Wine coolers usually consist of an open acrylic cylinder with air-filled double walls. The double walls provide thermal insulation, which prevents warm air from reaching the beverages.
A wine cooler on my patio table, keeping the initially chilled bottle cold.
First, you place a chilled bottle in the wine cooler. It must be cool, as this then prevents the temperature from rising to a certain extent. The bottle creates a pool of cold air, and because the density of colder air is higher than that of warmer air, the cooler air stays inside the wine cooler. The air in the insulating walls is slightly warmer than the air surrounding the bottle, but cooler than the air surrounding the cooler. The walls prevent the exterior warm air from reaching the inner chamber of the cooler.
I performed a quick test at home by measuring the air temperature inside my wine cooler with the chilled bottle inside. The temperature measurements confirmed that the temperature inside the cooler quickly drops from the ambient temperature to a much lower temperature, indicating that this type of simple wine cooler works — at least initially. According to some wine cooler manufacturers, the bottle should stay cold for at least an hour (even up to three hours) without requiring any additional cooling method, such as ice cubes or a refrigeration system.
To evaluate if a wine cooler can keep a beverage cold for a certain amount of time, we first need to determine how much the beverage’s temperature can increase before we no longer consider it cold. White wine, for example, has a typical recommended serving temperature that varies between 6 and 12°C (48.2°F and 53.6°F). Since the beverage warms up in the glass when served, I would say that 10°C (50°F) is a suitable limit, above which the beverage is no longer cold.
Let’s use COMSOL Multiphysics to show if these manufacturers’ claims are reasonable. In addition, how will the wine cooler perform in outdoor temperatures that exceed normal room temperature?
For the COMSOL Multiphysics® model, we can take advantage of the cylindrical shape of the wine cooler and create an axisymmetric model, which is computationally efficient compared to a full 3D model and in line with the desired level of details. We also assume axially symmetric conditions (that is, the model does not account for external effects such as wind or thermal radiation from the sun). The model geometry includes the beverage in the bottle, the bottle itself, the air inside the cooler, the cooler walls, and the insulating pocket of air in between the inner and outer cooler walls.
Geometry of the wine cooler model.
For the material data, the model uses the following materials from the built-in Material Library:
Heat transfer occurs in all parts of the cooler setup, modeled as heat transfer in solids (heat conduction) in the glass and plastic and heat transfer in fluids (heat conduction and convection) in the beverage and air. To model the more active natural convection around the bottle (because the bottle walls and the cooler walls have different temperatures), the air between the bottle and the cooler is represented using an increased thermal conductivity via a Nusselt number, Nu = 10 (see the settings for the Fluid node below). Similar Nusselt numbers could have been used in all fluids, but this part is where the mixing is the most important.
The settings for the Fluid node representing the air inside the cooler, with the setting for the Nusselt number (at the very bottom) representing mixing.
For the initial conditions, the bottle and the beverage inside it have a temperature of 6°C, representing a chilled bottle. The air and the cooler have their initial temperature set to 21°C, which is a typical room temperature.
For boundary conditions, COMSOL Multiphysics takes care of the axial symmetry automatically when you create a 2D axisymmetric geometry. The bottom of the cooler is considered thermally insulated. For the exterior of the cooler and the bottle, a convective heat flux describes the condition at the boundary, with the external temperature set to the ambient temperature. The open boundaries at the top of the bottle and the cooler are represented using a temperature condition: The temperature at those boundaries is set to the ambient temperature.
To measure the temperature of the beverage and the air inside the cooler during the simulation, we add two domain point probes that provide the simulated temperature at a point inside the bottle (for the beverage) and a point inside the cooler (for the air). It could also be of interest to use a domain probe to compute the average temperature for the beverage, for example.
Initially, the ambient temperature is also set to 21°C (a so-so summer day in moderately warm areas). In a parametric sweep, we will increase the ambient temperature in steps of 5° to 26°C and 31°C, representing a warm and a hot summer day, respectively. The parametric sweep shows us how sensitive the cooler is to the outside temperature.
The following image shows how to specify a parametric sweep with a set of values and an associated unit:
The Settings window for the Parametric Sweep node (top part only). This sweep includes three values for the ambient temperature (in degrees Celsius).
Finally, we can remove the cooler from the simulation to check how quickly the bottle would heat up if placed directly in the warm air. Doing so in COMSOL Multiphysics is easy. We remove the Heat Transfer interface from the cooler domains and assign the same convective heat flux condition as the other exterior boundaries for the part of the bottle’s boundary that was previously an interior boundary, between the bottle and the air inside the cooler. This part now becomes an exterior boundary exposed to the ambient temperature.
When you run an axisymmetric simulation in COMSOL Multiphysics, the solution is automatically revolved into a full 3D solution. The following 3D plot shows the temperature in the bottle and cooler after 1 hour with an ambient temperature of 21°C:
The temperature after one hour. The bottle and the surrounding air inside the cooler are still cold, except at the top.
From the probes, we can plot the measured temperature inside the bottle and in the air inside the cooler (at 10 cm above the bottom):
The temperature inside the bottle (green) and in the air inside the cooler (blue).
As the plot shows, the air inside the cooler quickly drops to about 10°C, which matches the measurements I did using a household thermometer. The temperature of the beverage stays below 10°C for almost 1.5 hours and below 12°C for over 2 hours. So, with an ambient temperature of 21°C, the promise of a beverage kept cold for at least an hour seems to be valid.
Now, we can run the parametric sweep. The ambient temperature increases by 5 and then by 10 degrees. The following plot shows the temperature for the beverage inside the bottle for the 3 ambient temperatures:
Temperature for the beverage in the bottle, with an ambient temperature of 21 (blue), 26 (green), and 31 (red) degrees Celsius.
As expected, when the ambient temperature gets warmer, the cooler cannot keep the beverage cold for as long. For the case of 26°C, the temperature of the beverage is slightly above 10°C after 1 hour. When it’s hot outside (31°C), the temperature of the beverage is warmer than 10°C after 1 hour, but still below 12°C. So the cooler is still doing a good job of keeping the beverage cold, albeit a couple of degrees warmer than the room temperature case.
