To keep the model simple and take advantage of the symmetry of a balloon, we can build a 2D axisymmetric geometry that consists of only a rectangle and an ellipse, together with slightly larger double versions to account for the rubber balloon. Our goal is to see what happens if we let the same amount of water into differentsized balloons. To do so, we can parameterize the geometry and use a scaling factor to change the initial size of the balloon, while the material thickness and the neck radius remain constant.
Geometry of two deflated water balloons of different sizes. The size is controlled by the stretching factor fact: fact = 1 (left) and fact = 2 (right).
The water balloon model makes use of new features in version 5.3a of COMSOL Multiphysics, including improved fluidstructure interaction (FSI) functionality and a realigned moving mesh.
As of COMSOL Multiphysics version 5.3a, FSI is modeled via a Multiphysics node. The node connects the physics from Fluid Mechanics and Structural Mechanics interfaces. In contrast to earlier versions of the software, where there was a separate FluidStructure Interaction interface, we can now use all of the available features from the twoway coupled physics.
The interfaces and the moving mesh after adding the FSI physics.
In this example, it is easy to take gravity into account. All we need to do is place a checkmark in the Laminar Flow interface settings. This activates the gravity of the earth, which in turn has an effect on the mechanical behavior due to the hydrostatic pressure in the water. We can expect that there will be a noticeable effect of gravity on the results, and this effect will be more significant in the larger water balloon, because there is more mass in the beginning.
On the mechanical side, the physics settings can likewise be set up quickly. We only have to define a suitable material model that describes the hyperelastic properties of the material of the balloon correctly. In the Application Library, the Inflation of a Spherical Rubber Balloon model contains a variety of hyperelastic materials. We can use the Ogden model here, because it reproduces the analytical solution most accurately.
Interested in details about fitting measured data to different hyperelastic material models? Check out this previous blog post.
By the way, copying model interfaces between different models is now very simple. Since version 5.3a of the COMSOL® software, interfaces and components can be exchanged via the copypaste functionality — even between two running COMSOL Multiphysics® simulations! This means that we can efficiently insert material settings from another model into the water balloon model.
Parameters of the hyperelastic Ogden material model used for the balloon.
Another improvement in COMSOL Multiphysics version 5.3a is the new positioning of the Moving Mesh interface. It is now found at a more prominent position under Definitions. One advantage of the new structure is that it helps avoid accidental overlaps between deforming and nondeforming areas. For the water balloon model, this improvement means that we have only two tasks for the mesh: selecting the balloon’s interior water as a Deforming Domain and adding a Prescribed Normal Mesh Displacement on the symmetry axis (to avoid unwanted movement away from this axis due to numerical inaccuracies).
The final step before solving the water balloon model is to set the water flow timing. Turning the tap on and off quickly with a defined amount of time in between can be expressed by a rectangle function. This function is multiplied to the inlet velocity of 15 cm/s, creating a flow of about 1.4 l/min.
The water inlet velocity is controlled via a rectangle function.
We can carry out a parametric sweep study to compare the simulation results for three different initial balloon sizes. All three balloons are filled with the same amount of water because the inlet velocity and the filling time period are the same. By far, the largest stress occurs in the smallest balloon. This is as expected, because the small balloon has the smallest surface and the largest relative volume increase.
Von Mises stress distribution in the balloon material after inflation for three different initial sizes. (Note: These plots were created with the Cividis color table, a color table that is optimized for people with color vision deficiency, new as of COMSOL Multiphysics version 5.3a.)
These results call for some animations! If we take a look at the transitional behavior of the inflation, we clearly see the influence of gravity on the largest balloon, because it oscillates before the water injection even starts. There is no prestress in the balloon, so it starts falling down a bit until the counterforce from the material is large enough to compensate for the gravity.
Animation of the von Mises stress in the smallest water balloon during inflation.
Animation of the von Mises stress in the mediumsized water balloon during inflation.
Animation of the von Mises stress in the largest water balloon during inflation.
The FSI functionality in COMSOL Multiphysics version 5.3a includes useful enhancements and is more user friendly than in previous software versions. With surprisingly little effort, it’s possible to set up a complex FSI model and solve it in a short time.
I am very curious to see how you use these new features to master your modeling challenges!
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You can save a lot of computational time when running a simulation if the impact of temperature variations on the flow field are negligible compared to the accuracy required for the solution. In such cases, we can compute the flow field in the first study step and then use it as an input for the heat transfer problem solved in the second study step, which is an easy thing to do in COMSOL Multiphysics.
Instead of solving a twoway coupled problem (flow ↔ transport), we solve a simpler oneway coupled problem (flow → transport). The reduction in computational time and memory is even higher if the solution of the flow field can be reused several times; e.g., when a parametric study for different heat transfer conditions is carried out for the same flow field.
The oneway coupling approach can be applied for all types of fluid flow, including turbulent regimes and flow in porous media. It is also possible to apply this technique to any advected field, provided that the coupling is weak; e.g., for chemical species transport in dilute solutions.
The important criterion for the validity of the oneway coupling approach is that the influence on the flow field, assuming a constant temperature, is much smaller than the accuracy required in the computation. We have to check that the variations in density and viscosity caused by temperature changes are small enough that their impact on the flow fields falls within the accuracy limit in the analyses. It is recommended to assume the flow mean temperature as the reference temperature for density and viscosity in the oneway coupled case.
The best way to check the validity of a oneway coupled approach is to solve a test model and compare the results to a twoway coupled solution of the same problem. Pick a few sample points in the analysis where the fully coupled problem is computed and verify the simplified approach against the full solution. If these points fall within the required accuracy, we can use the simplified approach for the bulk of the computations. The samples need to be selected wisely, as the verification points must fall inside the simulation window of operation. Ideally, these points should be the extreme conditions and all other computations should fall within the extreme points.
