During a period of time in which energy efficiency and sustainability are heavily emphasized, magnetic cooling has found its way into new technologies, from industrial to household applications. Based on the magnetocaloric effect, this cooling technology involves the phenomenon in which a temperature of a magnetocaloric material is altered by exposing it to an applied magnetic field. This applied field causes the magnetic dipoles to align, resulting in an increase in temperature. The removal of this magnetic field causes the atoms to become disorganized and the material then cools.

With continued research on the optimization of this technology, the potential to reduce energy consumption in homes and offices across the world has become a more realistic goal. This left one group of researchers wondering if this same method could be used to address another source of high energy consumption — heating and air conditioning in electric vehicles.

*Examples of electric cars (“Ride and Drive EVs Plug’n Drive Ontario” by Mariordo. Licensed under Creative Commons Attribution-Share Alike 2.0 via Wikimedia Commons).*

Using COMSOL Multiphysics, a team of researchers from the National Institute of Applied Science designed a magnetocaloric HVAC system for an electric vehicle.

These vehicles rely on energy from batteries for heating and air conditioning, just as they do for operation. The level of energy required is furthered by the vehicle’s lack of available heat waste from the thermal engine, which makes it easier to heat the internal space of conventional vehicles. The additional need for cooling to prevent overheating in the vehicle’s battery further contributes to the energy usage, while highlighting the importance of adequate cooling systems.

In their design, the researchers used a 2D model to analyze an active magnetic regenerator refrigeration cycle for a magnetic refrigeration system. In this case, the magnetocaloric regenerator was comprised of thin parallel plates, with micro-channels featuring heat transfer fluid alternating in between.

*The geometry of an active magnetic regenerator. Image by A. Noume, C. Vasile, and M. Risser and is taken from the presentation titled “Modeling of a Magnetocaloric System for Electric Vehicles“.*

As a means to optimize the efficiency of the system, the team simulated the behavior of the magnetocaloric regenerator coupled with the circulating fluid. They particularly focused on the convective heat transfer coefficient connected to the heat transfer between magnetocaloric material and coolant — an especially important parameter in the overall design.

During the refrigeration cycle simulation, researchers analyzed the hot- and cold-end temperature variation. The temperature span — the difference between the maximum and minimum temperature — was found to be around 8 K. Adding new materials and alloys was recognized as a potential method of optimizing thermal properties in future designs of these systems.

The results of this study provide a valuable foundation for the use of magnetic cooling technology in electric vehicles. Both rooted in the quest for lower energy consumption, the combination of these two innovative technologies could greatly enhance the autonomy of electric vehicles and make magnetic cooling more mobile.

Bell Labs, the research arm of Alcatel-Lucent, is committed to designing and implementing new technologies for significantly improving energy management for the next generation of telecommunications products. Working to meet this goal, Bell Labs founded the GreenTouch consortium, a leading organization of researchers dedicated to reducing the carbon footprint of information and communications technology. It is the goal of GreenTouch and Bell Labs to demonstrate the key components needed to increase network energy efficiency by a factor of 1,000 compared to 2010 levels.

The Thermal Management and Energy Harvesting & Storage Research Group within the Efficient Energy Transfer (ηET) Department at Bell Labs (led by Dr. Domhnaill Hernon) is one such group working towards this goal. The department focuses on two main areas. One focus of the thermal research group is to deliver game-changing thermal technology into Alcatel-Lucent products across all scales and across multiple disciplines ranging from reliable active air cooling to single and multiphase liquid cooling. One way that this is done is through research for improving the thermal management of laser light transmission in photonic devices by developing an approach to achieve a 50 to 70 percent reduction in energy usage per bit. This approach is explored here.

In addition, the department performs research into Alternative Energy and Storage solutions to enable power autonomous deployment of wireless sensors and small cell technology. In this blog post, we will focus on the production of an energy harvesting device used to power wireless sensors that can produce up to 11 times more energy than current approaches.

In order to improve energy efficiency in photonic devices, the Thermal Management department is using multiphysics simulation to model new designs for cooling photonic devices, which rely on the thermoelectric effect for cooling. Photonic devices used for communications contain a thermoelectric material that is used to cool the device.

In these materials, a temperature difference is created when an electric current is applied to the material, resulting in one side of the material heating up and the other side cooling down. When thermoelectric materials are used to control the temperature of photonics devices, the system is known as a *thermally integrated photonics system* (TIPS). Currently, a large thermoelectric (TEC) cooler is used to cool off the entire system within the photonic device. While TECs can be used for precise temperature control, they are highly inefficient. The group’s new approach improves thermal management by using an individual micro TEC (μTEC) to cool down each laser in the device.

*Schematic of the thermally integrated photonics system (TIPS) architecture, which includes microthermoelectric and microfluidic components.*

Using COMSOL Multiphysics, the team simulated a TIPS architecture to be used in new laser devices, including the electrical, optical, and thermal performance of the device. In addition to cooling, these devices are used by telecommunication laser devices in order to maintain the correct output wavelength, output optical power, and data transmission rates.

The team investigated temperature control and heat flux management in the integrated TIPS and μTEC architecture using simulation. In particular, they investigated how temperature control can be archived in these systems through the integration of μTECs with the semiconductor laser architectures. The simulation of the integrated laser and μTECs can be seen in the image below, on the right.

*Multiphysics simulation of a laser with an integrated μTEC where temperature (surface plot), current density (streamlines), and heat flux (surface arrows) are shown.*

Another project currently being conducted by Bell Labs is the design of an energy harvesting device that can convert ambient vibrations from motors, AC, and HVAC into usable energy. This would prevent the need for the replacement of batteries used in wireless sensors frequently utilized across the network. Applications for this new design include the monitoring of energy usage in large facilities, and in sensors for the future Internet of Things (IoT).