The longer the bottle stays in the cooler, the more it becomes affected by the higher ambient temperature. But what if we don’t use a cooler at all? In another simulation, we can analyze the bottle’s temperature without a wine cooler. In this case, we get the following results for the parametric sweep:
Temperature for the beverage in the bottle without a cooler, with an ambient temperature of 21 (blue), 26 (green), and 31 (red) degrees Celsius.
From the plot above, it’s clear that the cooler makes a big difference. Even for the room temperature case, the temperature of the beverage in the bottle does not stay below 12°C. For the higher outdoor temperatures, the temperature of the beverage rises very quickly. Now, we can say for sure that bringing a bottle of wine outside without any kind of cooler is not a good idea.
This simulation shows how fast you can set up a model in COMSOL Multiphysics to verify the order of magnitudes (and that they compare to your measurements, if available) and to explore other configurations, such as variations in the ambient temperature, using parametric sweeps.
The wine cooler simulation we discussed in this blog post demonstrates that you can, indeed, keep a bottle of your favorite beverage cold for a couple of hours, or at least close to an hour — even if it’s really hot outside. All you need to do is properly chill the bottle before bringing it outside and remember to put it in a wine cooler. Fundamental heat transfer mechanisms then do the work for you, so you can “chill out”.
Two topics frequently pop up on the COMSOL Blog: 3D printing and metamaterials. Their potential applications, such as generating customized medical implants, printing houses, and being used for cloaking technology, could transform the world around us.
A 3D printer. Image by Jonathan Juursema — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.
By combining these two technologies, we might be able to use direct-laser-writing (DLW) printing to fabricate complicated metamaterials, a process that could be challenging or impossible with other manufacturing techniques. Using this idea as inspiration, a team from the Karlsruhe Institute of Technology (Germany) and Université de Bourgogne Franche-Comté (France) investigated a metamaterial that displays the unique mechanical property of negative effective compressibility under stable and static conditions.
The researchers’ poroelastic metamaterial is a 3D manmade composite that experiences isotropic expansion when the hydrostatic pressure generated by the surrounding environment is increased. Most natural elastic materials react in the opposite way, reducing their volume when exposed to an increased hydrostatic pressure at a fixed temperature.
A sponge is a material that is affected by poroelastic phenomena.
So why does this metamaterial expand? To answer this, let’s take a look inside the metamaterial, which consists of a single ordinary constituent solid. Within the material are hollow 3D crosses with concealed internal volumes that contain air at a constant pressure. Each cross also has circular membranes attached to its ends.
These membranes warp inward or outward when the surrounding pressure is different than the pressure within the crosses. The bars asymmetrically connected to the membranes translate this warping into a cross rotation. If the hydrostatic pressure outside is greater than the pressure inside, then the individual rotations translate into an isotropic expansion of the structure, causing a negative effective compressibility.
A unit cell at zero pressure (left) and an elevated pressure (right), depicting the principle of negative compressibility. Image courtesy Jingyuan Qu and Muamer Kadic.
While this negative compressibility may appear to violate the laws of physics, the effective volume increase corresponds with an unseen volume decrease within the material. This ensures that the structure is stable.
To examine the detailed structure of the innovative metamaterial, the researchers turned to the COMSOL Multiphysics® software. When asked about the benefits of this approach, Jingyuan Qu — a member of the research team — noted how easy it was to implement.
The model of the metamaterial is a single unit cell. To see what happens when there is a difference between the pressure inside and outside the material, a pressure increase is implemented as a normal force on all of the outer surfaces of the model. Further, the model is simulated under periodic boundary conditions, enabling the researchers to successfully find the effective material parameters.
Note that built-in periodic boundary conditions are available in the Structural Mechanics and MEMS modules.
For their research, the team performed two main numerical experiments:
In their experiments, the team used the equations for standard linear elasticity:
Now, let’s take a look at the second numerical experiment.
To mimic an infinite material case, periodic conditions are implemented so that each side of the unit cell must contract or expand isotropically. First, selections for each side of the structure are created and named according to the directions x+, x-, y+, y-, z+, and z-. Then, probe variables were created, giving the average displacement on the “minus” sides (dispx, dispy, dispz), shown in the second screenshot below.
Examples for the x direction case, displaying the process of selecting the boundaries that connect to the next unit cell, shown for one of the six planes (top) and the boundary probe settings (bottom). Images courtesy Jingyuan Qu and Muamer Kadic.
Next, the probe variables are used as boundary conditions on both sides (prescribed displacement). That is, on the ‘x-’ boundaries, the x direction displacement is set to dispx, while on the ‘x+’ boundaries, it is set to -dispx. Similar boundary conditions are then set on the other periodic cuts. The idea is that the displacement dispx, still unknown, becomes part of the solution. Since the conjugate reaction force to the prescribed displacements must be zero, the structure will expand or contract in such a way that there is no net force.
Prescribing the probed displacement. Image courtesy Jingyuan Qu and Muamer Kadic.
Moving on, outer pressure is also applied. After selecting the outer boundaries of the geometry and using a high angular tolerance, the model shows that the inner boundaries in the concealed volumes are not selected, as seen below.
The outer boundary settings. Image courtesy Jingyuan Qu and Muamer Kadic.
The hydrostatic load is then applied as a boundary load with pressure (P).
The hydrostatic pressure is implemented as a normal force acting on all outer boundaries. Image courtesy Jingyuan Qu and Muamer Kadic.
The resulting structure of the poroelastic metamaterial at different angles. Images courtesy Jingyuan Qu.
As a point of comparison, the researchers also examined an ordinary porous structure and a cube of a continuous isotropic material. When exposed to an increased hydrostatic pressure, both of these structures shrink in volume. Under the same conditions, the porous metamaterial expands — highlighting its negative effective compressibility.
Thanks to their extensive research, the team was able to capture the metamaterial’s behavior, improve its design, and use this information to move on to the fabrication stage. While fabricating this material may not be possible with conventional machining techniques, 3D printing serves as an alternative option for creating negative compressibility metamaterials. 3D printers can form this metamaterial by using ordinary materials that shrink under hydrostatic pressure.
Qu notes that after it is realized, the metamaterial may find uses in high-pressure applications due to its ability to maintain a constant effective volume, even when subjected to high-pressure environments.
A frequent topic on the COMSOL Blog, heat sinks dissipate excess heat in electronic devices. Heat sinks have become more important with advancing technology. As electronic devices become more powerful, they often generate more heat. This can impair the performance of these devices and even reduce their lifespans.
