Humans have used drying as a method for preserving food since ancient times. Since then, the drying process has expanded from openair drying or sun drying to other drying techniques, such as solar drying, freeze drying, and vacuum drying. Drying is also a key process in many other application areas, from the pharmaceutical industry to plastics.
Today, we’ll focus on the chemical process of vacuum drying, which is particularly useful when drying heatsensitive materials such as food and pharmaceutical drugs. Vacuum dryers, commonly called vacuum ovens in the pharmaceutical industry, also offer other benefits. Because they require lower temperatures to operate, vacuum dryers use less energy and therefore, reduce costs. They also recover solvents and avoid oxidation.
A rotary vacuum dryer. Image by Matylda Sęk — Own Work. Licensed under CC BYSA 3.0, via Wikimedia Commons.
Vacuum dryers remove water and organic solvents from a wet powder. The dryer operates by reducing the pressure around a liquid in a vacuum, thereby decreasing the liquid’s boiling point and increasing the evaporation rate. As a result, the liquid dries at a quicker rate — another major benefit of this process.
For vacuum drying to be effective, we need to decrease drying times without harming the products, which means that we need to maintain a strict control of the operating conditions. To balance these goals and to understand how operating conditions influence the product, you can use the multiphysics modeling capabilities of COMSOL Multiphysics.
Today, we’ll analyze the vacuum drying process of a Nutsche filterdryer model. The dryer works by heating a wet cake from the bottom and the side walls of a container and by decreasing the pressure in the gas phase on the top of the cake. This example is based on a paper published by Murru et al. (Ref. 1 in the model documentation).
Let’s start by taking a closer look at our model. The vacuum dryer is comprised of a cylindrical drum filled with wet cake, which consists of three different phases: solid powder particulates, a liquid solvent, and a gas. As such, the cake’s material properties need to include the properties of all three individual phases, which vary depending on the proportion of each phase in the cake. The portion of each phase is determined by the volume fraction, which is one of our modeled variables.
The cake is modeled as a rectangular geometry with a radius of 40 cm and height of 10 cm in a 2D axisymmetric component. At the top, our model is exposed to a lowpressure head space. Meanwhile, heat flux boundary conditions at the filter dryer’s side and bottom boundaries account for a 60°C heating fluid.
The vacuum drying process in an axisymmetric Nutsche filter dryer.
Moving on, our tutorial combines evaporation and heat transfer modeling in order to study the cake’s liquid phase profiles and temperature. We calculate the cake’s solvent volume fraction with the Coefficient Form PDE interface and simulate heat transfer with the Heat Transfer in Solids interface. To solve the moisture transport in porous media, we use a predefined multiphysics interface in the Heat Transfer Module. We also include solvent evaporation by using both a heatsink and masssink term and approximate the solvent transport as a diffusion process.
Our model makes the following assumptions:
In these situations, we can use a step function to smoothly ramp both the evaporation rate and diffusion coefficient down to zero.
We see that our simulation results are as predicted. Let’s start by examining our analysis of the cake after 30 hours have passed. As seen below, the cake’s temperature is close to that of the heating fluid (60°C) at both the side and bottom boundaries, and the liquid phase’s volume fraction is lowest near these heated boundaries and highest at the cake’s center. Additionally, the apparent moisture diffusivity is highest at the cake’s center and almost zero in places where the liquid phase has evaporated. Considering our model’s assumptions, these results are all expected.
The cake’s temperature (left), volume fraction of the liquid phase (middle), and apparent moisture diffusivity (right) after 30 hours.
Switching gears, let’s expand our timescale to look at the evaporation rate after 10, 20, and 30 hours. This study also yields expected results, since it shows evaporation beginning at the heated walls and decreasing when the amount of solvent at these boundaries lessens. During this process, the evaporation front shifts toward the cake’s center.
The evaporation rate after 10 (left), 20 (middle), and 30 (right) hours.
The quantitative results generated by our simulation study are in good agreement with previous research, confirming their validity. As such, we can use this model to accurately predict how dry a product is as a function of time. Using this information, we can minimize the amount of time that a product is exposed to elevated temperatures. Additionally, we can change the dryer’s size if we want to reduce the drying time when working with heatsensitive products. Through multiphysics simulation, we can design more efficient and effective vacuum dryers for use in a variety of industries.
As a refresher, let’s begin by reviewing some of the key concepts behind modeling gears in COMSOL Multiphysics. A gear is defined in a Gear node as a rigid body with six degrees of freedom in the form of translations and rotations at the center of rotation. It is used in a Gear Pair node in the model tree in order to connect with another gear. Here, you can specify a finite stiffness for the gear mesh or gear tooth, either for individual gears or for the pair. A mathematical formulation is used to describe the connection between two gears, without any need for a defined, realistic gear geometry to detect the contact between the two gears. Therefore, you can represent a gear with either a realistic gear geometry or any similar geometry of a disc.
It is possible to compute the inertial properties of a gear from the geometry using its calculated mass density, or you can directly enter the properties in the form of mass and moment of inertia in the node’s edit fields. You can also apply external forces and moments on the gear as well as constrain certain degrees of freedom of a gear. For instance, when modeling torsional vibrations, all of the degrees of freedom except the axial rotation can be constrained.
COMSOL Multiphysics offers a number of standard gear types, each with its own merit and applications. As mentioned above, the gear is an abstract object, but if you want to add a realistic geometry for visualization, you can access the Part Libraries, where you can find various types of gears and racks.
In the following images, you can see the various types of gears and racks available and the geometrical parameters needed for their mathematical descriptions.
A Spur Gear (left) and Helical Gear (right) with their external gear mesh.
A Spur Gear (left) and Helical Gear (right) with their internal gear mesh.
A Bevel Gear (left) and Worm Gear (right).
A Spur Rack (left) and Helical Rack (right).
The inputs required to model each gear type are shown in the respective figures. They are as follows:
After selecting the appropriate gear type, you can then define the parameters controlling the size and shape of the gear teeth. As an example, these parameters are required to define a helical gear:
A screenshot showing the settings window for a helical gear. Various inputs required to model a helical gear, including gear properties, gear axis, center of rotation, and density are shown.
The next step is to define the position and orientation of the gear. The gear position is defined in terms of the center of rotation. This is the point at which the degrees of freedom are created and the rotation is interpreted. The forces and moments acting on the gear due to meshing with other gears are also interpreted about this point. By default, the center of rotation is set to the center of mass of the gear, but there are other ways to define it explicitly as well.
The gear orientation is specified in terms of the gear axis, which is the axis of rotation passing through the center of rotation. The gear axis is used when creating the gear local coordinate system. Also interpreted about this axis is the gear rotation, a degree of freedom in the Gear Pair node.
You can mount gears in one of two ways: on a flexible or a rigid shaft. These devices can be mounted either rigidly or with a finite stiffness using a fixed joint. Joints are the features used to connect two components by allowing certain relative motion between them.
When there is no clearance between the gear and the shaft in the geometry, the objects can be either in an assembly state or a union state. For a flexible shaft, gears are by default rigidly mounted on the shaft if both the gear and shaft are in a union state.
It is not necessary to model a shaft in order to mount gears, as the devices can be mounted directly to the ‘ground’ either rigidly or with a finite stiffness using a hinge joint. The prescribed displacement/rotation subnode of a gear can also be used for this purpose.
Note that it is also possible to support shafts on:
This can be done using hinge joints, which can be rigid or have a finite stiffness.
Figure showing gears with an actual geometry as well as those modeled through equivalent discs. Different mounting methods for gears and shafts are also depicted.
In order to connect the different types of gears that you have defined in your model, you can use a Gear Pair node. This node can connect spur, helical, and bevel gears. You can also use Worm and Wheel as well as Rack and Pinion nodes for their specific cases. These nodes connect two gears in such a way that there is no relative motion along the line of action at the contact point. The remaining displacements and rotations of the two gears are independent of each other.