The team’s design used the principles of the conservation of momentum and velocity amplification to convert vibrations into electricity using electromagnetic induction. The device uses a novel approach with multiple degrees of freedom to amplify the velocity of the smallest mass in the system. Simulation played a big part in the design, as parametric sweeps allowed the team to determine how different structural, electrical, and magnetic parameters would affect one another and the design as a whole. The figure below shows the novel design (left) along with the simulation of the device (right).

*Left: Prototype of novel machined-spring energy harvester. Right: Simulation of the energy harvester, showing von Mises stress.*

Although these new designs are not yet available on the market, researchers at Bell Labs believe that because of the accuracy achieved through their simulations, the devices should be ready for commercial production in as little as five years. Whereas previously these designs would have taken years of time-consuming physical testing, the Bell Labs team anticipates that these devices will be available with a much faster time-to-market, thanks to the use of multiphysics simulation.

For more detail, read the full story “Meeting High-Speed Communications Energy Demands Through Simulation“, which appeared in *Multiphysics Simulation* magazine.

There seems to be a general trend when it comes to construction. Offshore structures are constructed in deeper and deeper waters; buildings are constructed increasingly close to each other; offshore wind turbines are developed in deep waters far away from the coasts, which are likely to experience extreme loading conditions. Therefore, in recent decades, geotechnical engineers have developed numerical simulations to cope with this construction trend and ensure safe building methods.

*“Paris Metro construction 03300288-3″. Licensed under Public domain via Wikimedia Commons.*

Materials for which strains or stresses are not released upon unloading are said to behave *plastically*. Several materials behave in such a manner, including metals, soils, rocks, and concrete, for example. These give rise to an elastic behavior up to a certain level of stress, the *yield* stress, at which plastic deformation starts to occur.

The elastic-plastic behavior is path-dependent and the stress depends on the history of deformation. Therefore, the plasticity models are usually written connecting the *rates* of stress, rather than stress, and the plastic strain. The most widespread and well-known plasticity model throughout the industry is based on the von Mises yield surface for which plastic flow is not altered by pressure. Therefore, the yield condition and the plastic flow are only based on the deviatoric part of the stress tensor.

However, this model is no longer valid for soil materials since frictional and dilatation effects need to be taken into account. Let’s see how this can be worked out and briefly explain the different soil plasticity models available in the COMSOL Multiphysics® simulation software.

For materials such as soils and rocks, the frictional and dilatational effects cannot be neglected. This whole class of materials is well known to be sensitive to pressure, leading to different tensile and compressive behaviors. The von Mises model presented above is thus not suitable for these types of materials. Instead, yield functions have been worked out to take into account the behavior of frictional materials.

Let’s illustrate the frictional behavior and plastic flow for these materials by considering the block shown here:

The block is loaded as shown by a normal load N and a tangential load Q. Assuming that the block rests on a surface with a coefficient of static friction \mu, according to Coulomb’s law, the maximum force that the block can withstand before sliding is given by F=\mu N. Therefore, the onset of sliding occurs when the following condition is reached:

(1)

f=Q-\mu N=0

The direction of sliding is horizontal. For tangential loads such as f<0, the block will not slide, but as soon as f=0, the block will slide in the direction of the applied load Q. The *Mohr-Coulomb* criterion — the first soil plasticity model ever developed — is a generalization of this approach to continuous materials and a multiaxial state of stress. It is defined such that yielding and even rupture occur when a critical condition that combines the shear stress and the mean normal stress is reached on any plane. This condition is stated as below:

(2)

\tau=c-\mu\sigma

Here, \tau is the shear stress, \sigma is the normal stress, c is the cohesion representing the shear strength under zero normal stress, and \mu=\tan\phi is the coefficient of internal friction coming from the well-known Coulomb model of friction. This equation represents two straight lines in the Mohr plane. A state of stress is safe if all three Mohr’s circles lie between those lines, while it is a critical state (onset of yielding) if one of the three circles is tangent to the lines.

*Mohr-Coulomb yield behavior. The Mohr circles are based on the principal stresses \sigma_1, \sigma_2, and \sigma_3. As you can see, one of the circles is tangential to the yield surface, and so the onset of yielding is occurring.*

According to the figure above, the stress state is given by \tau=\frac{1}{2}(\sigma_1-\sigma_2)\cos\phi and \sigma=\frac{1}{2}(\sigma_1+\sigma_3)+\frac{1}{2}(\sigma_1-\sigma_3)\sin \phi. The yield criterion and Equation 2 can therefore be re-written in a generalized form as follows:

(3)

f_y({\bf\sigma})=\sigma_1-\sigma_3+(\sigma_1+\sigma_3)\sin\phi-2c\cos\phi

It can even be seen as a particular case of a more general family of criteria based on Coulomb friction and written by equations based on invariants of the stress tensor:

(4)

f_y({\bf\sigma})=F(J_2,J_3)+\lambda I_1-\beta

*Representation of the Mohr-Coulomb yield function. *

The *Mohr-Coulomb* criterion defines a hexagonal pyramid in the space of principal stresses, which makes it straightforward for this criterion to be treated analytically. But, the constitutive equations are difficult to handle from a numerical point of view because of the sharp corners (for instance, the normal of this yield surface is undefined at the corners).