A heat sink on a motherboard. Image by Adikos — Own work. Licensed under CC BY 2.0, via Flickr Creative Commons.
To analyze the cooling capacity of a heat sink, we can turn to the heat transfer modeling capabilities of COMSOL Multiphysics. Due to the flexibility of the software, we can approach the analysis in multiple ways. Let’s discuss two of these approaches using the simple example of electronic chip cooling.
For our example, we model an electronic chip with a heat sink that is cooled by the surrounding fluid (air). To create the model, we use the heat sink geometry parts available in the Part Library in the Heat Transfer Module, an add-on product to COMSOL Multiphysics. These parts, which are available as of version 5.3, make it easier to include a heat sink in a model geometry.
For more details, check out the Heat Transfer Module Release Highlights page.
In this case, the model consists of an electronic chip and aluminum heat sink. The heat sink is mounted within a channel that has a rectangular cross section. As for the electronic component, it dissipates the equivalent of 5 W of heat. This is distributed throughout the chip volume.
Model geometry of the heat sink (gray) and electronic chip (purple).
Air flows from the inlet (where the temperature is set) through the channel to the outlet. There, convection is the main form of thermal energy transport. A combination of conduction and convection transports the thermal energy in the cooling air, while conduction transfers thermal energy in the electronic component and heat sink.
The model enables us to solve for the thermal balance between the various components and to find the thermal contact between the chip and the heat sink when dealing with a thermally thick layer. We can estimate the cooling capacity of the heat sink and predict the temperature of the electronic component by solving the model equations. Next, let’s look at the two different approaches for solving this model, starting with the more computationally efficient approach.
For the first approach, we only model solid parts and use a convective cooling boundary condition on the heat sink boundaries instead of computing the flow velocity, pressure, and temperature of the air channel. This enables us to make rapid computations. However, the accuracy of these computations is tied to the reliability of the heat transfer coefficient used to define the convective cooling boundary condition. Here, we use an empirical value of 10 W/(m^{2}·K).
Using this approach, we test three configurations for thermal contact between the heat sink and chip:
Temperature plots for the ideal contact (left), air layer (middle), and thermal grease layer (right) configurations.
As expected, the lowest maximum temperature (around 84°C) is obtained for the ideal contact case. When we include an air layer between the heat sink and chip in the model, this rises to almost 95°C, causing the performance of the cooling system to decrease.
We can reduce the effect of this thermal resistance by swapping out the layer of air for a layer of thermal grease. The thermal grease layer improves the thermal contact between the base of the heat sink and the top of the electronic chip. The simulation predicts that the maximum temperature nears that of the ideal thermal contact case. From this, we can conclude that a thin layer of thermal grease helps to improve the heat sink design.
Moving on, let’s take a more computationally expensive approach that adds a domain representing an air channel to the model. This domain allows us to calculate the temperature and flow fields of the air while assuming nonisothermal flow in the channel. This approach is more general than the previous approach, which is helpful, since the heat transfer coefficient is generally unknown. This way, we can accurately simulate flow cooling without needing to approximate the heat transfer coefficient.
The results from this approach show that the maximum temperature of the electronic component is around 95°C. In addition, there is a hot wake behind the heat sink, indicating its convective cooling effects. This second modeling approach is also more precise. For example, we can visualize and predict the temperature difference on a fin at the edge facing the flow or at the opposite side.
The temperature field on the heat sink’s surface and channel walls (left). Visualizing the temperature difference on a fin (right).
We can enhance this approach by modifying the model to account for a large (close to 1) surface emissivity. Since surface-to-surface radiation should be considered when working with such large emissivities, we include it in our simulation. This addition lowers the maximum temperature to around 81°C, as seen below. This reaffirms that when the surface emissivity is large, radiative heat transfer should be considered.
The temperature field when surface-to-surface radiation is included.
As we’ve seen here, COMSOL Multiphysics offers tools that make it possible to use different approaches to analyze electronic cooling — an important step in the design of heat sinks for optimized electronic devices. Each modeling approach has its own benefits, whether it be a more computationally effective option or a method with higher accuracy.
When molten metal solidifies, grains begin to form. These affect the physical properties of the solid metal; for instance, a smaller grain size makes the metal stronger. The grain of a metal is affected by many factors, such as temperature and cooling time. Engineers can also adjust grain morphology during the metal solidification process by using the physical phenomenon of acoustic streaming (AS) to induce drag on particles.
Molten metal being processed. Image by Goodwin Steel Castings — Own work. Licensed under CC BY-SA 2.0, via Flickr Creative Commons.
In AS, an oscillating sonotrode sound emitter is placed in a liquid and generates a steady fluid motion. To produce a significant effect, the sound waves need a high amplitude and frequency, often in the ultrasound range. Thus, this technique can require an ultrasonic treatment.
To further improve and develop AS, engineers rely on costly physical experiments. Simulation provides an alternative, enabling those in the metal processing industry to build models to test AS treatments that use different materials and fluids. These models can then be validated via experimental testing.
Let’s take a look at how researchers from the Institute of Thermal and Fluid Engineering and Institute of Product and Production Engineering at the University of Applied Sciences Northwestern Switzerland investigated this possibility.
The research team’s goal was to build and experimentally validate an AS model that can analyze a variety of fluids by altering the parameters. The resulting 2D axisymmetric model represents an experimental setup of an oscillating sonotrode placed into a fluid and generating a harmonic acoustic pressure field. They simplified this model by assuming isothermal behavior, neglecting cavitation, and using an averaged stationary flow.
Acoustic streaming sample geometry. The numbered dots represent the boundary positions and the colored dots show the locations of the three tracing massless particles. Image by D. Rubinetti, D. Weiss, J. Muller, and A. Wahlen and taken from their COMSOL Conference 2016 Munich paper.
Since AS is a multiphysics phenomenon, the researchers accounted for two physics phenomena:
Note that accounting for the acoustics in the high-frequency domain requires a compressible fluid description, due to the coupling of density and pressure perturbations.
The researchers solved the model equations with three study steps:
The first study shows that the sonotrode acceleration causes a sharp rise in acoustic particle velocity. Using the acoustic velocity field found here, the force term for the second study is generated.
Frequency-domain results, showing the acoustic velocity field. Image by D. Rubinetti, D. Weiss, J. Muller, and A. Wahlen and taken from their COMSOL Conference 2016 Munich paper.