If it turns out that the oneway coupling is not a suitable simplification for a certain simulation task, using this technique can still be helpful. The approach of solving the decoupled problem first is a good option to get good initial guesses for the fully coupled problem for steady nonisothermal flows. There are cases where the flow field does not converge unless a decent initial guess is provided, which is what we can obtain with the approach discussed here.
Let’s try out the oneway coupling approach using the CrossFlow Heat Exchanger tutorial model. This type of heat exchanger is found in labonachip devices in biotechnology and microreactors, such as for microsized fuel cells.
The modeled part of the microsized heat exchanger.
The modeled system consists of two sets of channels, one hot and one cold, arranged in a crossflow pattern with five channels in each set, as shown in the figure above. The model is reduced due to the symmetry of the heat exchanger.
If we check the study nodes of the model, we find two stationary study steps. In the first study step, only laminar flow (spf) is selected for solving, while in the second study step, heat transfer (ht) is selected together with the multiphysics coupling nonisothermal flow (nitf1). The flow field is solved in the first study step and the result is automatically taken in the second step because of the applied coupling provided by the Nonisothermal Flow multiphysics node. This study setup is preset; available as of COMSOL Multiphysics version 5.3; and called Stationary, OneWay Coupled, NITF for stationary simulations and Time Dependent, OneWay Coupled, NITF for transient simulations.
We can compare the results of the oneway coupled approach with a twoway coupled version by adding a new study with a stationary, fully coupled study step. After computing both studies, it turns out that the results vary only slightly. The heat transfer coefficient, probably the most interesting result of the model, becomes 1547.8 W/(m^{2}K) for the twoway coupling and 1548.1 W/(m^{2}K) for the oneway coupling. The difference of less than 0.2‰ is probably a lot smaller than the numerical error in the two computations. Further, the computation time is halved — from about 3 minutes for the twoway coupled problem to less than 1.5 minutes for the oneway coupled problem.
A comprehensive comparison of the two approaches can be found in the slideshow presentation available with the model documentation.
Temperature results of the oneway coupling (left) and twoway coupling (right) stationary solutions.
If the transient behavior of the model is of interest, other study combinations are possible. For example, we can add two timedependent study steps, where the flow is solved first, followed by a transient heat transfer study step (Time Dependent, OneWay Coupled, NITF). We can also create a study sequence with a stationary flow and a transient heat transfer study if the flow conditions are not changed with time (except temperature). The table below gives an overview of the different study combinations and their respective computation times for a simulation time of 10 seconds.
Study Type  Computation Time (Seconds) 

Oneway coupled (stationary) 
77

Twoway coupled (stationary) 
183

Oneway coupled (time dependent) 
571

Twoway coupled (time dependent) 
806

Oneway coupled stationary flow and timedependent heat transfer 
246

The computation times of different study approaches on an Intel® Core™ processor E51620 @ 3.70 GHz machine.
As expected, cases where the transient heat transfer is oneway coupled with stationary flow fields are computed in less time. The demonstration problem is obviously small with respect to the computational time, but the simplified approach discussed in this blog post becomes a more important option as a problem grows.
Intel and Intel Core are trademarks of Intel Corporation in the U.S. and/or other countries.
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The diverse range of papers and posters presented at the COMSOL Conference 2016 Munich covered application areas such as the flow properties of shark skin, processes during cake baking, and the optimization of automotive loudspeaker systems, to name just a few. Altogether, over 150 highquality simulation projects were presented at the event.
Three papers were awarded Best Paper awards by the conference program committee and for the three Best Poster awards, over 300 conference attendees voted for their favorites. The winners were recognized during the awards ceremony at the conference gala dinner.
The awards ceremony at the COMSOL Conference 2016 Munich.
The conference program committee for the COMSOL Conference 2016 Munich consisted of 27 leading academic and industrial simulation experts who were tasked with reviewing the conference abstracts and choosing the top three conference papers.
The COMSOL Conference 2016 Munich Best Paper award winners. From left to right: KarlHeinz Leitz; Tobias Gasch; Iryna Tomashchuk; and Bernhard Fluche, managing director of Comsol Multiphysics GmbH.
The first Best Paper award was given to Tobias Gasch from KTH Royal Institute of Technology in Stockholm, Sweden for the paper “Cracking in QuasiBrittle Materials Using Isotropic Damage Mechanics“. Together with his colleague Anders Ansell, Gasch created an extension of the Solid Mechanics interface in COMSOL Multiphysics to analyze localized deformations of quasibrittle materials, which occur in cracking concrete, for example.
The research team used the Physics Builder to create a custom user interface and validated the implemented model using two literature examples of plain and reinforced concrete. The models show good agreement with experimental results.
KarlHeinz Leitz and his colleagues from Plansee SE in Reutte, Austria also won a Best Paper award for their paper titled “ThermoFluiddynamical Modelling of Laser BeamMatter Interaction in Selective Laser Melting“. In their research, they describe the development of a thermofluiddynamical simulation for laser beammatter interaction.
This comprehensive multiphysics model includes laser radiation absorption; conductive and convective heat transfer; as well as melting, solidification, evaporation, and condensation. The model is used to compute the process dynamics of selective laser melting (SLM), but the researchers emphasize that it could easily be adopted for the simulation of other laserbased manufacturing processes.
The final Best Paper award went to Iryna Tomashchuk and her colleagues at the Laboratoir Interdisciplinaire Carnot de Bourgogne in France for the paper “Multiphysical modelling of keyhole formation during dissimilar laser welding“. In their paper, the researchers describe the implementation of a timedependent multiphysics simulation of the pulsed and continuous laser welding of dissimilar metals, based on the moving mesh approach.