In order to avoid the issue associated with the sharp corners, another yield criterion of this family, the *Drucker-Prager* yield criterion, has been developed by modifying the von Mises yield criterion to take into account the Coulomb friction, i.e., incorporate a hydrostatic pressure dependency:

(5)

f_y({\bf\sigma})=\sqrt{J_2}+\alpha I_1-k

This represents a smooth circular cone in the plane of principal stress, rather than a hexagonal pyramid. If the coefficients \alpha and k are chosen such that they match the coefficients in the *Mohr-Coulomb* criterion, as follows:

(6)

\alpha=\frac{2}{\sqrt{3}}\frac{\sin\phi}{3\pm\sin\phi},\quad k=\frac{2\sqrt{3}c\cos\phi}{3\pm\sin\phi}

the *Drucker-Prager* yield surface passes through the inner or outer apexes of the Mohr-Coulomb pyramid, depending on whether the symbol \pm is positive or negative. The plastic flow direction is taken from the so-called “plastic potential”, which can be either the same, associative plasticity, or different, non-associative plasticity, than the onset of yielding (the yield function). Many different non-associative flow rules can be developed.

Using an associative law for the Drucker-Prager model leads the volumetric plastic flow to be nonzero. Therefore, there is a change in volume under compression. However, this is contradictory to the behavior of many soil materials, particularly granular materials. Instead, a non-associative flow rule can be used such that the plastic behavior is isochoric (volume preserving) — a much better reflection of the plastic behavior of granular materials.

*Representation of the Drucker-Prager yield function.*

Next, I will show you how to use a non-associative law for soil plasticity in COMSOL Multiphysics. Non-associative plastic laws can be used regardless of the plasticity model used in the software.

If you’re using the Mohr-Coulomb model, there are basically two different approaches to handling non-associative plasticity. The plastic potential can either be taken from the Drucker-Prager model or be the same as the *Mohr-Coulomb* yield function but with a different slope with respect to the hydrostatic axis, i.e., the angle of friction is replaced by the *dilatation angle* (see screenshot below).

Moreover, when using the Drucker-Prager matched to a *Mohr-Coulomb* criterion, it is easy to adapt the dilatation angle to match with the non-associative law that you want to use. For instance, the non-associative law presented above can be worked out by taking the dilatation angle null.

Last but not least, a useful feature called *elliptic cap* has been developed to avoid unphysical behavior of the material beyond a certain level of pressure. Indeed, real-life material cannot withstand infinite pressure and still deform elastically. Therefore, to cope with this, we can use the elliptic cap feature available in COMSOL Multiphysics.

*Soil Plasticity feature settings window.*

Let’s try to put into practice everything we’ve learned so far by analyzing the example of a tunnel excavation. This will also be an opportunity to figure out what the effects of the different features we mentioned above are.

The simulation of a tunnel excavation process is especially important in predicting the necessary reinforcements that the workers need to use to avoid the collapse of the construction.

The following model aims to simulate the soil behavior during a tunnel excavation. The surface settlement (i.e., the vertical displacement along the free ground surface) and the plastic region are computed and compared between the different soil models used to carry out this simulation. The geometry we’ll use is presented in the figure below. To make our model realistic, infinite elements have been used to enlarge the soil domain, while keeping the computational domain small enough to get the solution in a relatively short time.

*The geometry consists of a soil layer that is 100 meters deep and 100 meters wide plus 20 meters of infinite elements. A tunnel 10 meters in diameter is placed 10 meters away from the symmetry axis and 20 meters below the surface.*

First of all, we need to add the in-situ stresses in the soil before the excavation of the tunnel. Then, we can compute the elastoplastic behavior once the soil corresponding to the tunnel is removed. The in-situ stresses must be incorporated in this second step. This is fairly straightforward to set up in COMSOL Multiphysics.

We can begin by adding a stationary step where the in-situ stresses will be computed. Then, in a second step but still within the same study, we add a soil plasticity feature. Finally, we compute the solution. In order to get the pre-stresses incorporated into the second step, we should add an Initial Stress and Strain feature under the *Solid Mechanics* interface, as shown below.

*Initial Stress and Strain feature used to incorporate the in-situ stresses from the first step as initial stresses for the second step, during which excavation occurs. The variables solid.sx, solid.sxy, etc. are the *x*-components of the stress tensor, the *xy*-components of the stress tensor, etc.*

The first plot shows the in-situ stresses computed from the first step. These stresses result from the gravity load.

*The von Mises stress in the soil before the excavation of the tunnel.*

The second plot shows the stress distribution after excavating the tunnel. In-situ stresses are taken from the first step. Note, as expected, the increase in the von Mises stress around the tunnel as well as the deformation of the tunnel shape.

*The von Mises stress in the soil after excavation of the tunnel.*

As mentioned previously, while removing the tunnel domain, a plasticity feature is added and the soil experiences a plastic behavior. This is depicted in the figure below of a Drucker-Prager model with associative plastic flow. The plastic region is concentrated around the near surroundings of the tunnel. The analysis of this region is quite important in gaining insight into how the soil is more likely to deform. Therefore, it allows us to handle the necessary reinforcements in order to avoid collapse and get the desired tunnel shape.

*Plastic region after excavating the tunnel.*

This tunnel excavation simulation has been carried out in four different cases in order to compare the different soil models presented in the previous section as well as understand the influence of the cohesion on the soil’s behavior. The results are shown by taking the surface settlement as the criterion.

Below, we have a 1D plot from which we can observe the following relationship: The lower the cohesion, the greater the deformation. We can also note that the Mohr-Coulomb model tends to, somehow, make the soil stiffer than the Drucker-Prager model. The non-associative law with a null dilatation angle prevents the soil from dilating under compression and so the surface settlement becomes greater.