Each Gear Pair node adds two degrees of freedom:
The following constraints are added by the Gear Pair node in order to connect two gears:
For a line contact model, one more constraint is added to restrict the relative rotation about a line joining the two gear centers. If friction is included, frictional forces are obtained using the contact force, which is computed as the reaction force of the contact point constraint. These frictional forces are then applied on both gears in a plane perpendicular to the line of action.
In a Gear Pair node, you can select any two gears defined in the model. But in order to achieve proper tooth meshing, a set of gears must fulfill the following compatibility criteria:
All these checks are automatically performed and an error message is issued during equation compilation if the two selected gears are not compatible.
Examples of incompatible gear mesh. In the figure on the left, the gears have different modules. In the figure on the right, the gears have different pressure angles.
A coordinate system for each gear is defined using the gear axis and center of rotation of both gears. The first axis of the coordinate system triad is the gear axis itself. The second axis is the direction pointing from the center of rotation to the contact point. The third axis is normal to the plane containing the first two axes. This coordinate system is attached to the gear and varies with the changes in gear orientation. Note, however, that it does not rotate with the gear rotation about its own axis.
A schematic showing coordinate systems and other parameters for both gears connected by a gear pair.
These quantities are illustrated in the above figure of a gear pair:
The gear tooth coordinate system is defined for both gears by rotating the gear coordinate system with the tooth angle matrix. This matrix is constructed using the helix angle and the cone angle.
The line of action, meanwhile, is defined as the normal direction of the gear tooth surface at the contact point on the pitch circle. This is the direction along which the forces are transferred from one gear to another. It is defined by rotating the third axis of the gear tooth coordinate system (gear tangent) about the first axis of the gear tooth coordinate system with the pressure angle (α). Based on the direction of the driver gear, the gear tangent can be rotated either clockwise or counterclockwise.
Two figures depicting the line of action and the direction of rotation of the driver gear. The line of action is defined due to the fact that the driver gear and tangent rotate in the clockwise direction (left) and counterclockwise direction (right).
The contact between the two gears is modeled through analytically founded equations. These are independent of the finite element mesh and thus much faster and more robust compared to meshbased contact methods. To compute contact forces and moments, you can choose one of two methods:
The point of contact on each gear is defined via the center of rotation, displacement vector at the center of rotation, contact point offset from the gear center, pitch radius, and cone angle. Based on the orientation of both gears, different gear pairs can be classified into one of two configurations:
For a parallel or intersecting configuration, the contact point offset from the pinion center is the input and the contact point offset from the wheel center is automatically computed. The contact model can be selected as either:
For a configuration that is neither parallel nor intersecting, the contact point offset from the pinion, as well as the wheel center, is automatically computed. The reason for this is that there is always a point contact and the contact point can be uniquely determined.
From left to right: Thin gears (point contact model), thick gears (line contact model), and thick gears with an axial offset.
Now that we’ve explored gears in further detail and how to connect them, let’s look at various examples of gear pairs classified based on their configurations. You can use many gear pairs together in order to model complex parallel and planetary gear trains.
Some examples of the parallel axis configuration are as follows:
Bevel gears, meanwhile, offer an example of an intersecting axis configuration.
Set of spur gears and parallel helical gears with an external gear meshƒ.
Set of spur gears, one with an internal gear mesh and the other with an external gear mesh, as well as a set of bevel gears.
Some examples of a crossed (neither parallel nor intersecting) axis configuration are as follows:
Set of crossed helical gears with an external gear mesh and the worm and wheel.
Rack and pinion with a straight gear mesh.
When it comes to modeling gears, there are many important elements to consider to optimize your simulation results. As we’ve demonstrated here today, the new features and functionality for gear modeling in COMSOL Multiphysics allow you to address such elements, providing you with more useful insight into how to improve your gear design.
In the next blog post in our Gear Modeling series, we’ll discuss how you can use advanced features on your gear pairs (i.e., gear mesh elasticity, backlash, transmission error, and friction) in order to perform simulations requiring greater fidelity. We’ll also show you how these parameters affect the dynamics of your system. Stay tuned!
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Arteries are blood vessels that carry oxygenated blood from the heart throughout the body. They are layered structures consisting of intima, with media and adventitia (also called externa) on the outside layers. The media and adventitia are the layers that are primarily responsible for mechanical behavior in healthy arteries.
Both of these layers are made up of collagenous soft tissues that exhibit noticeable strain stiffening behavior. Each layer has anisotropic properties because of families of collagen fibers. This means that these fiberreinforced structures allow the blood vessels to experience large strains.
An artery wall’s structure. Image by BruceBlaus — Own work. Licensed by CC BY 3.0, via Wikimedia Commons.
A reliable constitutive model for arterial wall mechanics is essential to investigate changes in the arterial system due to age and disease, and for designing prostheses, among other uses (Ref. 3). The HolzapfelGasserOgden (HGO) constitutive model (Ref. 2) captures the anisotropic mechanical response described above, which has also been observed in lab experiments on excised arteries. Typical experiments in the lab measure the response of arterial sections subject to combined axial stretch and internal blood pressure, and numerical examples try to match this data to better understand its mechanics.
As Bower points out in Applied Mechanics of Solids: “Finite strain viscoelasticity is not as well developed as finite strain plasticity, and a number of different formulations exist.” In COMSOL Multiphysics version 5.2a, we have implemented the Holzapfel model for largestrain viscoelasticity (Ref. 1, Ref. 3), which is well suited to combine with any of the predefined hyperelastic material models available in the COMSOL software.
The author proposes a generalized Maxwell model based on the splitting of the strain energy density into volumetric and isochoric contributions
Here, stands for the right CauchyGreen deformation tensor and stands for its isochoric counterpart. The free energy associated to the nonequilibrium state, , is a function of the isochoric right CauchyGreen tensor and internal strainlike variables (Ref. 1, Ref. 3).
The strain energy in the pureelastic branch is normally denoted with the superscript to denote the longterm equilibrium (as ).
Then, Holzapfel derives the expression for the second PiolaKirchoff stress in the hyperelastic and viscoelastic branches
and also defines the auxiliary stress tensors from thermodynamic considerations
The total second PiolaKirchoff stress (hyperelastic and viscoelastic branches) is then given by
A schematic representation of the second PiolaKirchoff stress for a generalized Maxwell model in largestrain viscoelasticity.
The evolution of the stresses in the viscoelastic branches is given by solving the rate equations
where is the relaxation time of the branch and is the corresponding second PiolaKirchoff stress tensor.
Holzapfel also assumes that there should be an isochoric strain energy density associated to the spring on each branch, , so
The main assumption in the Holzapfel formalism is that the isochoric strain energy per branch depends on the isochoric strain energy of the main hyperelastic branch
Here, the dimensionless coefficients are called strain energy factors.
The second PiolaKirchoff stress per branch then becomes
and the rate to solve becomes
The generalized Maxwell viscoelastic model in version 5.2a of COMSOL Multiphysics is available for all of the hyperelastic materials, and it also contains the same options for modeling thermal effects as implemented for linear viscoelasticity.
With the User defined option, you can neglect thermal effects, use the predefined WilliamLandelFerry or Arrhenius shift functions, or define your own shift function.
Let’s take a look at how largestrain viscoelasticity can be applied to biomechanics modeling.
To model the behavior of arterial walls after sudden changes in axial stress, we need to use a hyperviscoelastic material model, which goes beyond the HGO material model.
Take a look at the Arterial Wall Mechanics tutorial model for more details on how to set up an anisotropic hyperelastic material.
To start, let’s add viscoelastic behavior to this particular material model. As described in Ref. 3, a generalized Maxwell model of five branches added to the HGO model is suitable to model relaxation times ranging from 1 ms to 10 seconds. This model is suitable for quantitatively representing the viscoelastic response of circumferential segments of arteries (Ref. 3). To do so, we rightclick on the Hyperelastic material node and add a Viscoelasticity node (we can also combine this with thermal expansion or other effects).
By default, we get a generalized Maxwell model with one branch. However, we can also use the standard linear solid (SLS) model or the KelvinVoigt viscoelastic model.