The simulation results for the pulsed welding model were validated for a Ti6A14V alloy and Ti6A14V steel coupled material by comparing them to experimental well dimensions. The research team plans to test more dissimilar material couples in order to further understand keyhole formation.
All of the attendees at the COMSOL Conference 2016 Munich had the chance to vote for three Best Poster awards, which were presented at the event in addition to the Best Paper awards.
COMSOL Conference 2016 Munich Best Poster award winners. From left to right: Pascal Le Bideau, Christoph Hollenbeck, Kersten Kröger, Andrej Genschel, and Simon Bekemeier, with Bernhard Fluche.
The first Best Poster award went to Pascal Le Bideau from Université de Bretagne Sud in Lorient, France for a culinary engineering presentation. The poster “Numerical Model for Predicting Heat and Mass Transfer Phenomena during Cake Baking” gives an overview of a cake baking simulation study performed by Le Bideau and his colleagues.
The aim of the research is to provide a numerical model for predicting heat and mass transfer phenomena as well as the swelling encountered during the baking of cake contained in a mold. The medium is assumed to be a deformable porous medium containing three phases (solid dough, liquid water, and gaseous water and CO_{2}).
Christoph Hollenbeck received a Best Poster award on behalf of his research group from Imagine Engineering GmbH in Bergheim, Germany. The poster, titled “Simulation of the thermal expansion of an inductively heated gear wheel for shrink fitting purposes“, describes the simulation approach for a 3D coupled magnetic, thermal, and mechanical model.
The aim of the study is to identify suitable parameters for the coil current and the frequency of the alternating field in order to attach (and detach) a gear wheel on a shaft. The simulation results of the mechanical stresses enabled the researchers to evaluate whether process control concerning rapid heating is mechanically possible. They found that increasing the coil current could significantly reduce the process time for induction heating. This is because the tested type of steel has a yield strength five times higher than the maximum arising equivalent stresses.
The final Best Poster award was given to Simon Bekemeier and his colleagues from the University of Applied Sciences in Bielefeld, Germany for the poster “Beverage Refrigeration Simulation in Dependence on Container Shape, Material and Orientation“. This poster may have been so popular because the conference location in Bavaria pairs perfectly with the poster topic: beer.
Using the Application Builder, the research group created an easytouse simulation app, which enabled them to examine the influence of several input parameters on the cooling process of beverages. The poster details the functionality of the BEverage rEfRigeration (BEER) simulation app. The conclusion of their simulation research: “Do not bother buying PETbottled beer, glass bottles cool slightly faster.”
We would like to thank everyone who presented their work at the COMSOL Conference 2016 Munich and congratulate the six Best Paper and Best Poster award winners. It was a great experience learning about all of your interesting research topics.
The projects highlighted in this blog post are just the beginning. Stay tuned to access all of the posters and presentations from the COMSOL Conference 2016 Munich.
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At the COMSOL Conference 2016 Munich, over 300 attendees learned about multiphysics modeling, heard innovative ideas from fellow engineers and researchers, and connected with each other and COMSOL staff.
I had a great time at the conference this year and appreciate everyone who attended and presented at the event. Some highlights include watching COMSOL President and CEO Svante Littmarck tapping the beer barrel at Paulaner’s Wirtshaus, hearing keynote talks and user presentations in a wide variety of interesting subjects, and connecting with all of you at the gala dinner.
Browse the photo slideshow below for scenes from the COMSOL Conference 2016 Munich.
Thank you to all who attended and presented at the COMSOL Conference 2016 Munich. We’d also like to extend a special thank you to our Gold Sponsors for this year’s event: Dell, Simpleware, and Intel. As always, it was a pleasure!
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The branch of physics that describes a tank with a sloshing fluid is fluid dynamics. We account for the two fluids in the tank: the sloshing liquid and air everywhere else. This means that we need to use a twophase flow model that allows for a relatively large difference in density between the two phases. Due to the density difference between the liquid and air, the interaction and separation between the two phases is gravity driven. In this case, it simply means that when the shaking stops, the air tends to move towards the top of the tank, with the liquid at the bottom of the tank. For our model, let’s define the two fluids (phases) as water and air.
There are several possibilities to model twophase flow (the dynamics of two immiscible fluids) with COMSOL Multiphysics. The phase field and level set methods both support the topological changes of the free surface that we expect to see in a sloshing tank. The phase field method is more accurate than the level set method for this example, since the surface tension is obtained from the minimization of the free energy of the system, while in the level set method, it is explicitly added. If you want to know more about which multiphase flow interface you should use, check out this previous blog post.
Since discussing the simulation research of IAV, we have received several requests for the model files for the featured sloshing tank. The geometry of our sloshing tank is defined in COMSOL Multiphysics, but some of the geometry operations require the Design Module. If you do not have the Design Module, you can alternatively import the three different parts of the geometry (i.e., the tank and the sloshingreducing baffles) into COMSOL Multiphysics from the files we provide. Whether you import the parts or use the whole sequence, the final geometry sequence should contain two If + End If loops, where the sloshingreducing baffles can be applied to the tank with Boolean parameter settings.
Initially, the model can be built without these baffles; i.e., by setting walls = 0 and box = 0 in the parameter list. The geometry sequence also includes a parameterized work plane that is used as the initial shape of the free surface between the two fluids. The work plane is tilted, which gives an initially unstable state, and the system immediately starts to slosh from this state.
The geometry sequence, with imported parts and the resulting tank geometry.
From the physics definition point of view, setting up a twophase flow model with two separated fluids sloshing in a box is really simple. All we need to do is:
Knowing that the transient computation of fluid flow can already be computationally demanding, it is obvious that twophase flow models, with the description of the free surface, present an even bigger challenge. To make it less computationally demanding, we can choose a coarse mesh and switch to P1 + P1 discretization to generate the numerical model equations. (This sets linear elements for both the velocity components and the pressure field. The defaults are secondorder elements for the velocity components and linear elements for the pressure field.) The solution to the numerical equations may give a less accurate estimate of the real solution compared to the default settings, but we get quicker results and a rough idea of the performance of the system.