The fluid flow study investigates the flow pattern, beginning with the axial jet exiting from the actuating sonotrode tip. The jet continues to the bottom wall, where it is deflected and creates vortices in the bottom corners. The generated flow is almost at a standstill in the open interface zones and has the highest value beneath the sonotrode.
Left: Stationary velocity field for an aluminum melt with a frequency of 20 kHz and 30 µm amplitude. Right: Comparison of the velocities of the three massless particles, with the colors matching those in the sample geometry. Images by D. Rubinetti, D. Weiss, J. Muller, and A. Wahlen and taken from their COMSOL Conference 2016 Munich paper.
When looking at the dispersal of the three separate tracing particles, the model demonstrates that the particles beneath the sonotrode (depicted as the black line in the right plot above) have a high acceleration, which increases the number of cycles.
To experimentally validate their simulations, the researchers created a small-scale laboratory model involving an aluminum sonotrode partially submerged in a fluid-filled crucible. Next, they performed experiments at a frequency of 20 kHz and at three different amplitudes: 10, 20, and 30 µm. To track the fluorescent particles used in the experiment, the team relied on a combination of a high-speed camera, diode laser, and lasersheet. They then determined the correlated velocity field via particle image velocimetry.
The experimental setup. Original image by D. Rubinetti, D. Weiss, J. Muller, and A. Wahlen and taken from their COMSOL Conference 2016 Munich paper.
When testing a seed oil liquid, we can see that the axial jet is visible in both the simulation and experimental results, as shown below. While the results don’t completely line up, they do match on the right side of the crucible for the direction and location of the induced flow.
The velocity fields for the simulation (left) and seed oil test case (right) with a frequency of 20 kHz and amplitude of 30 µm. Images by D. Rubinetti, D. Weiss, J. Muller, and A. Wahlen and taken from their COMSOL Conference 2016 Munich paper.
We can also compare the velocities along the rotational axis in the simulation and experiment, which show good agreement near the tip of the sonotrode. The results begin to differ as we move farther away from the tip, with the simulation achieving a peak velocity (over twice that of the experimental maximum velocity) within 10 mm from the tip. The difference between the results decreases when the axial difference increases, with both the simulation and experiment showing a decline behavior.
Comparison of the simulation and experimental results for the velocity magnitude along the rotational axis. Image by D. Rubinetti, D. Weiss, J. Muller, and A. Wahlen and taken from their COMSOL Conference 2016 Munich paper.
The deviation between the results may be due to a few reasons, including nonaccurate optical measurements (the experimental data is difficult to collect) and the team’s simplified simulations. The underlying issue may be that the flow is not steady in reality. As their experimental plot above confirms, momentum dissipates a lot more in the experiments than the model. This suggests that there are unsteady smaller eddies that transfer momentum at a rate that is not described by the averaged steady flow used in the model.
Through their experimental testing, the research team concluded that their AS model gives a qualitative description of the flow, except for the small region close to the sonotrode where the description seems to be fairly accurate. As a reasonable approach for analyzing AS and predicting fluid flow behavior, it can save time and money by minimizing the required amount of physical experiments.
Simulation is also a good choice for testing various fluids, parameters, and geometries. This enables engineers to efficiently study different AS treatments by tailoring the model to fit a specific case. The researchers also note that the model is versatile enough to be used to study other sound-driven fluid motion applications.
Under plane strain conditions, no expansion in the out-of-plane direction is allowed. There are usually stresses in that direction caused by the coupling to the in-plane strain through a nonzero Poisson’s ratio. On the other hand, when a thin sheet is studied, the plane stress assumption is more useful. In this case, the material is free to contract or expand in the out-of-plane direction and the transverse stress is zero.
If the structure is long in the transverse direction when compared with its in-plane size, but is still not restrained in the transverse direction, then neither of these assumptions is good. This is where a generalized plane strain condition becomes useful.
A possible generalization of the plane strain formulation is to assume that the strains are independent of the out-of-plane coordinate. In the COMSOL Multiphysics® software, this generalization can be implemented using a 2D geometry of the cross section and the Solid Mechanics interface, in which the plane strain formulation is a default option.
The strain tensor components are assumed to be functions of only the in-plane coordinates x and y (and possibly time):
(1)
Under the small-strain assumption, the strain tensor components are related to the displacement field as:
(2)
The above equations have the following 3D solution:
(3)
where a, b, and c are constant coefficients.
The corresponding out-of-plane strains are:
(4)
This strain state differs from the standard plane strain assumption only by the fact that the normal out-of-plane strain is nonzero and can vary linearly over the cross section. At the cross section z = 0, the deformation is in-plane and fully characterized by the in-plane displacement components u(x,y) and v(x,y).
The coefficients a, b, and c in the expression for the normal out-of-plane strain can be introduced as extra degrees of freedom (DOF) that are constant throughout the model (global variables). The extra strain contribution can be incorporated using the External Strain feature available in the Solid Mechanics interface.
A generalized strain formulation is important is when analyzing stress-optical effects, such as birefringence in waveguides composed of several layers of different materials (e.g., silicon-on-insulator waveguides). This stress-optical effects tutorial model shows such a case.
To illustrate the efficiency of this approach, let’s consider a simple beam-like structure composed of two layers with square cross sections of 1 cm. The layers are made of materials with significantly different elastic and thermal properties: aluminum and nylon. The data is taken from the built-in Material Library in COMSOL Multiphysics. The length in the out-of-plane z direction is L = 20 cm. The structure is assumed to be manufactured at an elevated temperature. Due to the mismatch in the thermal expansion properties of the materials, a residual thermal stress builds up in the structure when it has cooled down to the operating temperature. This makes the structure bend slightly in the out-of-plane direction.
The following figure shows plots of the total displacement together with the deformation for a full 3D model and a 2D generalized plane strain condition:
The 2D solution computed for u(x,y) and v(x,y) within the cross section has been extruded in the out-of-plane z direction using the analytical solution for the corresponding 3D displacement field given above.
The 3D solution requires around 32,000 DOF, while the 2D solution only needs around 250 DOF.
The following plots show the variation of the out-of-plane strain and stress along one of the edges.
Strain (left) and stress (right) along the z-axis.