Then, we add the five branches with the corresponding energy factors and relaxation times as described in (Ref. 3). We are able to retrieve (and also save) such parameters from a text file.
After refurbishing the HGO hyperelastic material with five viscoelastic branches, we can simulate what happens in an artery section when subject to a constant axial strain for four minutes.
The stresses in the viscoelastic branches. Note the different relaxation times for the stresses in the five branches of the generalized Maxwell viscoelastic material.
The axial stress relaxes to the steady state for times longer than the highest relaxation time. The ability to model largestrain viscoelasticity in COMSOL Multiphysics enables us to easily investigate and understand different biomedical materials and applications.
Let’s say, for example, that you’re almost done drawing the geometry of a muffler model. It’s nothing too fancy, just a couple of cylinders representing the pipe and an extruded oval forming the main chamber. Then it strikes you: Some of the sections in the pipes and the baffles separating the chambers need to be perforated — and not just by placing a hole here or there. In this case, you could use the Array tool to draw an arbitrarily large collection of holes, but meshing and solving the resulting geometry would take too much time and memory.
Detail of muffler geometry, including perforated sections consisting of a few thousand holes.
Fear not — there are better solutions. The simplest and most convenient approach is to draw the contours of the perforated regions and apply an Interior Perforated Plate condition. We then supply properties, such as the hole diameter and plate thickness, and get a partially transparent surface that represents the perforate.
In many cases, the accuracy of this method meets our simulation needs. However, the accuracy can be compromised if we push the limits of validity for the underlying engineering relation. For example, holes that are very small, lie too close to each other, or have a noncircular shape produce less reliable results.
The same muffler geometry with the perforations now replaced by boundaries for applying a Perforated Plate or Interior Impedance condition.
A more general alternative to the Interior Perforated Plate condition is the Interior Impedance condition. This allows us to specify a complexvalued transfer impedance, which represents the ratio between the pressure drop across the perforate and the normal particle velocity through it. The Interior Perforated Plate condition is a special predefined version of the Interior Impedance condition. The value of the impedance can be based on imported measurement data or an analytical expression. If we lack trustworthy measurements of the transfer impedance or a good analytical expression, the impedance can be based on numerical results. This method, as described below, makes it simple to model the impedance numerically.
The figure below shows the considered perforate in such a model, described in our Transfer Impedance of a Perforate tutorial. The principle behind this model is rather simple: We send a plane wave towards the hole and then calculate the transfer impedance from the resulting pressure difference across it and the average velocity through it.
Representation of the perforate with the model domain colored according to the local acoustic velocity field.
The example model takes advantage of available symmetries, occupying only a quarter of the hole itself as well as half of the distance between holes. The holes in this case have a diameter of 1 mm, which is small enough that we must consider the thermoviscous boundary layers and losses. To account for these factors, we can use the Thermoviscous Acoustics interface.
To send the plane wave, we utilize the Background Acoustic Fields node, a staple of the Pressure Acoustics interface that has recently been added to the Thermoviscous Acoustics interface. Its Plane wave option sends a pressure, velocity, and temperature distribution corresponding to a plane wave with viscous and thermal attenuation. We cap the model by adding perfectly matched layers above and below the model domain.
For even more on transfer impedances in perforated mufflers, we recommend the Thermoviscous Acoustic Impedance Lumping model. In addition to computing the transfer impedance of a perforate, this example demonstrates how to use the result in a model of an entire muffler, thereby summing up and driving home the message of this blog post.
If you’ve ever flown on a commercial airplane, it’s likely that your flight was powered by a turbofan engine. Turbofan engines function by capturing air and sending part of this into a compressor. The compressed air then enters a combustion chamber, where it is ignited with fuel, and then the released products propel the plane forward.
Left: A turbofan engine schematic. Image by K. Aainsqatsi — Own Work. Licensed under CC BYSA 3.0, via Wikimedia Commons. Right: A realworld turbofan engine. Image by Sanjay Acharya — Own Work. Licensed under CC BYSA 3.0 via Wikimedia Commons.
In recent years, the design of turbofan engines have vastly improved, with a particular emphasis on noise reduction. To understand why, consider once again being a passenger on a flight — it can be rather unpleasant to listen to a loud engine. And for those people who live near airports, loud noise from planes as they land and take off can disturb sleep patterns. Reducing the noise generated by airplanes and their engines has therefore been a key point of focus in the aviation industry.
Reducing the excess fan noise that comes from turbofan aeroengines offers one potential solution to this issue. In the COMSOL Multiphysics® software, you can analyze and optimize the radiated noise from a turbofan engine to meet such goals. To learn more, let’s take a look at our simplified tutorial model of a jet pipe.
To analyze a turbofan aeroengine, we can focus on specific elements of its design. In this case, we’ll investigate the radiation of fan noise generated by a turbofan aeroengine’s annular duct. Let’s start by looking at our axisymmetric model geometry, which has a symmetry axis at the engine’s centerline. The model geometry mimics the outlet nozzle of the jet engine (see the schematic above). The gray area in the following schematic represents the interior of the engine in the nozzle. The model obviously uses a very simplified geometry and focuses on the physical principles and model setup.
Turbofan motor geometry. The gray zone indicates the internal machinery of the engine. Air flows through the jet (M_{1}) as well as around the jet (M_{0}).
In this model, air flows inside and outside of the duct as uniform mean flows with a Mach number of M_{1} = 0.45 inside and M_{0} = 0.25 outside, respectively. This corresponds to the red and pink regions in the initial schematic of the turbofan engine. Since the air surrounding the engine moves at a slower speed than inside the jet, a vortex sheet (indicated by the dashed lines in the image above) results in the jet stream, which separates the air flows along the extension of the duct’s wall. Using our model, we can calculate the nearfield flow on both sides of the vortex field.
When solving our jet pipe model, we used the Linearized Potential Flow, Frequency Domain interface in the Acoustics Module to describe acoustic waves within a moving fluid. It’s important to note, however, that the field equations are valid only when working with an irrotational velocity field. Since this is not the case across a vortex sheet, the sheet has a discontinuous velocity potential. To model such discontinuity, we applied the builtin Vortex Sheet boundary condition on the interior boundaries. As for the acoustic field within the duct, we described this element using the sum of the eigenmodes propagating within the duct and then radiating into free space. This is a common approach when setting up sources in this type of simulation.
For our study, we utilized a boundary mode analysis to find the inlet sources. The first step was to investigate circumferential wave numbers (m = 4, 17, and 24) and generate various eigenmodes that correspond to different radial mode numbers. The second step was to use three eigenmodes as incident waves inside the duct: (m,n) = (4, 0), (17, 1), and (24, 1). The results indicate that the largest eigenvalue for a given m corresponds to the radial mode n = 0. The smallest eigenvalue, meanwhile, corresponds to n = 1.
Plot of the eigenmodes featuring circumferential mode shapes m = 4, 17, and 24 and radial modes n = 0 and 1.
As part of our analysis, we also investigated the source velocity potential. As depicted in the plot below, we used a revolved geometry that included the circumferential wave number contribution to see its spatial shape.
Model showing a boundary mode of (m, n) = (4, 0).
To gain further confidence from the results of our analyses, we compared our simulation findings to those results presented in the paper “Theoretical Model for Sound Radiations from Annual Jet pipes: Far and Nearfield Solution” (see Ref. 1 in the model documentation). The plots below, for instance, showcase the nearfield pressure from different source eigenmodes in our simulation study. All of the results are solved for a Mach number of M_{1} = 0.45 inside the pipe and M_{0} = 0.25 outside of the pipe.
From left to right: The nearfield solution for (m, n) = (4, 0), (17, 1), and (24, 1).
Further, we analyzed the nearfield sound pressure level and the revolved geometry’s nearfield pressure. The results from these studies are highlighted in the plots below, respectively.
Left: Nearfield sound pressure level for (m, n) = (24, 1). Right: Nearfield pressure shown in the revolved geometry for (m, n) = (4, 0).