For more accurate investigations of sloshing, we need to refine the coarse mesh and discretization settings. (A mesh refinement study can be run afterward to ensure the accuracy of the solution.) The same thing can be said about setting the Convective term in the Advanced Settings section: You can use either the conservative or the nonconservative formulation of the level set or phase field equation. The conservative form perfectly conserves the mass of each fluid, but the computational time is generally longer — this is why we relax the strict enforcement of conservation for this problem.
There are some settings in the solver that we need to change for better convergence of the solution of the model equations. First, we change the Time Stepping method under Study 1 > Solver Configurations > Solution 1 > Time Dependent Solver 1 from the default backward differentiation formula (BDF) to Generalized alpha. Then, we change the steps taken by the solver to Intermediate and reduce the initial step to 0.0001 seconds. This setting allows the timedependent solver to take finer time steps during the simulation.
After, the Direct solver subnode needs to be enabled in the Time Dependent Solver 1, which automatically disables Iterative 1. The direct solver may be switched from MUMPS to PARDISO because it converges faster for this problem. (More information on the Direct and Iterative solvers can be found here). With these settings, the model should solve between 1 and 1.5 hours on an average workstation.
The timedependent solver settings for the sloshing tank model.
Results from transient simulations of fluid flow are great for creating animations, since these animations are intuitive and easy to interpret. This way, the dynamics of the system can be qualitatively validated. In the animation below, water is visualized with the rainbow color legend, while air is kept transparent. This provides a great visualization of the free surface that separates water and air.
To create this plot, the plot group denoted Water Phase contains a volume plot that displays the variable for the volume fraction of fluid 1: tpf.Vf1. An additional filter is added to reduce the displayed range to volume fraction values larger than 0.5. As a result, we get a plot of the water phase and a clear picture of the free surface in the tank.
An animation of the water phase for one second of sloshing in the tank.
Even though we used P1 – P1 elements with a coarse mesh for our sloshing tank model, we have already established a qualitatively reasonable result (it looks natural). The fluids inside the tank tend to move into their equilibrium state, with water at the bottom, and overshoot, causing sloshing to occur. The next step for a more accurate solution of the model equations would be to use a denser mesh or higherorder elements. In the model itself, we may consider checking the effect of adding damping inner walls.
Over the last few decades, the automotive industry has come to rely more and more on simulationbased techniques for fostering design advancements. These analyses enable designers and engineers to study a wide range of components within vehicles, from the smallest screw to the engine system as a whole, while saving on both time and costs.
With the growing complexity of vehicles and reduced development time, the demand for simulationbased design has grown particularly high. While many variables are important points of focus within this segment of the automotive industry, one of the most requested details concerning noise, vibration, and harshness (NVH) optimization is reducing sloshing forces in car tanks.
Sloshing is a term that refers to liquid moving inside of another object. Modern vehicles contain a whole string of fluids in tanks and, when a vehicle accelerates (or brakes) rapidly, inertia forces cause movement in all of them. Such movement of liquids produces transient longitudinal and transversal forces that disturb vehicle dynamics and cause splashing, which can in turn generate unwanted noise. The latter problem grows even more important as other sources of noise such as the vehicle’s combustion engine or tires become quieter.
Rather than building and testing a variety of prototypes, simulation engineers at IAV, one of the world’s leading engineering companies in the automotive industry, created a numerical model and used it to develop and design a solution to this problem. The purpose of their study was to find out how internal walls in the tank can best reduce the sloshing of fuel. This process involved solving a twophase flow problem in COMSOL Multiphysics. The design that IAV ultimately developed with this method has been rather successful and is easily applicable to a variety of sloshing problems in automotive tank systems.
Using an arbitrary tank geometry that we designed, let’s take a closer look at the principles behind their force reduction approach.
The situation presented here starts after the vehicle’s braking process, when the free fluid surface is inclined and about to return to a horizontal position because of gravity. It can take several seconds for the fluid inside of the tank to settle down.
The initial distribution of liquid and gas in the tank.
In the series of animations below, we can see the impact that the use of internal walls has on the movement of the fluid in the tank. When no wall is used, the fluid moves noticeably more freely and takes longer to settle. Partially equipped approaches (one uses only the boxy structure with gaps or one uses only the two vertical double walls) somewhat help to calm down the movement of the fluid. A fully equipped approach that utilizes both of these internal components is the most effective at damping the wavy movement of the fluid. We can conclude that these modifications to the tank’s internal geometry have a large influence on the fluid system.
Animations illustrating the impact of internal walls on sloshing in a fluid tank design.
The graph shown below compares the fluid’s kinetic energy over time. Without the use of the walls and the boxy structure, the initial peak of kinetic energy is significantly higher with damped, but lasting, oscillations due to the sloshing liquid. The peak and the oscillations are damped most effectively when all of the internal wall components are included in the tank’s geometry. Through this analysis of the liquid’s kinetic energy, we can quantify and assess the impact of modifications to the tank’s geometry at an early stage in the design process.
Kinetic energy of the sloshing fluid.
The majority of fluid tanks in vehicles are designed to fit into free spaces between other important components. This can often result in complex tank geometries that vary greatly depending on the vehicle type in which they are used.
Here, we have presented a model of a sloshing tank that offers a solution for all conceivable designs. Designers and engineers simply need to modify the tank geometry to fit their needs and recompute the solution, allowing them to easily compare different design configurations and identify the one that is optimal.
If you have any questions pertaining to your own simulations, please contact your COMSOL support team.