Around 80% of the true 3D structure has stress and strain fields similar to those predicted by the generalized plane strain theory. Only near the free ends, where the stress tends toward zero, does the strain field start to deviate from the linear distribution within the cross section.
One way to incorporate the changes needed for the generalized plane strain approximation is to start with a 2D component and the Solid Mechanics interface and then add the following nodes in the Model Builder tree:
The Model Builder tree, showing the nodes needed to implement a generalized plane strain condition.
In addition to the standard settings for a 2D problem, you must perform the following steps. First, in the Global Equations node, add the a, b, and c coefficients as DOF. Note that you do not set up any equations for those variables here. Thus, all input fields other than the variable names are kept at their default values.
The Global Equations node, showing the a, b, and c coefficients.
In the Variables node, define the out-of-plane normal strain component eZ in terms of a, b, and c.
The Variables node, showing the expression for the variable eZ.
Next, incorporate the extra strain component into the stress-strain relation in the External Strain node. Note that any expression you enter in this node is subtracted from the total strains before the elastic stresses are computed from strains using Hooke’s law. Usually, this node can be used to incorporate inelastic effects; for example, strains caused by various electromechanical multiphysics effects. Here, we use it simply as a mechanism to inject an extra strain component that is zero by default in the plane strain formulation.
The External Strain node, showing the extra strain component.
Lastly, in the Weak Contribution node, include the extra virtual work done by the out-of-plane stress. This sets up equations (in the weak form) to determine a, b, and c. Here, solid.d
is the thickness in the z direction, as defined in the Solid Mechanics interface.
The Weak Contribution node, showing the weak expression.
You can also skip the third and fourth steps above and insert the strain variable eZ directly into the equations of the Linear Elastic Material node. To do this, make sure Equation View is enabled.
The settings for Equation View.
This way, the new strain in the z direction is directly part of the material model and goes into the weak expression that is already generated in the Linear Elastic Material node.
We have shown how to use the functionality available in COMSOL Multiphysics to model elongated structures that are free to extend in the out-of-plane direction. The use of the 2D generalized plane strain approximation allows us to reduce the computation effort significantly while reproducing the possible out-of-plane bending of the structure — a 3D effect that can be important in applications such as piezoelectric devices and optical waveguides. It is also possible to incorporate out-of-plane shearing of the structure, which can be important in some piezoelectric applications.
The complex interaction of a stationary background flow and an acoustic field can be modeled using the linearized Navier-Stokes physics interfaces in the Acoustics Module. The interfaces allow for a detailed analysis of how a flow, which can be both turbulent and nonisothermal, influences the acoustic field in different systems. This includes all linear effects in which a background flow interacts with and modifies an acoustic field. The linearized Navier-Stokes interfaces do not induce flow-induced noise source terms. Basically, the equations solve for the full linear perturbation to the general equations of CFD — mass, momentum, and energy conservation.
Being able to model and simulate the details of how a background flow influences an acoustic field is important in many industries and areas of application. In the automotive industry, the acoustic properties of exhaust and intake systems are altered when a flow passes through them. For example, the transmission loss of a muffler changes depending on the magnitude of the bypass background flow. In aerospace applications, the study of how liners and perforates behave acoustically when a flow is present is of high interest. The detailed acoustic properties (absorption, impedance, and reflection coefficients) of these subsystems influence the system-level behavior of, for example, a jet engine.
In the muffler and liner examples, the attenuation of the acoustic signal by the turbulence present in the background flow can also be captured with the linearized Navier-Stokes equations. Moreover, the background flows in these models are often nonisothermal in nature.
Example of an automotive application. Results from the Helmholtz resonator with flow example presented below. In front, the color surface plot is of the sound pressure level. At the back, the streamlines are of the background flow.
The linearized Navier-Stokes interfaces have a built-in multiphysics coupling to structures. This enables an out-of-the-box setup of fluid-structure interaction (FSI) models in the frequency domain (or in the linear regime in the time domain). The interaction of flow, acoustics, and structural vibrations is important in many applications. One application example is for flow sensing in a Coriolis flow meter. In general, these interfaces are suited for the analysis of the changed vibrational behavior of structures when under a fluid load by a background flow.
Example of FSI in the frequency domain: the movement of a Coriolis flow meter actuated at the fundamental frequency. The surface shows the structural deformation (the phase and amplitude are highly exaggerated for visualization) and the open cut-out section of the pipe shows the acoustic pressure on the pipe’s inner surface.
Other applications of the linearized Navier-Stokes interfaces include the study of combustion instabilities and general in-duct acoustics as well as more academic applications like analyzing the onset of flow instabilities or studying regions prone to whistling.
The interfaces now include the Galerkin least squares (GLS) stabilization scheme, enabling more robust simulations. This new default setting better handles the numerical and physical instabilities introduced by the convective and reactive terms included in the governing equations. Moreover, the reformulated slip boundary condition is now well suited when solving models with an iterative solver. This is crucial in cases where large industrial problems have to be solved.
The linearized Navier-Stokes equations represent a linearization to the full set of governing equations for a compressible, viscous, and nonisothermal flow (the Navier-Stokes equations). It is performed as a first-order perturbation around the steady-state background flow defined by its pressure, velocity, temperature, and density (p_{0}, u_{0}, T_{0}, and ρ_{0}). This results in the governing equations for the propagation of small perturbations in the pressure, velocity, and temperature (p, u, and T) — the dependent variables. In perturbation theory, a subscript 1 is sometimes used to express that these variables are first-order perturbations. The governing equations (with subscript 0 on the background fields) read:
(1)
where Φ = ∇u : τ_{0} + u_{0} : τ is the viscous dissipation function; M, F, and Q represent possible source terms; κ is the coefficient of thermal conduction (SI unit: W/m/K); α_{p} is the (isobaric) coefficient of thermal expansion (SI unit: 1/K); β_{T} is the isothermal compressibility (SI unit: 1/Pa); and _{p} is the specific heat capacity (heat capacity per unit mass) at constant pressure (SI unit: J/kg/K).
In the frequency domain, the time derivatives are, in the usual manner, replaced by a multiplication with iω. The constitutive equations for the stress tensor and the linearized equation of state (density perturbation) are given by:
(2)
where τ is the viscous stress tensor (Stokes expression), μ is the dynamic viscosity (SI unit: Pa s), and μ_{B} is the bulk viscosity (SI unit: Pa s).
