.*

Behind the app, there is an underlying fully parameterized COMSOL Multiphysics model. When changing the geometry in the app, the mesh, physics, and results adapt automatically. When you press the *Compute* button, it starts four studies that find the performance parameters and results that are of interest. In addition, the app provides information about the computational time. We can also create reports and generate PDF documentation directly from the app.

The Thermoelectric Cooler app serves as a starting point and inspiration for your own app-building process. Although this app is limited to a single-stage thermoelectric cooler within a certain size range, you can modify and reuse the existing geometry sequence for a second- or third-stage cooler. This way, it takes little effort to extend the app to a multistage thermoelectric cooler.

*The temperature distribution in a two-stage thermoelectric cooler.*

With just a few steps, you can also enable users to input user-defined material data or provide additional predefined materials. For more extensive analyses, you can set intervals in the app so that users can perform multiple parametric analyses at once. There are almost endless possibilities for extending the capabilities of this app and all of the apps you can access in the Application Library.

Try it yourself and let us know what you come up with!

- Find more app inspiration by browsing the Applications category of the COMSOL Blog
- These previous posts are especially helpful for app design:
- Watch our introductory video series on the Application Builder

When designing power plant and other utility boilers, it’s important to find ways of increasing their lifetime and thermal efficiency while decreasing their pollutant emissions. Optimizing the performance of large power plant boilers, for instance, is a common area of study. Today, we focus on one boiler design element in particular: the furnace. Within a furnace, chemical energy is converted into heat energy, which greatly affects the energy conversion process and thus the boiler’s thermal efficiency.

*A retired scotch marine boiler with two coal-burning furnaces. Image by Andy Dingley — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.*

Radiation is the key heat transfer mechanism in boilers. Thus, radiation behavior, such as the radiative heat flux on the combustion chamber’s furnace walls, must be accurately accounted for when studying these devices. Radiation is difficult to predict, and its complexity and dependence on the enclosure geometry means that analytical solutions exist only for very simple problems. Further, experimental modeling of furnace enclosures is expensive.

As an alternative, engineers can build numerical models to properly analyze such enclosures and evaluate thermal efficiency. In the next section, we go over an example model designed for studying radiative heat transfer in a utility boiler’s furnace.

Let’s take a look inside our utility boiler model, which contains five very thin obstructions. This is a practical choice, as utility boilers often include thin obstacles, like panels, hanging in the radiation chambers. For our simulation, we model the obstructions as baffles with zero thickness to reduce the mesh and computational cost. The obstacles each contain an emitting-absorbing medium. Surrounding these baffles, which represent superheater panels, is a 3D enclosure that acts as the utility boiler’s combustion chamber and furnace walls.

*The obstructions in the utility boiler model.*

For our assumptions, we draw from existing research by P.J. Coelho, J.M. Goncalves, and M.G. Carvalho, titled “Modelling of Radiative Heat Transfer in Enclosures with Obstacles” (Ref. 1 in the model documentation). The model’s main assumption is to use existing temperatures and properties, as seen in the table below, inside both the volume and surface zones.

Coordinate (M) | Absorption Coefficient (1/M) | Temperature (K) |
---|---|---|

z ≤ 5 | 0.20 | 1600 |

5 < z ≤ 10 | 0.25 | 2000 |

10 < z ≤ 20 | 0.20 | 1600 |

20 < z ≤ 30 | 0.18 | 1200 |

As for solving our model, we use the discrete-ordinates method (DOM). This method is well suited for cases that involve radiation absorption and scattering in a cavity with a moderate optical thickness. We use the S4 DOM to calculate the distribution of incident radiation in the furnace and heat flux on the enclosure’s side walls. We end up with a set of 24 discrete directions that represents radiative intensity transport.

For more details on how we set up this model, including our use of the radiative transfer equation (RTE), boundary conditions, and the different quantities used in the model, refer to the model documentation.

Using the model presented here, we can easily find the radiative heat flux behavior both inside the furnace and on its surfaces as well as compute the radiative intensity in the participating media. As we can see in the images below, the largest amount of incident radiation happens when the medium’s temperature and absorbing coefficient are at their highest point (the boiler’s burner level).

*Incident radiation in a boiler (left) and on the front of a boiler (right).*

Further, we can compare our results to those from the reference for various configurations. For example, the simulation below, which analyzes the predicted outgoing heat flux, is in good agreement with the published data. This helps confirm the validity of our simulation analysis.

*The outgoing heat flux on the boiler walls.*

- Browse through these blog posts on modeling heat transfer:
- See how to simulate radiation with COMSOL Multiphysics in this archived webinar