]]>Stirling engines, or heat pumps, are systems that are able to work on incredibly low temperature differences. In fact, some types of Stirling engines only need human body heat in order to operate. Here, we explore the dynamics of this interesting machine that you can build at home and demonstrate how to model it using COMSOL Multiphysics.
Let’s begin by taking a step back in the history of the Stirling engine. Denominated the “engine of the future,” the Stirling engine was first developed by Robert Stirling nearly 200 years ago in 1816. While the technology never really came out on top, this type of heat engine has found extensive use in many modern applications. For example, the solar Stirling engine is used to directly transform solar heat into mechanical energy, which is in turn applied to powering a generator and producing electrical power. Additionally, analogous approaches exist that are based on geothermal energy or use industrial waste heat. The most astonishing modern application of the Stirling engine may be its operation in Swedish submarines — the absence of air is not an issue for a Stirling’s propulsion.
We’ve touched upon some of the applications of Stirling engines, but how does this machine operate? In a Stirling engine, thermal heat is converted into mechanical work (or the other way around) in a cyclic process. This can be realized in different ways, but the principle remains the same: The engine cycles through the four processes of cooling, compression, heating, and expansion. A gas is used to transport the heat from a permanent hot side to a cold side. The efficiency of the engine is restricted by the efficiency of the Carnot cycle.
In contrast to conventional engines, Stirling engines do not need to reach high temperatures to operate. Some Stirling engines only need a small Kelvin temperature difference between the hot side and the cold side. Furthermore, the sound level and subsequent energy losses are very low because there are no explosions and no exhaust. However, Stirling engines are most suitable for applications where constant power is required, as controlling dynamic power regulation would be an extensive task. This is likely the most prominent reason as to why there are still no cars powered by Stirling engines.
A Stirling engine operates using the heat from a human hand. (“A Stirling engine that works solely with the energy taken from the temperaturedifference from the surrounding air and the palm of the hand” by Arsdell — Own work. Licensed under Creative Commons AttributionShare Alike 3.0 via Wikimedia Commons).
For those of you with some experience in handcrafting, it is possible to build your own Stirling engine at home, without the need for professional tools or experience. Several video tutorials are available on YouTube that include instructions designed to guide you through the building process. The easiest of these examples is probably the version consisting of a cola can and some domestic odds and ends.
While easy to build, this model of a Stirling engine is likely not optimized from an efficiency point of view. Creating a numerical model of the engine offers a better solution.
With a numerical model of a Stirling engine, we are able to find and test sets of materials as well as parameter adjustments. The involved physics are heat transfer and fluid flow, and the mechanical process can be simplified by solving the equation of motion as an additional ODE.
The 2Daxisymmetric model consists of a main cylinder that includes the working fluid (air) and the displacer. The small cylinder on top contains the power piston. Both the displacer and the power piston are connected in parallel and joined 90º out of phase on a crankshaft, which is not featured as part of the model. The whole setup corresponds to a gamma type Stirling engine.
A model of a Stirling heat pump.
Here, the heat transport within the working gas is solved. The mechanical part is realized via a moving mesh (ALE) approach: The displacer and the power piston are free to move in the zdirection. The displacement is prescribed here, corresponding to a heat pump. Mechanical work is used to move thermal energy opposite to the direction of spontaneous heat flow. The other way around — a Stirling engine — can also be modeled by applying a heat source and solving for the resulting pressure forces at the power piston and the displacer. In any case, the system runs through a certain chain of processes that can be divided into the four steps of the Carnot cycle:
Thermodynamic processes that are acting on the working fluid.
While its efficiency is nowhere near the theoretical Carnot cycle diagram, the resulting pressure/volume graph shown below does correspond to experimental results.
A pressure/volume graph of the Stirling cycle.
The real advantage of the model is that we can analyze the physics within the heat pump. For example, the animation below indicates velocity distribution during the operation of the heat pump.
Velocity distribution during the heat pump’s operation.
Since the piston brings mechanical energy needed to pump heat, we can also investigate the dynamic temperature distribution within the heat pump during operation.
An animation depicting temperature distribution.
If you are looking to improve the efficiency of the Stirling engine, the goal is to maximize the enclosed area in the pressure/volume graph. This area corresponds to the work that is done by the engine. The engine’s overall efficiency can be improved in multiple ways. If we choose a working gas that features a high specific gas constant (i.e., a low molar mass), this will maximize the isothermal expansion (and therefore the work ability) of the engine. This is why hydrogen and helium are preferred as working gases. Another possibility is to maximize the heat that is transported by the displacer through using a porous medium as a regenerating displacer (see this paper).
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The development of drilling techniques during the last century provides the capability of spreading wells into several branches, among other investigations. This enhanced method allows a much wider reservoir area to be covered, either for oil and gas or for geothermal energy production.
Natural resource recovery: Oil is produced from a natural reservoir via mechanical pumps that raise the oil to the surface. A sidebranch of the drilling reaches into an adjacent reservoir by a junction.
However, branching junctions are also less stable than single boreholes and have to be stabilized with a liner or casing quite often. Even if the borehole would stay open by itself after a drilling procedure is completed, it undergoes additional stress when the production process is engaged by starting pumping through the wellhole. Borehole casing is very expensive, it adds significantly to the overall cost for a drilling project. If possible, one would rather avoid the deployment of additional mechanical stabilization.
The question is: Is the open borehole stable without casing?
A chain is only as strong as its weakest link. To answer the question of stability, it is sufficient to reduce the investigation to the most critical point: the branching junction (or one of the junctions, if there are more branches). If the well is stable here, we may assume that it will hold up everywhere else. As a first approach, we assume only elastic deformation. Thus, all we need to build the model is the Subsurface Flow Module and the Solid Mechanics physics interface in COMSOL Multiphysics.