By comparing our findings to the established literature highlighted above, we were able to further confirm the validity of our results. Such accuracy speaks to benefits of using COMSOL Multiphysics to help reduce noise pollution in turbofan engine designs and thus facilitate important advancements within the aviation industry.
Consider a jet engine in a plane flying overhead or a wind turbine with blades turning in the wind. Both of these structures include rotating machinery, the study of which is referred to as rotordynamics. In more specific terms, rotordynamics is the study of rotor vibrations and their dependence on rotor speed. Rotordynamic systems generally consist of a mechanical assembly that has a supporting structure (a stator) and one or more rotating structures (rotors). The relative motion of the stator and rotor, as noted in a previous blog post, is key to the operation of a rotating machine.
A jet engine is just one application that relates to rotordynamics. Image by Todd Huffman — Own Work. Licensed under CC BY 2.0, via Flickr Creative Commons.
Rotordynamics is used to improve the functionality, safety, and efficiency of rotating machines and to avoid failure and design problems. For instance, when a machine’s rotational velocity increases, it can cause the frequency of vibration to pass through its critical speed. This may result in excessive vibrations that generate wear and tear, machine failure, and even human injury. Instability or rotational imbalance due to faults or suboptimal designs are other important issues to consider as well.
When developing rotating machinery, engineers need to understand the potential problems highlighted above, among others, and find ways to prevent them. Rotordynamics is a powerful tool for addressing such challenges and thus optimizing the safety and performance of the structures.
The relevancy of rotordynamics continues to grow as machine speeds become quicker and the potential for issues involving critical speeds and rotor stability increases. Further, a push toward developing smaller devices and advancing performance means that these designs need to constantly be improved. So what types of modern applications can benefit from such analyses? Here are some examples…
To increase the efficiency of light rail transit, one group of researchers is looking to utilize flywheel technology. Such technology uses a rotor that increases its rotational speed when electricity is added. Then, when the energy is needed again, it can convert the rotational speed back into electrical energy. In terms of trains, this means that the energy generated when trains slow down and stop can be stored and used as electrical energy in the future.
An example of light rail transit. Image by haljackey — Own Work. Licensed under CC BY 2.0, via Flickr Creative Commons.
Nanoscale machines are also starting to utilize moving parts and rotor mechanisms. For instance, scientists have recently designed the world’s most complex rotary structure that uses DNA origami techniques. The technology uses a rotor mechanism that is formed by interlocking 3D DNA components. As the technology advances, it could be used for a range of purposes, including propelling drugdelivery vehicles.
Rotordynamics, as we’ve highlighted here, can help advance the design of various structures. Bringing these analyses into the simulation environment further enhances such studies by generating results quickly and removing the need for building physical prototypes at every step of the design workflow.
Let’s consider one very simple example. In rotating machinery, high rotational speeds can produce large centrifugal forces. This can, in turn, cause two counteracting effects: stress stiffening and spin softening. The combination of these effects alters the natural frequencies of a blade connected to a rotating shaft.
Left: Geometry of a rotating blade. Right: A Campbell plot comparing frequency and rotational angular velocity.
By studying the impact of these forces via simulation, you can gain further insight into the design of a rotating machine, modifying its configuration as needed to achieve optimal performance. As simulation tools continue to advance, more complex and intricate studies can be performed on rotating machinery, optimizing their performance in current applications and opening up the door to new uses.
In order to transport food and other perishables, refrigerated trucks are designed to maintain a cold temperature. If these products are not kept properly cooled, they can become heated above an acceptable temperature and thus harmed while intransit. This is especially true in cases where products are exposed to heat for a prolonged period of time.
A refrigerated truck.
While it may seem straightforward to keep a truck properly cooled, there are a few important elements to consider. For one, the cooling system’s design must be optimized in order to preserve the correct temperature. Additionally, the truck’s walls need to be sufficiently insulated to help maintain this desired temperature, which requires choosing the right materials for the job.
Now consider when goods are loaded or unloaded from the truck. The open and closeddoor cycles that refrigerated trucks undergo makes addressing the above elements even more challenging. It is, however, important to address these periods in order to obtain realistic results for the vehicle’s cooling efficiency.
Let’s see how SIMTEC, a COMSOL Certified Consultant, helped Air Liquide, a leader in technologies, gases, and services for industries and health, model this process.
When it comes to choosing the best thermal insulation materials for an efficient aircooling system, it is important to have a thorough understanding of the aerothermal configuration inside a truck’s refrigerated box. As we’ll demonstrate here, this can be achieved by coupling CFD and heat transfer in COMSOL Multiphysics.
For their simulation studies, the team at SIMTEC created an aerothermal model to simulate heat transfer during the normal operation of a refrigerated truck. Their goal: Predict the temperature and air velocity distribution inside the truck and find out what happens upon closing and opening the rear door.
We can begin by taking a closer look at their model geometry. The engineers modeled the refrigerated area of the truck (the refrigerated box) as a parallelepipedic box with a cooling system. The cooling system is comprised of two circular apertures used for air extraction and a rectangular aperture used to blow refrigerated air into the main box. Since the model geometry features a yz symmetry, it is possible to fully describe the physical phenomena occurring in the entire refrigerated compartment by using only a model of half of the box.
Geometry of the refrigerated truck. The entire refrigerated box geometry, the halfbox geometry, and a closeup view of the ventilation and cooling system are shown. Image by Alexandre Oury, Patrick Namy, and Mohammed YoubiIdrissi and taken from their COMSOL Conference 2015 Grenoble presentation.
For their analyses, the team investigated two stages of a refrigerated truck’s operating cycle, using two different computational methods to predict temperature and air flow distribution. In the first stage, which lasts for about three hours, the truck’s rear door is closed and the cooling system is turned on. Both the fans and refrigerating unit operate in order to cool the refrigerated box. The air from the refrigerating system is initially at a temperature of 27°C and later heats up when coming into contact with the warmer box walls. During this cooling period, the engineers partially decoupled their simulations to minimize the system’s degrees of freedom.
Cool air moving through a refrigerated truck with its rear door closed. Image by Alexandre Oury, Patrick Namy, and Mohammed YoubiIdrissi and taken from their COMSOL Conference 2015 Grenoble presentation.
In the second stage, which lasts around ten minutes, the truck’s rear door is opened after the cooling period, and both the refrigerating unit and ventilation system are switched off. Unlike the first stage, this step fully couples CFD and heat transfer to solve the laminar CFD and heat transfer equations. The team then used these equations to analyze the airflow into the box as well as the temperature change during this time period.
To answer this question, let’s begin by looking at the simulation findings from the first stage, when the truck’s rear door is closed. The figure below illustrates an air flow streamline of the local velocity after 10,000 seconds (approximately 2 hours and 45 minutes). At this point in time, the air reaches its maximum velocity at the roof of the box facing the inlet and along the door wall. The velocity decreases rapidly as the air flows through the rest of the box.
Streamlines showing the air’s local velocity after 10,000 seconds in the closeddoor stage. Image by Alexandre Oury, Patrick Namy, and Mohammed YoubiIdrissi and taken from their COMSOL Conference 2015 Grenoble paper.
The temperature field has a very similar distribution to the air velocity, with the coldest areas corresponding to highvelocity regions and vice versa. The plot below indicates that the warmest region is the recirculation zone located at the bottom of the box, where the air is greater than 0°C.
Streamlines showing the air flow’s local temperature after 10,000 seconds in the closeddoor stage. Image by Alexandre Oury, Patrick Namy, and Mohammed YoubiIdrissi and taken from their COMSOL Conference 2015 Grenoble paper.
The engineers also calculated the heat losses within the refrigerated box. By doing so, they found that air cools down inside the truck and global heat losses increase over time. Most of the heat is lost through the lateral and rear walls, both of which feature similar loss profiles. Because the floor is composed of the thickest materials and is the most insulated surface, a very limited amount of energy is lost through its surface.
Averaged heat loss for each of the five walls during the closeddoor cooling period. Image by Alexandre Oury, Patrick Namy, and Mohammed YoubiIdrissi and taken from their COMSOL Conference 2015 Grenoble paper.