The Fourier heat conduction law is used in the energy equation. A detailed derivation of the equations can be found in the Acoustics Module User’s Guide. The equations can be solved in the time domain or frequency domain using either the Linearized Navier-Stokes, Transient interface or the Linearized Navier-Stokes, Frequency Domain interface.
By taking a closer look at the governing equations presented in (1), you can see that they contain different types of terms:
Because of the general nature of the equations solved in the interfaces, they naturally model the propagation of acoustic (compressible) waves, vorticity waves, and entropy waves. The latter two types of waves are only convected with the background flow velocity and do not propagate at the speed of sound. As an acoustic wave propagates, it can interact with the flow (through the reactive terms) and energy can be transferred to and from an acoustic mode to both the vorticity and entropy modes. The reactive terms in the governing equations are responsible for this flow-acoustic-like coupling. This is in the sense that the vorticity and entropy waves are nonacoustic (CFD-like) perturbations to the background flow solution, so to some extent, they model the linear interaction between CFD and acoustics.
In many aeroacoustics formulations, the reactive terms are disregarded, as they are also responsible for the processes that generate the Kelvin-Helmholtz instabilities. These can be difficult to handle numerically. On the other hand, if the terms are disregarded, accurate modeling of sound attenuation and amplification is lost. The reactive terms are fully included in the linearized Navier-Stokes interfaces.
The growth of the instabilities is handled in two ways in COMSOL Multiphysics. The temporal growth of the instabilities can be handled by selecting the frequency-domain formulation rather than the time-domain formulation. The spatial instabilities, which can arise if the vorticity modes are not properly resolved, are efficiently handled by the GLS stabilization scheme.
Depending on the application modeled with the linearized Navier-Stokes equations, it may be necessary to resolve the acoustic, viscous, and thermal boundary layers. These are naturally created on solid surfaces for an oscillating flow, when no-slip and isothermal boundary conditions are present. Typically, it is not necessary to include the details of the losses in the boundary layers in large models (when compared with the boundary layer thickness). The thermal boundary layer can also often be disregarded in liquids but should be included with equal importance in gasses. The two effects can be disregarded by selecting either the slip or the adiabatic options on the wall boundary conditions.
It should be mentioned that one more indirect coupling between the background flow and the acoustics is possible. When an acoustic wave propagates through a region with turbulent background flow, it is attenuated. This effect can be included in the model by coupling the turbulent viscosity from the CFD RANS model to the acoustics model. This effect is important, for example, when analyzing the transmission loss of a muffler system in the presence of a flow.
Solving the linearized Navier-Stokes equations, which falls under the field of computational aeroacoustics (CAA), poses numerical challenges that need to be considered, understood, and handled carefully. As mentioned above, the governing equations are prone to both physical (Kelvin-Helmholtz) and numerical instabilities. Because the interfaces use stabilization, the remaining main numerical challenge is to avoid the introduction of numerical noise in terms involving the background field variables (p_{0}, u_{0}, T_{0}, and ρ_{0}). This is especially true in the reactive terms involving the gradient of the background flow variables.
The likelihood of this problem increases if different meshes are used for the CFD and acoustic models and/or different discretization orders are used for the background flow and acoustics problem. Note that using different meshes or discretization orders is well motivated by the fact that the two problems need to resolve different physics and length scales. To prevent this, a careful mapping of the background flow data from CFD to acoustics is necessary. This is a well-understood and described step in CAA modeling. Additionally, the mapping step can be used to smooth the CFD data. This can be an overall smoothing or a local smoothing of certain details, like the hydrodynamic boundary layer, if its details are not important for the acoustics model.
In COMSOL Multiphysics, the mapping between the mesh is performed by an additional study step. The details of this step are described in the Acoustics Module User’s Guide and in tutorial models using a linearized Navier-Stokes physics interface.
When performing simulations with a linearized Navier-Stokes physics interface, the following points should be considered:
Helmholtz resonators (used in exhaust systems) attenuate a narrow and specific frequency band. When a flow is present in the system, it changes the resonator’s acoustic properties as well as the subsystem’s transmission loss. The Helmholtz resonator tutorial model investigates the transmission loss in the main duct (the resonator is located as a side branch) when a mean flow is present.
To calculate the mean flow, the SST turbulence model is used for Mach numbers Ma = 0.05 and Ma = 0.1. The Linearized Navier-Stokes, Frequency Domain interface is used to solve the acoustics problem. Next, the acoustics model is coupled to the mean flow velocity, pressure, as well as turbulent viscosity. The predicted transmission loss shows good agreement with results from a published journal paper (Ref. 1). For the resonances to be located correctly and the amplitude of the transmission loss to be correct, the model must balance convective and diffusive terms properly. This is achieved in the model.
Transmission loss through the resonator as a function of frequency and Mach number of the background flow.
The pressure distribution inside the system at 100 Hz for Ma = 0.1. A plane wave is incident from the left side upstream of the flow.
In the Acoustic Liner with a Grazing Background Flow tutorial model, the acoustic liner consists of eight resonators with thin slits and the background grazing flow is at Mach number 0.3. The sound pressure level above the liner is calculated and shows good agreement with results from a published research paper (Ref. 2). This example computes the flow via the SST turbulence model in the CFD Module and the acoustic propagation with the Linearized Navier-Stokes, Frequency Domain interface. The acoustic boundary layer is resolved and the default linear discretization is switched to quadratic to improve the spatial resolution near walls.
The curves show the sound pressure level on the surface above the liners for four driving frequencies. The colored part of the curves highlights the extent of the liner. These results show good agreement with the experimental results from the referenced research paper.
The acoustic velocity fluctuations as a plane wave propagates above the liners, showing the first four liners. The driving frequency is 1000 Hz. The color plot shows the velocity amplitude and the arrows show the velocity vector. Near the holes at the surface of the liner, vorticity is generated by the flow-acoustics interaction.
Coriolis flow meters — also called mass or inertial flow meters — can measure the mass flow rate of a fluid moving through it. This device can also compute the density of the fluid, hence the volumetric flow rate. The Coriolis Flow Meter tutorial model demonstrates how to model a generic Coriolis flow meter with a curved geometry.