*Surface settlement comparison between different plasticity models and material properties.*

There are also a couple of other plasticity models for soil, rocks, and concrete available in COMSOL Multiphysics. Please check out the links below to get further information about geotechnical simulations and the Geomechanics Module of COMSOL Multiphysics.

Also, be sure to watch the video on how to build a model of an excavation:

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While you may or may not hear the term on a regular basis, lithium-ion batteries certainly play an active role in keeping you connected to others in your day-to-day life. These lightweight, rechargeable batteries are commonly used within various consumer electronics, including laptops and cell phones. With their high energy densities, lithium-ion batteries have even grown in use for industrial and transportation purposes as well.

*A lithium-ion battery from a cell phone. (“NOKIA® Battery” by Kristoferb — Own work. Licensed under Creative Commons Attribution-Share Alike 3.0 via Wikimedia Commons.)*

As the uses for these devices grow, so does the concern surrounding their safety. As mentioned in a previous blog post, a lithium-ion battery overheated and caught fire last year in a new Boeing 787 Dreamliner, forcing the temporary grounding of all Dreamliner planes. Last year, Design News reported on the overheating of lithium-ion batteries within Mitsubishi vehicles (read their story here).

Two different headlines raising one common issue — the effect heat has on the safety and longevity of lithium-ion batteries.

In order to address this concern, it is important to understand the reasoning behind it.

Let’s first begin with the design of the battery. A lithium-ion battery is composed of two electrodes and a nonaqueous electrolyte, which allows for ionic movement. During charging, lithium ions move from the cathode to flow through the electrolyte and then become captured within the crystalline structure of a carbon-based anode. When discharged, the process reverses and these ions flow back, resulting in the reverse electrical flow of current to power the device’s circuit.

As this process, which is similar to electric current flowing through a wire, occurs, internal resistance is created within the electrolyte to bring about Joule heating. In the design of a lithium-ion battery, it is important that this heat disperses quickly enough so that the cell does not reach a high enough temperature for decomposition. As noted in this white paper on modeling the lithium-ion battery, the decomposition reaction is exothermic, meaning that once this process begins, the temperature will continue to rise and feed the decomposition reaction — a phenomenon known as *thermal runaway*. This spread of heat can be a potential source of fire hazard.

With the help of COMSOL Multiphysics, you can visualize and better understand temperature distribution within a lithium-ion battery. The Thermal Modeling of a Cylindrical Li-ion Battery model from the Batteries & Fuel Cells Module couples heat transfer with the lithium-ion battery chemistry and the flow of ions. The *Conjugate Heat Transfer* interface is used to investigate the air cooling of this 3D thermal model of a lithium-ion battery.

*The components of the thermal model.*

The model below shows the battery’s temperature and the flow’s streamlines after 1,500 seconds of charging. The highest temperature is located in the active battery material, toward the end that is thermally insulated. Thus, this area of the cell is more prone to aging and degradation.

*Temperature distribution within the Li-ion battery.*

Modeling and simulation are useful resources in optimizing the design of lithium-ion batteries. By analyzing how heat is transferred during the battery’s operation, researchers and manufacturers are able to improve the performance of the battery and pave the way for a safer, longer-lasting technology.

Download the model files and run your own simulations: Thermal Modeling of a Cylindrical Li-ion Battery in 3D

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As I’m writing this, it’s Friday and we’ve had a long and very productive week. Like me, you’re probably wondering what type of IPA you will order with your nachos later. We are quite lucky to have such a wide selection of beers available in the U.S., thanks to the recent rise of craft beers. But rather than going to our local micro brewery, we could take it a step further and attempt to make our own beer.

Here, we give you a crash course on beer brewing and explain how you can improve the brewing process with CFD simulations. The main purpose is to find out how to cool down five gallons of boiling water as quickly as possible. But, first: Some context.

Making your own beer is relatively straightforward. If you can make tea, you can probably make beer.

Here’s my list of seven steps to brewing beer:

- The process starts by steeping cracked malted barley in hot water around 150°F-160°F for an hour. For home brewers, this is typically done in a cooler, so that the mixture remains at a constant temperature. Steeping the barley cracks down the starch into sugars to make it easier for the yeast to consume it during the fermentation phase.
- After an hour, the resulting liquid, the
*wort*, is moved to a kettle where it is boiled for an hour (or up to two hours for some hoppier IPAs). Hops is added during the boiling process. Additions made early in the boiling process contribute to the bitterness of the beer, while hops added later contribute to the hoppy flavor that most of us love. For the hoppiest beers, even more hops is added right after the boiling step or a few days into the fermentation process. Hops is also used as a preservative agent. - Next, we will cool down the wort as quickly as possible, before adding the yeast to begin the fermentation process.
- Now, we need to be patient for a few days and let the yeast work its magic.
- After the allotted time, we add some sugar to the result, bottle the beer, and give it a name. Let’s call it
*hopsol 1.0*for now. - Another boring step… We need to wait a few weeks while the added sugar carbonates the bottles.
- Finally: We can taste our product, brag about it, and repeat the process.

That’s the standard procedure. But, why settle for standard when we can improve the process? Most of the above steps can be optimized with simulations. Let’s start with the most crucial one — Step 3: Cooling the wort.

In Step 3, the boiling wort needs to be cooled down very quickly for several reasons.

First of all, it needs to be cool enough for the yeast to survive. Second, cooling it limits the production of sulfur compounds and other contaminants during the cooling process. These compounds are associated with off-flavors in the finished beer. Finally, some proteins need to be thermally shocked in order for them to precipitate.