Shifting gears to the second stage… With the cooling and ventilation system both off, the engineers stressed that the only driving force for air is the natural convection caused by the difference between outer and inner air temperatures. Since the temperature within the box is much cooler, warmer air flows into the box.
As illustrated in the following simulations, hot air rapidly enters the box at first, but after 50 seconds, the average air velocity drops below 10 cm/s. At 500 seconds, the average air velocity is as low as 2 cm/s, which may be due to the fact the temperature difference between the box and the outside environment is greatly reduced.
The average air velocity in the truck box at 2, 10, 50, and 500 seconds after opening the rear door. Images by Alexandre Oury, Patrick Namy, and Mohammed YoubiIdrissi and taken from their COMSOL Conference 2015 Grenoble paper.
As for the temperature, about 10 seconds after the truck’s rear door opens, the temperature for most of the box matches the outside temperature (around 25°C). One exception is the area around the walls, where a thermal inertia helps the surrounding air stay cool.
The temperature in the truck box at 2, 10, 50, and 500 seconds after opening the rear door. Images by Alexandre Oury, Patrick Namy, and Mohammed YoubiIdrissi and taken from their COMSOL Conference 2015 Grenoble paper.
The team later compared their thermal simulation results to physical experiments where the rear door of the truck was opened and closed multiple times. They found that the model predictions were in reasonable agreement with the experimental temperatures. The simulation results, however, do show oscillations occurring during the opendoor periods that were not observed in the physical experiments. An explanation suggested by the engineers for this is that the temperature was calculated in a different location in the model and physical sensor. Another possible cause is the sensor’s intrinsic inertia, which may have a small leveling effect on the temperature. The model itself shows an instantaneous temperature of air.
A plot comparing simulation results with experimental data. Image by Alexandre Oury, Patrick Namy, and Mohammed YoubiIdrissi and taken from their COMSOL Conference 2015 Grenoble presentation.
With COMSOL Multiphysics, the team at SIMTEC found it easy to couple turbulent CFD and heat transfer in order to perform aerothermal simulations of a refrigerated truck model. These findings have the potential to serve as a powerful tool for designing the next generation of refrigerated trucks, identifying the optimal materials for their walls and indicating ways to enhance the power specifications and locations of their cooling systems.
To learn more about our COMSOL Certified Consultant SIMTEC and their services, visit their website.
Solid objects change their size, shape, and orientation in response to external loads. In classical linear elasticity, this deformation is ignored in the problem formulation. As such, the equilibrium equations are formulated on the undeformed configuration. In many engineering problems, the deformations are so small that the deformed configurations are not appreciably different from the undeformed configuration. Ignoring the changing geometry therefore makes practical sense as this yields a linear problem that is easier to solve.
On the other hand, for problems like metal forming, where the deformation is large, the equilibrium equations have to include the effect of changing geometry. Updating the equilibrium equations to include the effect of changing geometry introduces a nonlinearity known as geometric nonlinearity.
Model geometry for a sheet metal forming process, where deformations can be rather large.
When geometric nonlinearity is included in structural analysis, the COMSOL Multiphysics® software automatically makes a distinction between Material and Spatial frames. The material frame corresponds to the undeformed configuration, while the spatial frame corresponds to the deformed configuration. The software allows us to make a new geometry out of the deformed configuration; what we refer to as remeshing a deformed configuration. We can use this geometry as part of a new geometry sequence. Drill a hole in it, subtract it out of a bounding object, or simply add other geometric objects. Finally, solve a new physics problem on the composite domain. The new physics can be applied to the same COMSOL Multiphysics model in a different component, or in a different model. This is the first point that we will address.
If geometric nonlinearity is not included in structural analysis, the software does not distinguish between the material and spatial frames. Does that mean that if you do not want to include the effects of geometric nonlinearity in the equilibrium equations, you can not remesh a deformed configuration? The answer is no. You can split the two frames and force linear strains in the equilibrium equations. This is the second item that we will address.
For threedimensional problems, there is an additional option. Surface plots can be exported as STL files. These files can be imported and used for solid modeling. In this process, we do not need to split the material and spatial frames. This is the third and last item we will discuss in today’s blog post.
Please note that remeshing a deformed configuration means simply obtaining the deformed shape computed in structural analysis. When we use this deformed geometry for a later analysis, we are not considering residual stresses. If the second analysis is another structural analysis, keep in mind that the remeshed configuration is being used as a stressfree configuration for subsequent studies.
To consider effects of finite deformation in structural analysis, we have to select the Include geometric nonlinearity check box in the settings window of the study step. In some cases, COMSOL Multiphysics automatically enables geometric nonlinearity, such as when you include hyperelastic or other nonlinearly elastic materials, largestrain plastic/viscoelastic materials, or add any contact boundary conditions.
After we complete our structural analysis, we use the Remesh Deformed Configuration command to get the deformed shape. This is done in the meshing section of the Model Builder. Finally, the deformed mesh can be exported and imported back as a geometry object.
We demonstrate the above steps in the following sections.
Let’s consider the problem of squeezing a circular pipe between two flat stiff indenters. Because of the large deformation involved, geometric nonlinearity is included in the structural analysis, as shown in the screenshot below. Because of symmetries, we consider only a quarter of the geometry.
The original geometry (outline) and the deformed geometry.
The next step is to remesh the deformed configuration. This can be done by rightclicking on the data set (Study 1/Solution 1 in this example) and choosing Remesh Deformed Configuration. Alternatively, we can use Results > Remesh Deformed Configuration from the menu while the data set is highlighted.
In either case, this adds a new mesh to the mesh sequence and opens the Deformed Configuration settings window. Next, we click on Update. Note that for parametric or timedependent problems, we have to pick a parameter value or time step.
Each parameter of a parametrized data set has its own deformed configuration.
Finally, we go to the new mesh under Meshes > Deformed Configuration and build it.
Remeshing a deformed configuration creates a new meshing sequence.
One possibility is to reuse the deformed configuration in the same model file. To do so, we add another component and import the deformed mesh in the Geometry node of the new component, as highlighted in the screenshot below.
A deformed mesh from one component can be imported in the geometry sequence of another component.
We can now add more items to the geometry sequence. Let’s cut out from the bent pipe. We probably do not need the rigid indenter once we have used it to squeeze the pipe and will therefore get rid of it. The result is shown in the screenshot below. A new physics can be added to the second component.
Deformed objects resulting from structural analysis can be used as part of a new geometry sequence.
To use the deformed configuration in a different model file, export it to a separate file first.
If the Include geometric nonlinearity check box is unchecked, the spatial frame stays the same as the material frame. Therefore, we can not remesh the deformed configuration. If we do select the check box, COMSOL Multiphysics will include nonlinear terms in the strain tensor. What if we have a problem with infinitesimal strains and do not want to include expensive and unnecessary nonlinear strains in the equilibrium equations? The solution is to select the Include geometric nonlinearity check box in the study step, while ignoring the nonlinear strain terms by selecting the Force linear strains check box in the material model.
Splitting material and spatial frames while keeping only linear strains in the equilibrium equation.
The procedure for remeshing the deformed configuration remains the same as in the previous section.
The above method, including geometric nonlinearity and remeshing the deformed configuration, can be applied to both 2D and 3D problems. In 3D cases, we have an additional option via STL files. Any 3D surface plot can be exported as an STL file. This file can then be imported in the geometry sequence of another component or model file. By adding a Deformation node to a surface plot before exporting, we can get the deformed geometry. Do not include geometric nonlinearity, unless your problem is a large deformation problem.
Add a deformation to a 3D surface plot and export the surface plot in the STL format.
We can edit the X, Y, and Zcomponents of the displacements in the Deformation settings window of the above screenshot to introduce anisotropic or nonuniform scaling of the displacements. In fact, these quantities do not need to be structural displacements. By typing any valid mathematical expression for deformation components, we can subject the original geometry to arbitrary transformations.
To use the deformed geometry in a new file or component, the STL file generated in the above step can be imported in a geometry sequence, as shown below.