There is no need to take the fluid flow within the well into account. All we need to know is the pressure level at the well boundaries. This is prescribed by the pump and the depth of the junction. Also, the local symmetry of the part of interest allows us to reduce the geometry to one half and apply symmetry boundary conditions.
Model geometry of the multilateral well branching junction. Due to symmetry, only one half of the junction needs to be considered.
The subsurface flow within a porous media reservoir can be described by a combination of Darcy’s Law and a continuity equation:
with the permeability, , the dynamic viscosity, , and the oil pore pressure, . The pressure in the pores will change when the production process starts; a lower pressure at the walls of the well due to pumping leads to flow in the pores and a pore pressure drop.
Changes in the pressure load effect a deformation because the Cauchy stress tensor, , is related to :
with the elasticity matrix , which depends on Young’s modulus, , and Poisson’s ratio, , the strain tensor, and the BiotWillis coefficient, . This relation couples the fluid flow with the equation for the quasistatic deformation
where denotes any external body forces.
The only exit for the oil is the well. Thus, the boundaries perpendicular to the well axis (top and bottom) are assumed to provide No Flow conditions. As I mentioned above, the symmetry face provides symmetry conditions in both the flow and mechanical interfaces. The model constrains movement at all external boundaries, while the well opening is free to deform.
Variable  Description 
Value 

Fluid density 
0.0361 lb/in^{3} 

Solids density 
0.0861 lb/in^{3} 

Fluid dynamic viscosity 
1·10^{7} psi·s 

Permeability 
1·10^{13} in^{2} 

Young’s modulus 
0.43·10^{6} psi  
Poisson’s ratio 
0.16  
Reservoir pressure 
122.45 psi  
Well pressure 
0 psi 
Model properties.
To get an impression of the results, let’s have a look at a remirrored result plot showing the total displacement due to the pressure drop and the resulting velocity field. So far, we know that the most critical part seems to be the area right above the branching. But we still do not know if the well is stable.
Surface displacement (color plot) and velocity field (arrow plot) of the branching junction.
The results of the model show how the pressure change affects the stress and strain distribution around the well. To find out if the hole is stable or not, we need a failure criterion. The 3D Coulomb criterion (a linearized form of the Lade criterion) reproduces the laboratory data of rock failure experiments well, but note that it requires calibration data.
The criterion relates rock failure, the three principal stresses (, , and ), and the fluid pressures as follows:
where
with the Coulomb cohesion, , and the Coulomb friction angle, . For different results of the function, various scenarios can be expected: predicts stability; indicates the onset of rock failure; and denotes catastrophic failure.
Evaluation of the fail function (equation anchor: failure). The highest potential for failure is right above the branching.
When the fail values become increasingly negative, the potential for failure is higher. As expected, the fail function estimates show the greatest potential for failure just above the branching in the well. The failure potential can be reduced by adding a support force, e.g., a casing.
Below, you can see the results of a parametric sweep study. With increasing well support force, the failure potential decreases.
Evaluation of the fail function for different well support forces from a parameter study (support force from left to right: 5e^{4}, 7.5e^{4}, 1e^{5}, 2.5 ).
Here, we have showed you how to model a branching junction of an openhole multilateral well. As the stability of open wells is an essential piece of information, this model is a good example of how numerical simulations can significantly save money. The important estimation of stability can be clarified in the early project phase.
The presented approach focuses on elastic deformation, while related analyses can also be conducted for elastoplastic materials, for example. These are automated in the Geomechanics Module and in the Structural Mechanics Module. This model could also easily be expanded to study the influence of temperature by adding thermal expansion. This would be especially of interest for studies concerning geothermal energy approaches.
Modern building construction requires efficient climate control, preferably using sustainable energy resources. Such resources can be solar collector systems for warmwater support or ambient air or geothermalbased heat pump systems that can be used for cooling in the summer and heating in the winter.
The geothermal applications mostly consist of pipe installations with water or brine as fluid to provide the heat exchange between the subsurface and a heat pump. There are two main types of these socalled closedloop systems: vertical borehole heat exchangers (BHEs), which are installed in boreholes (see part 1 of this series); and horizontal heat collectors, which are deposited on large areas and in very shallow depths of only one or two meters.
While the vertical BHEs are often simplified as infinite thermal line sources or sinks, the horizontal systems are more complicated from a modeling point of view. This is because they have to be arranged in patterns to cover large surface areas, and the use of this simplification does not match anymore. The total length of the pipes can easily reach 100 meters or more. The flow regime is preferably turbulent because of the resulting lower thermal resistivity between the fluid and the subsurface. You can imagine that a CFD simulation approach of systems with these dimensions is a challenge (despite the computational performance of modern HPC clusters). Fortunately, there is a solution for this kind of task.
Let’s have a look at how the Pipe Flow Module can be used in the growing sector of geothermal heat collector systems.
Thankfully, we do not have to care about the computational difficulties that may arise when simulating long pipe systems. The Pipe Flow Module provides all necessary capabilities to calculate pressure drops, velocities, and foremost — the heat transport between the pipe fluid and the subsurface, using prescribed 3D curvededge correlation functions. All we need to do is to define the inlet conditions (temperature, velocity) and the pipe, fluid, and subsurface properties. Therefore, we can completely focus on questions like, “What do you think is the best horizontal pipe arrangement underneath your garden?”
Let’s find out.