We won’t go into these reasons in detail, as we prefer to focus on the engineering aspect of the process. The size of a typical home brew batch is between five and ten gallons. The problem of cooling down the liquid quickly becomes even harder in an industrial set-up, where the amount of wort to be cooled down is massive.

I’ve seen on YouTube that you might try throwing ice in the boiling kettle. I would not recommend this as it might introduce contaminants in the beer and will dilute it.

How about placing the boiling kettle in an ice bath in our kitchen sink? While this is the cheapest way, it is not the most efficient. I will let you find that out on your own using our *Conjugate Heat Transfer* interface (included in the CFD Module and the Heat Transfer Module.)

The cooling of the kettle in an ice bath can modeled very similarly to the Free Convection in a Water Glass model tutorial, found in our Model Gallery.

*Free convection in a glass of water.*

This model treats the free convection and heat transfer of a glass of cold water heated to room temperature. Initially, the glass and the water are at 5°C and are then put on a table in a room that’s 25°C warm. The boiling kettle cooling problem could be modeled the same way by setting the initial wort temperature to 100°C and the external wall temperature to 0°C.

Alternatively, we could use a wort chiller. Basic wort chillers consist of a long helicoidal pipe that you immerse in the kettle at the end of the boiling process. You will then run cold water from your sink into the pipe to cool down the wort.

*Sketch of a wort chiller.*

As you can deduce by the shape, the modeling procedure for this wort chiller is identical to the one found in our geothermal heating problem model of a pond loop:

In this example, a pond is used as a thermal reservoir and fluid circulates underwater through polyethylene piping in a closed system. The model finds out how much heat is transferred from the pond to the working fluid in the pipes. To this end, the *Non-Isothermal Pipe Flow* interface sets up and solves the equations for the temperature and fluid flow in the pipe system.

In the pipe flow physics interface, the pipes are represented by 1D lines, rather than actual 3D pipes, which drastically reduces the computational load of such a model. The following snapshot shows a possible design for the wort chiller, immersed in the boiling kettle, and the corresponding temperature field within the pipes:

We could also combine method 2 and 3 for faster results, i.e., use a wort chiller while the kettle is in an ice bath.

Another option is to use a flat plate or counterflow heat exchanger. A heat exchanger is a device that transfers heat from one fluid to another. Water, initially at a low temperature and used as the coolant, is being heated up while the wort is being cooled down. The following picture shows a flat plate heat exchanger.

*Flat plate heat exchanger used to cool down beer at a local “brew-your-own-beer” establishment. (You might recognize it from a similar picture on our Instagram account, COMSOL_.)*

These types of heat exchangers are very popular due to their compact size. Many brewers also use *counterflow heat exchangers*.

You can model these devices by following the step-by-step instructions in the Shell-and-Tube Heat Exchanger model, which shows the basic principles of setting up a heat exchanger model. In the model, two separated fluids at different temperatures flow through the heat exchanger, one through the tubes (tube side) and the other through the shell around the tubes (shell side).

Using a heat exchanger is not only the fastest way to cool down your wort, it is also the most efficient. Indeed, most of the heat taken form the wort is transferred to the water. This water can then be reused to steep the next batch of malted barley. This way, no energy is wasted!

Here, we have discussed conjugate heat transfer problems, a pipe flow model, and a heat exchanger model. I encourage you to try modeling these different cooling strategies in COMSOL Multiphysics and find out what works best. You can set up and solve these models using the CFD Module, Heat Transfer Module, and Pipe Flow Module.

After all this modeling, your beer must be pretty tasty and your friends are probably asking for more. It’s time to scale up and use a larger mixer tank, so that you don’t need to stir it manually anymore. Mixing tanks can be modeled using the Mixer Module, an add-on to the CFD Module that allows you to analyze fluid mixers and stirred reactors.

*Model of a turbulent mixer with a three-bladed impeller. The model also considers the shape of the free surface.*

If you have any questions about the models that I’ve presented here, contact our Technical Support team. If you are not yet a COMSOL Multiphysics user and would like to learn more about our software, please contact us via this form — we’d love to connect with you.

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Conducting the structural analysis of a model is an imperative step in the design process. The Structural Mechanics Module, an add-on to COMSOL Multiphysics, offers a virtually limitless amount of capabilities for you to do just that.

This video introduces you to the Structural Mechanics Module and walks you through the entire model-building process for setting up and solving a mechanical problem. This includes demonstrations on how to create parameters, named selections for different parts of your geometry, local variables to implement complicated expressions defined in the model, custom meshes, and tabulated results. To demonstrate the workflow for building and solving a structural mechanics problem, the COMSOL Multiphysics version 4.4 tutorial model of a static bracket assembly is used.

The smallest components, while often overlooked in design, can be the most instrumental ones. Brackets serve as a core component of support for many mechanical devices in numerous industries. In this model, a bracket assembly is fixed in place through eight mounting bolts. A load is applied on the two arms of the bracket, which is representative of a pin being placed between the holes in the bracket arms. As a result, the two bracket holes will experience a loading from this pin. After an initial analysis is complete, the direction of the pin load is varied through a parametric sweep to see the variations in force exertion, stress distribution, and deformation.