Importing an STL file to a geometry sequence.
COMSOL Multiphysics allows seamless coupling of different physics effects. If you want to couple structural analysis with another physics on the same domain, you will find builtin tools within our software that enable you to do so. The Moving Mesh and Deformed Geometry interfaces are often used together with physics interfaces to solve problems on evolving domains.
However, if you want to use the deformed configuration from a structural analysis as part of a new geometry sequence, where you add new objects to the deformed shape or include it in Boolean operations, you can apply the strategies demonstrated above.
As always, if you have any questions, please feel free to contact us.
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As the human population grows, so too does the amount of trash we produce. In fact, the amount of solid waste we generate may almost double by 2025. A large portion of the waste ends up in landfills, some of which are the same size as entire towns. Land, however, is a finite resource and eventually there will be nowhere to put our trash. Furthermore, these growing landfills can negatively affect their surrounding environments.
A landfill. Image by Alan Levine — Own Work. Licensed under CC BY 2.0, via Flickr Creative Commons.
As an example, let’s look at a common type of landfill: a drytomb landfill. As its name suggests, this landfill entombs waste in order to keep out moisture, resulting in reduced microbial growth and activity. While drytomb landfills are generally cost effective, if moisture enters, it can cause a variety of issues. For instance, landfills produce methane, a greenhouse gas much more potent than carbon dioxide, due to the resulting microbial activity. In the United States, landfills are even the third largest source of methane emissions. Landfills can also generate a harmful leachate that, if not treated, seeps into the water table and causes environmental and health problems. For these reasons, researchers are investigating alternative landfill designs, one of which is the aerobic bioreactor landfill.
There are two types of bioreactor landfills that correspond to with and without free air, respectively: aerobic and anaerobic. Here, we’ll focus on aerobic bioreactor landfills, which push air and moisture into a landfill in order to enhance aerobic microbial activity. This, in turn, increases the biodegradation rate, accelerating waste decomposition and more rapidly creating space for additional trash. This method also minimizes hazardous leachate and methane production when compared to anaerobic landfills.
We can turn existing landfills into aerobic bioreactor landfills by injecting air and recirculating the leachate to create an even distribution of bacteria and nutrients. The resulting aerobic landfill needs to be continuously monitored to ensure that the oxygen and moisture levels are ideal for aerobic microbial decomposition. Without outside intervention, the oxygen supply will be too low to provide a sufficient amount of aerobic bacteria, creating a lethal anaerobic environment. The hazardous leachate that is produced as a result may poison groundwater and harm the surrounding environment.
Before implementing this conversion process, further analysis needs to be done. Research using physical experiments could take years before reaching conclusions. For a more timeeffective method, researchers at the University of Western Ontario used simulation to analyze the conversion process.
For their study, the researchers modeled waste as a porous medium. They used both heat transfer and chemical reaction simulations to gain a better understanding of how aerobic landfills operate. With COMSOL Multiphysics, they were also able to include biological kinetic equations using distributed ordinary differential equations (ODEs).
The researchers created a 2D model, seen below, consisting of a 20 m by 20 m landfill cell with air injection wells at the corners and an extraction well at the center. With this model, the researchers studied how different key factors affected the landfill conversion process.
The landfill cell geometry. Image by Hecham M. Omar and Sohrab Rohani and taken from their COMSOL Conference 2015 Boston paper.
First, the team investigated the role of temperature in their simulations, using it as an indicator of successful biodegradation within the landfill. This is possible because aerobic biodegradation is exothermic, generating heat during the process. Therefore, when the landfill stays around ambient temperature, only minimal biodegradation occurs, whereas an increase in temperature indicates successful biodegradation.
It’s important to keep in mind that an increase in temperature is only beneficial in moderation. If the landfill isn’t monitored, the temperature will continue to rise and kill the aerobic bacteria. Thus, a major goal for researchers is to keep an aerobic bioreactor landfill at the perfect temperature, not too hot and not too cold.
With this in mind, the researchers investigated two methods for controlling the temperature:
For the first method, they increased the airflow rate fourfold. Instead of greatly decreasing the temperature, the air flowed farther into the waste before heating up to the waste’s temperature, and the cell temperature became more homogeneous due to convection.
The temperature after one day has passed for different airflow rates. The starting temperature for all three was 293 K. Images by Hecham M. Omar and Sohrab Rohani and taken from their COMSOL Conference 2015 Boston paper.
As for the second method, we see below that the temperature is significantly reduced by injecting leachate. This indicates that the leachate flow rate is more effective at controlling landfill temperature than the airflow rate. Here, leachate recirculation functions as a heat sink for the heat generated by the biomass, helping to ensure that the heat doesn’t reach unacceptable levels. Without this temperaturecontrol method, the biomass would overheat and begin to die, reducing its concentration to almost zero and slowing the biodegradation process.
The temperature after one day has passed for different leachate injection rates. The starting temperature for all three was 293 K. Images by Hecham M. Omar and Sohrab Rohani and taken from their COMSOL Conference 2015 Boston paper.
Another factor affecting the conversion of anaerobic landfills to aerobic bioreactor landfills is the initial aerobic biomass concentration. In the following images, you can see how various biomass concentrations affect the temperature. When there is a low initial aerobic biomass concentration, the biomass grows slowly and produces less heat. On the other hand, when the initial biomass concentration gets too high, the aerobic bacteria grows too rapidly and produces a harmful amount of heat. From this, the researchers concluded that the initial aerobic biomass concentration was a key factor in the landfill conversion process.
The temperature after one day has passed for different initial aerobic biomass concentrations. The starting temperature for all three was 293 K. Images by Hecham M. Omar and Sohrab Rohani and taken from their COMSOL Conference 2015 Boston paper.
With multiphysics modeling, the research team gained a better understanding of the factors that affect the aerobic landfill conversion process more quickly than they could have with experimental testing. Simulation also enabled them to study scenarios that would be difficult or impossible to study physically.
Although their model is consistent with both the researchers’ expectations and existing literature, the team notes that their model still needs to be validated against experimental and industrial data. Looking ahead, the aerobic bioreactor landfill design studied here can also be improved through other methods such as injecting aerobic sludge into the initial aerobic biomass concentration to help speed up the landfill conversion process.
When it comes to describing the velocity and pressure fields inside the system you are analyzing, there are many equations that could be appropriate. You could, for example, adequately describe a fluid slowly moving in a porous bed with Darcy’s law. But if the fluid moves rapidly, you may need to use the Brinkman equation. While there are many options available, today we will focus on the NavierStokes equations, as they are the most common in fluid flow analysis. Note that most of the explanations and practices highlighted here will also apply to the equations referenced above.
The first step is to characterize the type of flow that you are modeling based on fluid density. All fluids are compressible, that is, their density depends on absolute pressure and temperature through a thermodynamic relation, . However, from a practical point of view, most liquids can be safely described as having a density that depends uniquely on temperature, . Density is, of course, a function of an alternate element in some cases — for example, salt concentration in the Elder problem.
In the SinglePhase Flow interface available in COMSOL Multiphysics, there are three possible formulations for momentum and mass conservation equations: Compressible flow (Ma < 0.3), Weakly compressible flow, and Incompressible flow. You can easily select from these compressibility options within the Laminar Flow settings, as highlighted below.
Choosing a compressibility option in COMSOL Multiphysics.
In general, the various properties of a fluid are not constant and may depend on a number of quantities. Whether it is necessary to account for such dependencies in your modeling processes is up to you. Since the focus is on mass, momentum, and energy conservation equations in this blog post, we will review how COMSOL Multiphysics deals with viscosity , density , thermal conductivity , and heat capacity for the different compressibility options.
We’ll begin our discussion with isothermal flow simulations and later turn our attention to nonisothermal cases.
Pumps, mixers, airfoils, and multiphase systems… These are just some of the devices that are often modeled as isothermal. Isothermal flow simulations assume that , , , and are not dependent on temperature. If the properties are defined as a function of , they are evaluated at the reference value. This means that the energy equation is weakly coupled with the other two equations through the convective term, and it can be computed at a later time, if needed. It is, however, not always possible to make such an approximation.