As a fluid travels through an elastic structure (a curved duct, for instance), it interacts with the movement of the structure when vibrating. The Coriolis effect causes a phase difference between the deformation of two points on the duct, which can be used to determine the mass flow rate.
To model this, the Linearized Navier-Stokes, Frequency Domain interface is coupled to the Solid Mechanics interface via the built-in multiphysics coupling. As for the background mean flow, it is simulated with the Turbulent Flow, SST interface. Using this approach, FSI can be efficiently modeled in the frequency domain.
The phase difference between upstream and downstream points (red dots on the animation below). This curve represents the calibration results needed to run a Coriolis flow meter.
The movement of the Coriolis flow meter for three mass flow rates. The flow meter is actuated at the natural frequency of the structure, f_{d} = 163.5 Hz. The deformation amplitude and phase are exaggerated for visualization. As the flow rate increases, the phase difference upstream and downstream increases.
E. Selamet, A. Selamet, A. Iqbal, and H. Kim, “Effect of Flow in Helmholtz Resonator Acoustics: A Three-Dimensional Computational Study vs. Experiments”, SAE International Journal, 2011.
C. K. W. Tam, N. N. Pastouchenko, M. G. Jones, and W. R. Watson, “Experimental validation of numerical simulations for an acoustic liner in grazing flow: Self-noise and added drag”, Journal of Sound and Vibration, p. 333, 2014.
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.
The rotor system in this example is a simple rotor with a uniform cross section throughout its length. It is supported at both ends by bearings and there are three mounted components called disks at different locations of the rotor.
You can model this rotor using the Beam Rotor interface in the COMSOL Multiphysics® software. The inertial properties and offset of the rotor components are modeled with the Disk node. The bearing support is modeled by an equivalent stiffness-based approach via the Journal Bearing node provided in the Beam Rotor interface.
For more information about the geometric properties and model setup, check out the references in the model documentation.
Geometry of the beam rotor example.
Two types of analyses are commonly used to study rotor vibration characteristics: eigenfrequency and time-domain analyses. As mentioned in a previous blog post, critical speeds of the rotor strongly depend on the rotor’s angular speed. Therefore, while performing the eigenfrequency analysis, you need to consider the variation in the rotor speed to get the correct critical speeds. A time-domain analysis is performed when you want to look at the system response under time-varying excitation.
Now, let’s look at what type of information each analysis provides as well as the steps involved to perform these analyses.
Eigenfrequency analysis is used to determine the natural frequencies of a system. In a rotordynamics scenario, this analysis can be used in two different ways.
First, for the operating speed of a system that is not fixed, you can perform an eigenfrequency analysis of the system for the range of operating speeds and choose the one that is furthest from the critical speed of the system and meets other design considerations. If you cannot find a suitable operating speed for the current system, you might need to make certain design modifications in the system to get a stable operating speed that meets all of the requirements.
In the second type of analysis, the operating speed of the system is fixed. In such a case, you need to perform an eigenfrequency analysis at the given operating speed to check that any of the natural frequencies of the system are not close to the operating speed. If any of the natural frequencies fall closer to the operating speed, design modifications are a must.
The design modifications in the rotor system require an understanding of what kind of modifications will produce the desired effect and at what cost. This is where simulating simple systems to understand the effect of design modifications is very helpful. Simulation can provide guidelines for design modifications, thus reducing the number of iterations in the design process.
Consider the first case, in which the operating speed of the system is not fixed, to understand the analysis steps. In this case, you need to perform a parametric eigenfrequency analysis for the angular speed of the rotor. This requires two steps in the Study node: Parametric Sweep and Step 1: Eigenfrequency, shown below on the left. Settings for the Parametric Sweep node for a sweep over a parameter Ow representing the angular speed of the rotor are shown below in the center. This parameter is used as an input in the Rotor Speed section of the Beam Rotor node settings, shown below on the right.
Steps in the Study node (left), settings for the parametric sweep (middle), and rotor speed input (right).
After performing the analysis, you get a whirl plot of the rotor as the default, shown below. The whirl plot shows the whirling orbit and the deformed shape of the rotor for the given rpm and natural frequency combinations.
Whirl plot of the rotor.
The deformed shape of the rotor also gives you an idea of how strongly the natural frequency will depend on the angular speed of the rotor. If the disks move away from the rotation axis without significant tilting, then the split in the frequency in the backward whirl (opposite to the spin) and forward whirl (same direction as the spin) is not significant. Alternatively, if the disks do not move significantly far from the rotation axis and rather have significant tilting, then the split in the frequency of the backward and forward whirl is noticeable.
To understand this concept in depth, you can plot the variation of the natural frequency for different modes against the angular speed of the rotor, which is often called a Campbell diagram. The Campbell plot for the simply supported rotor example is shown below. You can see the strong divergence of the eigenfrequencies with rotor speed for certain modes; whereas for others, particularly the modes with low natural frequencies, the divergence is not significant. If you look at the mode shapes corresponding to these frequencies, they confirm the behavior previously discussed. Critical speeds of the rotor can be obtained from the Campbell plot by looking at the intersection of the natural frequency vs. angular speed curve with ω = Ω curve. These are the speeds near which a rotor should not be operated, unless sufficiently damped.
Campbell plot of the simply supported rotor system.
The damping in the respective modes can be accessed by plotting the logarithmic decrement with the angular speed of the rotor. The logarithmic decrement is defined as
where A(t) is a time-varying response and ω is the complex eigenfrequency of the system. T is the time period given by .
Logarithmic decrement for different bending modes in the simply supported rotor system.
In the plot above, you can see a logarithmic decrement variation for the different bending modes with the angular speed for the simply supported rotor. The notation ‘b’ and ‘f’ is used for the backward and forward whirl modes, respectively. A logarithmic decrement of zero means that the system is undamped, a negative value indicates an unstable system, and a positive value indicates a stable system.
You can also note the pattern change for some of the curves. The reason is that the modal data is arranged in increasing order of the natural frequency. But we know that the rotor’s natural frequencies decrease in the backward whirl modes and increase in the forward whirl modes. Due to this, there is crossover of the natural frequencies between the higher backward whirl and lower forward whirl modes beyond a certain angular speed. This upsets the initial order of the modes, resulting in switching of the patterns across the crossover points.