- Shown in the video: Bracket — Static Analysis (version 4.4)

*Today, we will be learning how to model a structural mechanics problem, in COMSOL Multiphysics. We will conduct a static analysis of a bracket assembly, and in the end, perform a parametric sweep to analyze a bearing load at different angles. So let’s get started.*

We start our modeling by opening COMSOL Multiphysics, bringing us to the New window. Here we have two options for setting up our model. Use the Model Wizard as a guide for specifying the dimension, physics, and studies you want, instead of starting with an empty model. Here we select our space dimension. When modeling in Structural Mechanics, we can work in 3D, 2D, or 2D axisymmetry, but not 1D or 0D. With our model being three-dimensional, choose 3D for the space dimension.

We are brought to the Select Physics window in the Model Wizard. Here we can add the physics that our model will exhibit. When it comes to choosing physics, this is completely dependent on the model and what meaningful information you are trying to extrapolate from the results. It is advantageous to familiarize yourself with all the physics interfaces available to you, because you may want to add more physics that are relevant to your model later on. Since we are doing a structural analysis of a 3D solid, we go under the Structural Mechanics branch, select Solid Mechanics, and add it to our model. Click Study to enter the Select Study window.

There are several different types of studies to choose from, as shown, depending on the physics interfaces used within the model. Selection of the study type is completely dependent on your analysis objectives. For example, in the case of our bracket model we want to compute deformations and stresses at static equilibrium; so the properties are time-independent. Therefore, under Preset Studies we select Stationary. After clicking Done, you are brought to the COMSOL Multiphysics desktop.

To create parameters and constants in COMSOL Multiphysics in the ribbon, add Parameters. Here we define the parameters and constants that we’ll be using later in this model, which are stored in this table.

Before we add the parameters let’s take a look at the problem we are modeling to have an understanding of where our parameters are derived from. We have a bracket, where the mounting bolts of the assembly are assumed to be fixed, and securely bonded to the bracket itself.

*A pin is placed between the two holes in the bracket arms, and the inner surfaces of the two bracket holes will experience a loading from this pin. We will want to vary the direction of the pin load to see the variations in stress distribution and deformation. After an initial analysis is done with the pin load applied to the bracket holes along the negative *y*-axis at zero degrees, we will perform a parametric sweep of the pin load direction, starting at 0 degrees, and rotating 45 degrees, up to 180 degrees. “theta0” will be used to specify the main direction of the load, and will be the parameter used later in our parametric sweep. “P0” is the peak load intensity applied to the bracket holes. Lastly, “y0” and “z0” are the coordinates of the centers of the bracket holes. It’s good practice to use parameters instead of just the numerical values. When you change these global parameters, they will update throughout the entire model.*

Creating a geometry in COMSOL Multiphysics can be done three different ways. The geometry can be manually created within COMSOL, it can be imported from a file, or you can synchronize the geometry you have open in a CAD program to COMSOL, through any of the LiveLink interfaces. In this example, the assembly of both the bracket and mounting bolts, are available to be imported. In the Geometry section of the ribbon, click Import. Change the Geometry import type to COMSOL Multiphysics file and Browse to where the “bracket.mphbin” file is stored on your computer. This should be located in your COMSOL folder under models, Structural Mechanics Module, and then Tutorial Models. Select the file, click Import, and the geometry will appear in your graphics window.

In the Model Builder Window, under the Geometry node, you can see the Form Union node which is the default setting for finalizing your geometry into COMSOL Multiphysics. Since we are dealing with a set of domains that are assumed to be perfectly bonded to each other, and will not move relative to each other, we use the default Form Union to finalize the geometry. Click Build All.

Creating definitions in COMSOL Multiphysics will help simplify your model, especially when working with large and or complicated geometries. Let’s take a look at a few of the options.

Go to the ribbon, and under the Definitions tab, in the Selections section, add a Box. Box selections allow you to create groups of geometric entities partially or completely inside the box, that would have the same features applied to them. This makes the process of changing materials, model equations, boundary conditions or constraints to different parts of your model much easier to do. In this example we want to make two box selections: the first is for the bolt domains, and the second is for the load-bearing boundaries of the bracket holes.

In the Box settings window find the Box Limits section. Here we can change the limit values which will serve as the dimensions of the box. We want to change these limit values so that the bolts are contained within the box. Under the Output Entities section, in the Include entity if list, choose Entity inside box.

Click the Wireframe Rendering button on the Graphics toolbar and we see in fact that the bolt domains are selected.

We can add a second box, or a cylinder selection to select the bracket hole boundaries, but we will instead add an explicit selection. From the level list, choose Boundary, in the graphics window, select any one of the interior boundaries of one of the bracket holes. Now we can check the box for Group by continuous tangent, and the rest of the interior boundaries will automatically be selected. In the graphics window, select any one of the boundaries of the other hole, and all four of the boundaries will be added.

*Now that we’ve added selections to our model, we can define expressions for adding the boundary load. Local Variables can be used to introduce short and descriptive names for the complicated expressions defined in the model. Go to the ribbon and under the Definitions tab click Local Variables. In the table to define the load, we need an expression for the angle and load intensity, so we enter the following. The angle variable is used to help define the load intensity. This expression evaluates the radial angle, based on its position along the global *z*-coordinate. Since our loading direction will change in only in the *y* and negative *z* directions; or equivalently the 3 ^{rd} and 4^{th} Cartesian quadrants, we can have COMSOL Multiphysics solve for the angle, by computing the four-quadrant inverse tangent. This enables calculating the arctangent in all four quadrants. The load that the bracket holes experience will be sinusoidal in nature, so the sine function is used. This last part of this expression is added to make sure that the load is only applied to the bottom half of each bracket hole.*

*COMSOL uses a global Cartesian coordinate system by default to specify material properties, loads, and constraints in all physics interfaces and on all geometric entity levels. For this model we want to define the orientation of the load applied to the bracket holes. Since the load direction will be rotating about the negative *z*-axis, we need to create a rotated coordinate system. In the Coordinate Systems section of the ribbon, choose Rotated System. This creates a rotated coordinate system, relative to the global system, that defines the orientation of the load applied to the bracket holes. Under the Euler angles subsection, in the beta field type “-theta0”.*

COMSOL Multiphysics comes with a Material Browser, complete with built-in material properties for common materials, as well as materials for specific applications, and any materials created by you, the user. The addition of the Material Library grants users access to the entire COMSOL Multiphysics database of materials. Under the Built-In node, scroll down to select Structural Steel, click Add to Component, and we are done. The material has been automatically assigned to all domains. Here we can see the properties of the newly assigned material. You are free to create your own materials using the New Material function, and you can also use the Add Material button to stay within the main user interface.