Given the freedom that the user interface (UI) of COMSOL Multiphysics grants us, we can first study and solve a single physics problem and then build a multiphysics problem on top of the initial solution. Keep in mind that neglecting the energy conservation equation, even if you’re not directly interested in the temperature field, is valid only below a certain Mach number ().
Let’s take a look at how to select the appropriate compressibility option for your modeling case.
Compressible flow (Ma < 0.3), the most general case, makes no assumptions for the system that is being solved. COMSOL Multiphysics takes into account any dependency that the fluid properties may have on the variables. In isothermal flows, temperature is typically uniform and fluid properties (density and viscosity) remain constant, respectively, evaluated at the reference value. Even so, the properties can still vary with pressure or other quantities, such as concentration. With this formulation, which is the most computationally expensive, we can model any kind of flow and also describe incompressible situations. Our Flow Around an Inclined NACA 0012 Airfoil tutorial model provides one example of how to use the compressible flow formulation.
For the Weakly Compressible flow option, new with COMSOL Multiphysics® version 5.2a, the equations look the same as they do for the Compressible flow (Ma < 0.3) option. The only difference is that if the density is pressure dependent, the density will be evaluated at the reference absolute pressure. In this case, all the other dependencies of the density, like species concentration, are accounted for, so we can still use this formulation to account for volume forces given by concentration gradients.
The Incompressible flow formulation, meanwhile, is valid whenever can be regarded as constant (i.e., when modeling isothermal liquids or gases at low velocities). This option is also used in immiscible two or threephase flow simulations where density is constant. When applying the Incompressible flow formulation, COMSOL Multiphysics automatically uses the reference temperature and pressure to evaluate . In addition, it uses the reference temperature to evaluate . Of course, there are many applications in which the quantities mentioned above are dependent on another variable, like a specy concentration. In these cases, the density must be explicitly evaluated at a reference value for these variables within the interface. To learn more, download our Water Purification Reactor model example.
Density specification in the case of isothermal weakly compressible and incompressible flows.
Also note that the form of the equations, which are shown within the Equation section, change according to the selected option.
The different formulations for compressible and incompressible NavierStokes equations.
Nonisothermal flow simulations typically relate to cooling and heating applications, namely conjugate heat transfer. These simulations can refer to systems that are governed by natural, forced, or mixed convection.
Depending on the type of system that is being analyzed and the hypothesis that is assumed to be true, any of the compressible options can be appropriate for nonisothermal simulations. Since Compressible flow (Ma < 0.3) is the only meaningful formulation for gases subject to high pressure changes, we will focus here on systems that are below the Mach number limit, and those with fluid properties that are uniquely dependent on temperature. (There is a dedicated interface available for modeling high Mach number systems, as highlighted in this Sajben diffuser tutorial model.) The system of equations — mass, momentum, and energy conservation — is completely coupled, as the velocity appears inside the energy equation. Meanwhile, the pressure will appear explicitly in the momentum equation; the temperature will appear explicitly in the energy equation; and both temperature and pressure may be inside the fluid properties in these two equations.
Couplings of momentum and energy equations. In the case of natural convection, a part of the volume force depends on temperature gradients.
For convective heat transfer simulations, the Compressible flow (Ma < 0.3) option can be used to analyze forced and natural convection.
Forced convection refers to when the properties of the fluid vary in a nonnegligible way from pressure and temperature. This is the case for highspeed systems where the pressure changes are nonnegligible in their influence on density. As previously noted, the density of liquids rarely depends on pressure, which makes this exactly the same as the Weakly compressible flow formulation. See our ShellandTube Heat Exchanger tutorial model to learn more.
In natural convection, the driving force is the buoyancy force due to temperature gradients. The Compressible flow (Ma < 0.3) option must be used for gases in closed cavities in order for the system of equations to be consistent. In fact, if the volume cavity and total mass are constant, then the average density needs to be constant. Pressure changes help balance out density variations that are caused by temperature variations. Interested in modeling such a system? Refer to our Free Convection in a Light Bulb tutorial, which exemplifies the set up of transient conjugate heat transfer models with radiation.
The Weakly compressible flow option comes with a simplified formulation that usually leads to increased computational speed. It is the default option when the predefined Nonisothermal Flow or Conjugate Heat Transfer coupling is opened in COMSOL Multiphysics. This formulation, which basically neglects the green arrow coupling shown in the previous image, can be used to analyze forced and natural convection.
In the case of forced convection, the Weakly compressible flow option can be applied to the simulation of water or other fluids and is often valid for modeling gases in open systems (see this heat sink model example). The same considerations are valid for forced convection, as demonstrated in our vacuum flask tutorial.
The Incompressible flow option can be applied to both forced and natural convection as well. The initial case applies when you want to make simulations in cascade. For instance, sometimes it is interesting to compute the flow field at a reference temperature and then compute the temperature field in a second simulation. This can provide a powerful approximation when fluid properties do not vary much within the simulation’s temperature and pressure range. One good candidate for this type of modeling is a heat exchanger that includes liquids. You can also apply this approach to obtain more consistent initial values for highly nonlinear stationary problems. After computing the flow field and temperature field with ‘frozen velocity’, it is possible to gain considerable convergence improvements by using the initial values for the fully coupled simulations.
For the former case, COMSOL Multiphysics implements the Boussinesq approximation. The reference temperature and pressure, specified in the interface, are used to compute density, viscosity, heat capacity, and thermal conductivity. Further, the software automatically computes the coefficient of thermal expansion for the fluid, , taking the derivative of the density around the reference temperature, , and uses it to impose the buoyancy force, , where denotes the gravity vector. You also have the option to enter the coefficient directly, as shown below.
Options for specifying the density.
When utilizing the gravity option, an important concern is the need for consistent boundary conditions and initial values, particularly when dealing with natural convection simulations. This could be a nontrivial task since a domain force, the buoyancy force, is working inside the system, and we need to account for its presence. Consider, for instance, a system like a pipe where a hydrostatic head is present. Here, it is clear that we simply can’t impose a constant pressure as a boundary condition if the boundary itself is not perpendicular to the gravity vector.
Besides enabling you to model all of the above systems and situations, COMSOL Multiphysics helps to address initial values and boundary conditions for each case. To learn more, check out our Gravity and Boundary Conditions tutorial model.
For forced convection, the flow and temperature coupling is taken care of at the Multiphysics node level. Within the NonIsothermal Flow interface, the equations are coupled and fluid flow and heat transfer properties are synchronized (see the screenshot below). Depending on the compressibility option that is chosen, COMSOL Multiphysics will operate in the background to make the appropriate changes for the fluid properties, making them consistent with the selected formulation. Additionally, the NonIsothermal Flow interface takes care of implementing thermal wall functions and computing turbulent heat conduction.
Settings window for the NonIsothermal Flow interface.
If it is necessary to include a buoyancy force due to temperature or concentration gradients, then you must select the Include gravity check box. This will generate values in the Reference Values section that can be used to compute the hydrostatic pressure approximate, alongside reference temperature and pressure. Selecting the Include gravity check box also causes a new subnode to appear: Gravity. Here, you can specify the direction of the acceleration acting on the system. When the Gravity subnode is added, the hydrostatic contribution taken at the reference temperature and pressure is automatically added inside the boundary conditions, if appropriate.
To model natural convection, it is simply necessary to use both the Gravity feature and NonIsothermal Flow interface. Together, they model flow and temperature fields coupled in the presence of gravity acceleration.
Changes prompted by selecting the Include gravity check box.
The following simulation plots correlate with our vacuum flask tutorial model, which evaluates the thermal performance of a bottle holding hot fluid. This system is composed of a gas, air, outside the flask that is flowing in an open system — the flask is leaning on a table in a wide room. Such considerations make this example a helpful resource for understanding the use, assumptions, and results of the various formulations. The Compressible flow (Ma < 0.3) option, for instance, is always applicable. Since air is flowing in an open system, the Weakly compressible flow option is also applicable. And lastly, because the density changes are small, the Incompressible flow option can represent the system appropriately as well.