The heat transfer example, provided in a recent blog post, shows how a prescribed heat extraction rate can be realized in the form of a numericallycalculated inlet temperature boundary condition. By doing so, any desired heat extraction rate can be given and the corresponding inlet temperature can be computed. Assuming that the fluid properties are temperatureindependent, a simplified version of the inlet temperature can be written as:
(1)
Thus, it is a function of the outlet temperature (), density (), heat capacity (), and volumetric flow rate () of the working fluid in the pipes. The temperature difference between the inlet and outlet temperature is controlled via the heat extracted by the heat pump, , which is a function of time because heat pumps (for house heating purposes) usually do not run all day. A typical heating rate of a onefamily house heat pump is 8 kW, where 2 kW of electric power is needed to run it and 6 kW are to be extracted from the subsurface. To reach a daily heat demand of, say, 48 kWh during the winter season in Germany or North America, the heat pump needs to run six hours per day.
6 kW of heat extraction for six hours per day, in a threeday cycle.
The number of possibilities of how the heat collector pipes can be arranged in the subsurface is endless. Here, we will have a look at three randomly chosen patterns. Let’s call them snake, snail, and meander designs. The heat collector is embedded in the subsurface with typical thermal properties that you could find in the uppermost soil layer in your garden. The subsurface temperature distribution corresponds to that of the temperature in Germany during the month of January.
Three different layer designs for a geothermal heat collector system: 1) snake, 2) snail, and 3) meander design.
The COMSOL Multiphysics feature, geometry subsequence, allows us to create all of the three different pipe geometries within the same model. It also allows us to perform a parametric study that solves for each particular subgeometry. We can compare the different designs by having a look at the resulting outlet temperature as a function of time.
Comparison of the outflow temperatures of the three collector designs.
The three collectors each show a different thermal behavior. This is due to the varying distances between the pipes, their different lengths, and thereby the surface available for heat transfer. Even though the freezing point of the waterglycol mixture inside the pipes is at about 13°C, the outflow temperature is — for environmental reasons — legally limited to stay above 5°C in Germany.
The snake design tends to go below this point only after a few days and would, thus, not be the preferred design here. However, a geothermal engineer could now proceed to optimize a design (modify the pipe length, pipe diameter, or the covered area to reach a maximum outlet temperature, for example.) One could, of course, also easily perform long time runs to study the impact of seasonal temperature variations or even use the pipes for cooling purposes in the summer.
Once the optimal design of the geothermal heat collector in your garden is found, why not go on and model the modern underfloor heating system in the house? The procedure is straightforward. There is only a change of sign because heat is injected here.
Combined model of a garden heat collector together with an underfloor house heating system. The collector extracts heat from the subsurface at 6 kW, while the house heating system injects heat into the floor of the house at 30°C.
One of the greatest challenges in geothermal energy production is minimizing the prospecting risk. How can you be sure that the desired production site is appropriate for, let’s say, 30 years of heat extraction? Usually, only very little information is available about the local subsurface properties and it is typically afflicted with large uncertainties.
Over the last decades, numerical models became an important tool to estimate risks by performing parametric studies within reasonable ranges of uncertainty. Today, I will give a brief introduction to the mathematical description of the coupled subsurface flow and heat transport problem that needs to be solved in many geothermal applications. I will also show you how to use COMSOL software as an appropriate tool for studying and forecasting the performance of (hydro) geothermal systems.
The heat transport in the subsurface is described by the heat transport equation:
(1)
Heat is balanced by conduction and convection processes and can be generated or lost through defining this in the source term, . A special feature of the Heat Transfer in Porous Media interface is the implemented Geothermal Heating feature, represented as a domain condition: .
There is also another feature that makes the life of a geothermal energy modeler a little easier. It’s possible to implement an averaged representation of the thermal parameters, composed from the rock matrix and the groundwater using the matrix volume fraction, , as a weighting factor. You may choose between volume and power law averaging for several immobile solids and fluids.
In the case of volume averaging, the volumetric heat capacity in the heat transport equation becomes:
(2)
and the thermal conductivity becomes:
(3)
Solving the heat transport properly requires incorporating the flow field. Generally, there can be various situations in the subsurface requiring different approaches to describe the flow mathematically. If the focus is on the micro scale and you want to resolve the flow in the pore space, you need to solve the creeping flow or Stokes flow equations. In partially saturated zones, you would solve Richards’ equation, as it is often done in studies concerning environmental pollution (see our past Simulating Pesticide Runoff, the Effects of Aldicarb blog post, for instance).
However, the fullysaturated and mainly pressuredriven flows in deep geothermal strata are sufficiently described by Darcy’s law:
(4)
where the velocity field, , depends on the permeability, , the fluid’s dynamic viscosity, , and is driven by a pressure gradient, . Darcy’s law is then combined with the continuity equation:
(5)
If your scenario concerns long geothermal time scales, the time dependence due to storage effects in the flow is negligible. Therefore, the first term on the lefthand side of the equation above vanishes because the density, , and the porosity, , can be assumed to be constant. Usually, the temperature dependencies of the hydraulic properties are negligible. Thus, the (stationary) flow equations are independent of the (timedependent) heat transfer equations. In some cases, especially if the number of degrees of freedom is large, it can make sense to utilize the independence by splitting the problem into one stationary and one timedependent study step.
Fracture flow may locally dominate the flow regime in geothermal systems, such as in karst aquifer systems. The Subsurface Flow Module offers the Fracture Flow interface for a 2D representation of the Darcy flow field in fractures and cracks.
Hydrothermal heat extraction systems usually consist of one or more injection and production wells. Those are in many cases realized as separate boreholes, but the modern approach is to create one (or more) multilateral wells. There are even tactics that consist of single boreholes with separate injection and production zones.
Note that artificial pressure changes due to water injection and extraction can influence the structure of the porous medium and produce hydraulic fracturing. To take these effects into account, you can perform poroelastic analyses, but we will not consider these here.
It is easy to set up a COMSOL Multiphysics model that features long time predictions for a hydrogeothermal application.