Eigenfrequency analysis gives the characteristics of the rotor system operating at steady state. However, before and after reaching the steady state, during the run-up and run-down, the angular speed of the rotor varies with time. In certain cases, the operating speed might be above the first few natural frequencies of the rotor. Therefore, during the run-up and run-down, the rotor will cross over the corresponding critical speeds. Also, there could be nonharmonic time-varying external excitation acting on the rotor. In such cases, the rotor response cannot be completely determined by an eigenfrequency or frequency-domain analysis. Rather, you need a time-dependent simulation to study the response of the system.
You can also perform a time-dependent analysis of the rotor at different angular speeds by performing a parametric sweep to see how the angular speed governs the response. An obvious extension of such an analysis is to evaluate the frequency spectrum of the time-dependent response of the rotor for all of the angular speeds and analyze what combinations of the angular speed and frequency result in a high amplitude response. A waterfall plot shows the response amplitude vs. angular speed and frequency and gives the distribution of the modal participation in the response at different speeds. Such an analysis can be set up using the three steps in the study node, as shown below.
Steps for a waterfall plot analysis. The Parametric Sweep study step is used to sweep the angular speed, a Time Dependent study step is used to perform a time-dependent analysis corresponding to each angular speed in the parametric sweep, and a Time to Frequency FFT study step takes the fast Fourier transform of the time-dependent data to convert into the frequency spectrum.
In the eigenfrequency analysis, bearings are modeled using constant stiffness and damping coefficients. However, in reality, these coefficients are strongly dependent on the journal motion. To highlight the effect of the nonlinearity for the time-domain analysis, a plain journal bearing model is used instead of the constant bearing coefficients. A plain journal bearing model is based on the analytical solution of the Reynolds equation for a short bearing approximation. The system in this case is self-excited due to the eccentric mounting modeled as a disk. To simplify the system, only the second disk is considered with small eccentricity in the local y direction.
The waterfall plot of the z-component of the displacement is shown in the figure below. You can observe three peaks clearly in the spectrum. The third peak, which falls along the ω = Ω curve, corresponds to a 1X synchronous whirl. This is in response to the centrifugal force due to eccentricity changing its direction with the rotation of the shaft. Other peaks correspond to the orbiting of the rotor due to the complex rotor bearing interaction. The reason is that the forces from the pressure distribution around the journal in the bearing have a cross-coupling effect with the journal motion. In other words, the motion of the journal in one of the lateral directions induces a component of the force in the lateral direction perpendicular to it. The effect of this phenomenon is a net force acting on the rotor in the direction of the forward whirl. This causes the subsynchronous orbiting of the rotor.
A waterfall plot shows the response amplitude vs. the angular speed and frequency of the rotor.
The orbit of the different locations along the length of the rotor at 30,000 rpm is shown below. The orbit curve changes its color from green to red with time. You can see that after the initial transient phase, the rotor undergoes a forward circular whirl in the steady state. Also, the second bending mode has the highest participation in the response.
Orbit of the rotor at different locations. The plot changes from green to red with time.
The time variation of the z-direction displacement of a point on the rotor at 30,000 rpm is shown below. Apart from the high-frequency variation, there is also a low-frequency component that envelops the response, but gets damped out with time.
Time variation of the z-displacement.
With this tutorial model, we have demonstrated the approach to set up different analyses in a rotor system, as well as how to plot and analyze the simulation results. Ready to give this tutorial a try? Simply click on the button below to access the MPH-file via the Application Gallery or open it via the Application Library in the COMSOL® software.
Probe tubes are attached to microphone cases in order to distance the device from the sound field being measured. When fitting hearing aids, the tube is inserted into the ear canal with the microphone worn on the outside of the ear. This system provides measurements that calibrate and verify the comfort and effectiveness of hearing aids, specifically if the device amplifies signals to the level that the patient needs. In fact, the American Speech-Language-Hearing Association and the American Academy of Audiology say that in-the-ear measurements are the preferred method for verifying hearing aid performance.
A probe tube microphone performing in-the-ear measurements. Image by Cstokesrees — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.
When adding a probe tube to a microphone, we have to consider how these two components interact with one another. For instance, we need to understand how the probe affects the sensitivity of the microphone, thus the measurements that the device delivers. As we show here, multiphysics simulation provides answers.
For this example, we use a time-dependent model that consists of a generic probe tube microphone configuration. It includes:
The probe tube is made of an elastic material with a Young’s modulus of 0.1 and a Poisson’s ratio of 0.4. In the schematic below, L represents the length of the tube, while D_{0} is its outer diameter. The cavity in front of the microphone is a cylinder with a radius of R and height of H. This cavity is connected to a cone that has a bottom radius of R and top radius of D_{0}. An external sound field with the wave vector k hits the probe tube. Note that this sinusoidal wave moves in the positive x direction and has an amplitude of 1 Pa.
The probe tube microphone configuration.
To model this probe tube microphone design, we use the Pipe Acoustics, Transient interface. In our analysis, the probe tube is treated as a 1D structure — a valid assumption as long as we neglect the interaction between this component and the incoming sound field. We also assume that no significant thermal and viscous boundary losses occur inside the tube. This holds for the current configuration in which the incident field is a monochromatic wave. Since the diaphragm is not a fully rigid structure, we assume a resistive loss that is consistent with the impedance of common condenser microphones. This gives us a fully coupled acoustics simulation, as the probe tube is connected to two separate 3D pressure acoustics domains.
When analyzing a probe tube microphone, an important parameter to consider is the relationship between the pressure at the tip of the probe and the pressure at the diaphragm. This is a necessary step for calibrating the measurement system. The plot below on the left shows that the solution, following an initial transient, becomes periodic after around 4 ms. The system then experiences a gain of about 1.4 and a phase shift takes place. These two factors are dependent on the frequency of the applied signal, which is a pure harmonic tone of 500 Hz. The plot on the right depicts the pressure distribution in the xz-plane at the end of the time interval.
Left: Diaphragm pressure vs. probe tip pressure. Right: Pressure distribution in the xz-plane at 8 ms.
These results show the potential of using the COMSOL Multiphysics® software to analyze probe tube microphone designs. With a better understanding of how the probe tube and microphone interact, it is possible to further improve the design of these systems for fitting hearing aids and for other applications.
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:
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.
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:
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.
We model the latent heat effect due to vapor condensation by adding the following flux in the heat transfer equation:
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:
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:
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 News 2017 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.