Defining the physics and boundary conditions in COMSOL is made as easy as possible, to let you focus on what matters, the physics. To start go to the ribbon and click the Physics tab. Each selection level comes with the various physical properties that can be applied. You can learn about each physical property by adding it, and clicking the Help button in the top right corner of the window.

We first want to set the constraints acting on the structure. Since the mounting bolts are fixed in place, click on the Domains button and add a Fixed Constraint. Under the Domain Selection section from the Selection list, choose Box 1. This assumes that the bolts are rigid and the displacements are perfectly constrained. Next, we want to define the loads acting on the structure. Since the inner surfaces of the bracket holes experience the pin load, in the Physics tab, click the Boundaries button and choose Boundary Load. Choose Explicit 1 for the Selection. Under the Coordinate System Selection section, from the Coordinate System list, choose Rotated System 2, setting the load orientation with a value of “theta0”. Under the Force section, specify the Load vector with the following.

Whenever building a finite element model, we may want to customize the mesh if we anticipate that higher accuracy is needed in some parts of the model. Although we can solve this model with the default mesh, I will demonstrate how to use the mesh settings to get a finer mesh in some regions. In the ribbon, go to the Mesh tab and select Mesh 1. This shows the bracket geometry with the default Normal mesh applied. Although the elements appear as having straight sides, the default mesh used for solid mechanics problems is a second order, or quadratic, mesh. This means that the elements are conformal to the curved geometry.

We will create a second mesh and customize the mesh via the Element size parameters. Click the Add Mesh button. In the Mesh settings window, change the sequence type to User-controlled mesh. This will generate a Size sub node under our second Mesh. Click the Size node. Under the Element Size section, click on Custom. This automatically expands the Element Size Parameters window where the element parameters can be changed. Reducing the Curvature factor to “0.3″ and rebuilding the mesh, results in a finer mesh around the bracket holes. Rotating the geometry, and zooming in on a bolt, we can see the mesh is quite coarse around these small curved faces. The Minimum element size parameter is preventing the changes from the curvature factor. Reduce it to “0.005″ and rebuild the mesh. The elements around the curved edges are now smaller, but the bulk of the geometry remains relatively unchanged.

You can continue to refine the mesh manually if you want. It is also possible to use Adaptive Mesh Refinement to let the software automatically refine and coarsen the mesh, if desired. For the purposes of this example, we will continue with the default Normal mesh size setting.

We continue our simulation with creating a mesh using the default options. Go to the ribbon and in the Home tab, click Build Mesh. Then, in the Study section, click on the eye glasses icon labeled Study 1. Notice the Generate default plots check box. This will create a plot automatically, based on the structural mechanics physics, so in this case, a Stress plot will be created. To solve a stationary study in COMSOL, it is as simple as clicking Compute. COMSOL Multiphysics also defines a solver sequence for the simulation based on the physics and the stationary study type.

After a model has solved in COMSOL, it is time to postprocess the results. We will show you how to add to an existing plot, create a new plot, and extract information from the results.

Here we see the von Mises stress in the bracket and an exaggerated picture of the deformation, which is occurring mostly in the bracket arms. We also want to visualize the plot with vectors, so we can better see the pressure distribution on the inner surfaces of the bracket holes. Under the contextual Stress tab, in the Add Plot section click Arrow Surface. In the Arrow Surface settings window you’ll see an Expression section. From the menu choose Solid Mechanics, Load, and then Spatial load. In the Coloring and Style section, under Number of arrows, the default setting is 200. Increasing the number of arrows will give you a larger volume of arrows that are smaller in size, but heavier in concentration, which allows you to better visualize the load on the bracket holes. Go ahead and experiment with the number to see this yourself. Three thousand seems to give a quality visual. You can now see the load that was applied is displayed.

In this model we’ll also be interested in any displacement that occurs within the bracket geometry. To make a plot showing this, go to the ribbon and in the Results tab under Plot Group, click 3D Plot Group. This will open the newly generated 3D Plot Group 2 contextual tab in your ribbon. In the Add Plot section, click on Surface. The plot for the total displacement experienced by the bracket is automatically generated. Go to the ribbon, and under the Results tab you’ll see different dimensional types for plot groups. In this example we stick to two plot groups, but you are virtually limitless as you can make as many 3D, 2D or 1D plot groups you want for any type of visualization desired.