Plots comparing velocity, temperature, and density fields (respectively) for simulations using the Compressible flow (Ma < 0.3), Weakly compressible flow, and Incompressible flow formulations. We performed these simulations by simply toggling between the three compressibility options.
Graphs comparing velocity, temperature, and density fields (respectively) for simulations using the Compressible flow (Ma < 0.3), Weakly compressible flow, and Incompressible flow formulations. We performed these evaluations at the red dashed line, shown in the plot on the right in the previous set of images, after a simulation time of 10 hours.
Choosing the correct compressibility option is key for solving your system in an accurate and efficient way. COMSOL Multiphysics provides you with functionality that allows you to model both natural and forced convection with ease, while still offering various modeling choices and giving you complete control over the simulations at hand. This results in an optimized approach to the numerical analysis of fluid flow and temperature fields, further advancing your engineering design.
Use the table below as a helpful guide for choosing the compressibility option that is most appropriate for your modeling needs.
Compressibility Option  Isothermal Flow  Nonisothermal Flow 

Compressible flow (Ma < 0.3) 


Weakly compressible flow 


Incompressible flow 


In principle, we can analyze mechanical devices with gears by explicitly including the contact interactions between gears as part of the simulation, but this method is computationally timeconsuming when performing a multibody dynamics analysis. Instead, we can implement a mathematical formulation to model the contact interactions between the gears.
With this formulation, we can include a realistic gear geometry, which provides accurate inertial properties when used in transient and frequencydomain studies. Realistic gear geometries from the Part Library can also be used to evaluate gear mesh stiffness in a static contact analysis and for multiphysics simulations. Note that the gear mesh stiffness is not analyzed through finite element analysis, but the stiffness of pairs of gear teeth are still in contact. Another benefit of having realistic gear geometries in a multibody dynamics analysis is that this provides better visualization when either setting up the physics or when postprocessing.
Geometry of a helical gear pair built using the Part Library.
We could manually build the geometry, but using builtin parts is both easier and faster. These parts are parametric in nature, which means that we can change their shape by readjusting the geometric parameters, and they come with optional features that can be added, such as shafts and fillets. The parts also have extensive checks to validate the input data as well as selections for the gear, shaft, and contact boundaries, therefore ensuring realistic physical entities and behavior. With the Part Library, it’s easy to specify the position and orientation of the gears as well as to align the gear mesh with their counterpart. These parts also contain robust geometric operations when creating complex gear geometries and the ability to manually change geometric operations.
The gear parts in the Part Library are divided into three categories based on whether they are gears with an external mesh, a gear with an internal mesh, or a rack. To learn more about the gear parts available in the Part Library, please read the previous blog post in our Gear Modeling series.
While the gear geometries in the Part Library are for individual gears or racks, gears are always used in pairs. Due to this, we need to build a gear train using individual gear parts. To illustrate the steps involved, we use a 2D spur gear pair example. The known quantities are as follows:
A spur gear pair showing the center distance of the two gears and the angular position of the second gear.
To place the second gear correctly, the first step is to compute the center distance ():
The position of the second gear () can be defined as:
Once the second gear is placed at the correct location, the next step is to align the teeth, or in this case mesh, of both gears. To accomplish this task, rotate the second gear with a mesh alignment angle () defined as:
where and are the mesh cycle of both gears, and they are defined as:
where and are the number of teeth of the first and second gear, respectively.
After computing the position of the second gear as well as the mesh alignment angle, we enter them as either expressions or numbers in the input parameter fields of the second gear, as shown below:
The input parameters of a 2D spur gear part with the gear center and mesh alignment angle highlighted.
For the gear tooth, we define the profile using an involute curve. The tooth shape and size are specific to the gear’s application, so a different application would require another type of gear tooth. Here is a list of input parameters through which we can control the shape and size of a gear tooth:
In the case that the fillet is not required in these places, we can set the tip or root fillet radius to zero.
An external gear tooth showing various input parameters.
The input parameters are mostly relative quantities for better scalability. We can compute different tooth profile parameters in terms of these input parameters:
Some applications require a specific type of gear tooth. Highpressure angle gears are better for highspeed applications as their wear rate is less than that of a standard tooth profile. Similarly, backlash is needed in highspeed applications because it provides space for a film of lubricating oil between the teeth, which prevents overheating and tooth damage. On the other hand, backlash is not desirable in precision equipment, such as instruments, machine tools, and robots. Backlash in these devices causes lost motion between input and output shafts, making it difficult to achieve accurate positioning.
Gears for different pressure angles and modules. Left: Gear with a standard tooth profile. Middle: Highpressure angle gear. Right: Highmodule gear.
After exploring the details of a gear tooth, we look at other parameters that influence the shape and size of a gear. The gear geometry is divided into three components: the gear teeth, gear blank, and shaft. For the gear shaft, the parameters are as follows:
Although the shaft is not an integral part of a gear, we can create one at the gear center with builtin gear parts. It is also possible to set the axial position of the gear on the shaft.
By default, a gear is placed at the origin and its axis is set to the zaxis, but it’s possible to control the position and orientation of the gear using the following parameters:
In order to align the gear mesh with the mating gear, we use a mesh alignment angle parameter to rotate the gear around its own axis.
A helical gear geometry showing different input parameters.
These input parameters, like the ones for the gear tooth, are relative quantities that we can use to calculate the gear parameters. They are as follows:
By default, a gear geometry comes with a set of features. Some of these are optional, and we can remove them by setting the appropriate input parameter to zero. It is possible, for example, to build a gear geometry without a shaft, gear blank ring, center hole, and fillets at the root and tip.
Geometry of spur gears where optional features are removed sequentially from (A) to (F). (A) Default geometry; (B) Without shaft; (C) Without gear blank ring; (D) Without center hole; (E) Without tip fillet; (F) Without root fillet.
While the gear blank shape is rather standard in all of the builtin gear parts, we can create a ring by removing the material in the gear blank. To customize the gear blank shape, we need to perform various manual geometric operations on the builtin parts.
Gears with customized gear blanks.
The builtin gear parts provide selections that we can use when setting up the physics or postprocessing. The available selections are for different components of the gear as well as for the gear teeth boundaries. We can use these boundaries to model contact between the two gears.
A spur gear where the geometry of the gear body, excluding the shaft, (left) and the gear teeth boundaries (right) are highlighted.
Since the gear parts are highly parametric, it is important to have an extensive set of checks to validate the input data. These checks ensure that the input parameters are correct independently as well as when combined with other parameters. We perform these checks before proceeding to build the geometry.
In the case that the set of input parameters is invalid, an appropriate error message is displayed. A few examples of nontrivial geometry checks, an external gear for instance, are as follows:
Next, we’ll look at some examples of gear geometries created using builtin parts.
The first example is a differential gear mechanism used in automobiles. This gear allows the left and right axles to rotate at different speeds. A differential gear uses five pairs of bevel gears, six bevel gears in total, to perform its operation.
Geometry of a differential gear mechanism.
The next example is a threestage wind turbine gearbox. The first stage is a planetary gear train, which has three planet gears, one sun gear, and one ring gear. The second and the third stages are parallel gear trains that each have a pair of gears. This gearbox uses eight pairs of helical gears, nine in total, to perform its operation. The typical gear ratio of this gearbox varies from 50 to 100.
Geometry of a wind turbine gearbox with the top and front view showing.
Designed to transfer rotary motion from one shaft to another, gears are important devices in a variety of machines, from automobiles to wind turbines. New functionality in COMSOL Multiphysics provides you with several possibilities for quickly building gear geometries. With these robust and highly parametric builtin parts, you can change the shape of a gear to create an applicationspecific gear geometry.
In the next blog post in our Gear Modeling series, we’ll show you how to simulate gearbox noise and vibration. Stay tuned! We encourage you to browse the additional resources below in the meantime.