*Because the mounting bolts are fully constrained, use a volume integration over those domains to accurately calculate the reaction forces. On the Results tab, click More Derived Values and choose Integration, Volume Integration. In the Volume Integration settings window, locate the Selection section and from the Selection list, choose Box 1 to add the bolts. Click Replace Expression here in the upper-right corner of the Expression section, and from the menu choose Solid Mechanics, Reactions, Reaction Force, and the *x* component of the reaction force. Click the Evaluate button. Let’s do this again for the *Y* and *Z* components as well. To save time you can edit the expression, in this case, by changing the component letter.*

*Click Evaluate and the results are shown in Table 1 under the Graphics window. They match what we would expect them to be; the entire load is in the *y* direction while negligible in the *x* and *z* directions.*

It’s often necessary to solve several iterations of a model to find the optimal properties for its design. Instead of manually changing parameter values, and resolving each time, a parametric sweep can be used. A parametric sweep allows you to change the values of a parameter by sweeping the parameter values through a range defined by the user.

Adding a parametric sweep to this model will enable us to solve for different load angles. Go to the ribbon, and in the Study tab, click Parametric Sweep. In the Parametric Sweep window, under the Study Settings section, click the plus sign button to add the load direction as a parameter. To the right of that, click the Range button to define the range for this sweep. We’ll start at zero degrees, and rotate the load forty-five degrees, up to 180 degrees. Click Add and then the Compute button to re-solve the model.

We are automatically brought back to our stress plot. In the 3D Plot Group window, under Data you’ll notice the Parameter value list. Now we have the five different solutions dependent on the angle of the load and can alternate between them by selecting the different values and then clicking Plot.

*After performing a parametric sweep, you can create a table that lists the solutions for each parameter value. This way you can view the different solutions all at once. In the Volume Integration 1 node, click Evaluate and then New Table. The reaction forces at the different parameter values are computed. The reaction force in the *x*-direction is always zero, while the *y* and *z* directions share the load, depending on the angle.*

Traditionally, the way to calculate the effective mass of a particle is to push on it and measure how it reacts to the applied force. One University of Alberta research team (Brad Hauer, Callum Doolin, Kevin Beach, and John Davis) uses simulation as an efficient and noninvasive tool to achieve thermomechanical calibration.

According to Hauer, “The proper calibration of resonators is extremely important, especially in industries where precision is nonnegotiable.”

Because of its accuracy, thermomechanical calibration enables equipment to function both correctly and optimally. The thermal motion of a resonator is proportional to its energy, which is in turn proportional to its effective mass and time-dependent displacement squared. The computation of the effective mass takes into account both the mass and mode shape, and consequently, the displacement of a resonator. Simply put, an accurate prediction of the effective mass of a resonator design allows for proper calibration.

Atomic force microscopy is one field in which very fine measurements are needed. Atomic force microscopy is a way for instruments to inspect surfaces. It works by creating high-resolution images of objects by running a physical probe along them. One downside to this process is that measurements can be completely thrown off by manufacturing errors in the equipment. A device as sensitive as an atomic force microscopy tip requires precise calibration.

The University of Alberta researchers analyzed the fundamental mode shape with the Eigenfrequency Study available in the Structural Mechanics Module. They then derived the effective mass by performing a volume integration of the resonator’s density multiplied by the normalized displacement squared over its entire geometry.

*Simulation of atomic force microscopy tip mode shapes, where light reflected off a cantilever is measured by a photodiode.*

With so many kinds of sensors in use that need to be calibrated, it is a huge benefit to be able to model all geometries in the same software. In the future, the researchers at University of Alberta will work on some cutting-edge designs involving optomechanics. Naturally, they will continue to use COMSOL Multiphysics® to model their designs.

There is a broad range of uses for companies working with nanostructures, nanostrings, and everything in between. The best part is that anyone with the Structural Mechanics Module can get the effective mass of nanoelectronic and nanomechanical devices in a more efficient and scalable way.

*A force transducer able to measure force in increments as small as Attonewtons (10 ^{-18}N).*

- Check out the University of Alberta paper: “Effective Mass Calculations Using COMSOL Multiphysics for Thermomechanical Calibration“
- Familiarize yourself with the Structural Mechanics Module

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?

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When a structure undergoes vibrations, its components experience stresses and strains, which are amplified by the excitation of the natural frequencies of the structure. In addition to potential damage to the structure itself, these oscillations can also be a source of discomfort and disruption for occupants.

Whether a rare or persistent hurdle, seismic and wind-induced vibrations, and their effects, are an important consideration in the design process. *Damping* is one influence that has proved valuable in helping to reduce such vibrations, particularly in tall buildings and bridges, and preserve the longevity of these structures.

Did you know that you can model viscoelastic structural dampers using COMSOL Multiphysics? To get you started, we have created a tutorial model. The Viscoelastic Structural Damper model can be found in the Model Gallery as well as within the Structural Mechanics Module Model Library.

The model first analyzes the frequency response of a viscoelastic structural damper. Comprised of two layers of viscoelastic material, the damper is restrained between steel mounting elements.

*Image depicts the two layers of viscoelastic material in bold, with the steel mounting elements shown in light gray.*

Here, two of the mounting elements are subjected to periodic forces, with frequencies ranging from 0 to 5 Hz. Meanwhile, one of the mounting elements remains fixed. The figure below shows the displacement of the damper at 5 Hz. The second figure highlights the relationship between the applied frequency and the storage modulus and loss moduli, a representation of the viscoelastic properties of the material.

*Displacement at 5 Hz.*

*Plot of storage (blue line) and loss moduli (green line).*

Next, we can run a transient analysis to find out the displacement field as a function of time, as seen in the figure below.

*Surface plot of the z-component of displacement field after 1 second of forced vibrations. *

Plotting the applied force versus the displacement at one of the loaded points shows hysteresis loops, which are characteristic for damped problems. Energy is dissipated since the force and displacement are out of phase with each other.

*2D plot relating the displacement to the applied force.*

*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