The model region contains three geologic layers with different thermal and hydraulic properties in a box with a volume V≈500 [m³]. The box represents a section of a geothermal production site that is ranged by a large fault zone. The layer elevations are interpolation functions from an external data set. The concerned aquifer is fully saturated and confined on top and bottom by aquitards (impermeable beds). The temperature distribution is generally a factor of uncertainty, but a good guess is to assume a geothermal gradient of 0.03 [°C/m], leading to an initial temperature distribution T_{0}(z)=10 [°C] – z·0.03 [°C/m].
Hydrothermal doublet system in a layered subsurface domain, ranged by a fault zone. The edge is about 500 meters long. The left drilling is the injection well, the production well is on the right. The lateral distance between the wells is about 120 meters.
COMSOL Multiphysics creates a mesh that is perfectly fine for this approach, except for one detail — the mesh on the wells is refined to resolve the expected high gradients in that area.
Now, let’s crank the heat up! Geothermal groundwater is pumped (produced) through the production well on the right at a rate of 50 [l/s]. The well is implemented as a cylinder that was cut out of the geometry to allow inlet and outlet boundary conditions for the flow. The extracted water is, after using it for heat or power generation, reinjected by the left well at the same rate, but with a lower temperature (in this case 5 [°C]).
The resulting flow field and temperature distribution after 30 years of heat production are displayed below:
Result after 30 years of heat production: Hydraulic connection between the production and injection zones and temperature distribution along the flow paths. Note that only the injection and production zones of the boreholes are considered. The rest of the boreholes are not implemented, in order to reduce the meshing effort.
The model is a suitable tool for estimating the development of a geothermal site under varied conditions. For example, how is the production temperature affected by the lateral distance of the wells? Is it worthwhile to reach a large spread or is a moderate distance sufficient?
This can be studied by performing a parametric study by varying the well distance:
Flow paths and temperature distribution between the wells for different lateral distances. The graph shows the production temperature after reaching stationary conditions as a function of the lateral distance.
With this model, different borehole systems can easily be realized just by changing the positions of the injection/production cylinders. For example, here are the results of a singleborehole system:
Results of a singleborehole approach after 30 years of heat production. The vertical distance between the injection (up) and production (down) zones is 130 meters.
So far, we have only looked at aquifers without ambient groundwater movement. What happens if there is a hydraulic gradient that leads to groundwater flow?
The following figure shows the same situation as the figure above, except that now there is a hydraulic head gradient of H=0.01 [m/m], leading to a superposed flow field:
Single borehole after 30 years of heat production and overlapping groundwater flow due to a horizontal pressure gradient.
During my time as a PhD student, I had to deal with different kinds of geothermal energy extraction techniques. The focus of my research was on shallow geothermal borehole heat exchangers, but within my team, I also came in contact with open borehole and deep geothermal approaches. To my surprise, I found that almost all of the processes involved in the various geothermal applications could be modeled with the Subsurface Flow Module.
Renewable energies are a growing industry and the geothermal energy branch is a hot topic of active research. Over the past few decades, different techniques were established to extract geothermal heat from shallow to deep subsurface levels. The closedloop borehole heat exchanger (BHE) is a standard approach for lower and middepth applications.
In a BHE, fluid circulates through pipes that are located inside of boreholes, resulting in indirect thermal contact between the fluid and the subsurface. This method was restricted by the closed nature of the heat exchanger with the effective heat exchange area limited by the equipment involved. This could could be significantly increased by extracting groundwater from aquifers, gaining thermal energy through the reinjection of hot fluid a distance from the injection point through hydrothermal doublets.
Enhanced geothermal systems (EGS) were developed to exploit geothermal resources in dry and impermeable rocks or hard dry rocks (HDR), by hydraulic fracturing, or fracking. This procedure involves pumping highpressure water into the desired layer to establish new cracks and widen existing cracks and fractures. Injected water can then travel through the fractures, heat up, and be extracted at a second borehole.
Geothermal heat extraction methods.
Heat transport in the subsurface proceeds by convection, dispersion, and conduction. You need to know the thermal properties of the geological layers in order to run simulations. However, we can usually only obtain vague estimations based on geological maps and core samples. Convective heat transport, either naturally driven by buoyancy forces or artificially established through the wells, can play a role in — and even dominate — the processes.
Depending on the local geology, the subsurface flow regime may be situated in fully or partially saturated porous media, or it can proceed in fractures. Although the different techniques for geothermal heat production differ fundamentally, the Subsurface Flow Module provides the required features to simulate the subsurface thermal development. You can easily couple heat transport with the velocity field.
In some situations, the coupling must be applied bidirectionally. If the temperature gradients are high, the temperature dependencies of involved parameters (hydraulic conductivity, for example) are no longer negligible and must be taken into account. Also, poroelastic processes may play a role in some cases, especially when it comes to fracking.
Let’s have a look at an example to demonstrate some of the features that are necessary to simulate geothermal processes. The following model solves for the heat transport around a shallow geothermal installation embedded in a geological domain. The domain is separated into various parts, representing different geological layers with their particular properties. The thermal influence of seasonal temperature changes at the surface is taken into account using the COMSOL Multiphysics internal ambient weather database.
A threebythree borehole heat exchanger (BHE) array 135meter deep, located in layered bedrock. Each BHE extracts heat at 20 W/mK throughout the year. Between 6070 meters is an aquifer where groundwater flow occurs, causing horizontal convective heat transport. Shown on the right are the borehole wall temperatures of the three BHEs in the middle of the array. Due to the thermal interaction between the heat sinks, the temperature of middle BHE (green line) is lower than the other two. In the region of the aquifer, the BHE further downstream than the other two (red line) is affected by the heat exchange occurring upstream of it, resulting in a lower temperature.
Simulation is necessary for the prediction of longterm impacts on the BHEs in order to identify whether the pipe will avoid freezing. The most straightforward way to quickly model the BHEs is to neglect the glow and heat transfer within the borehole and apply an appropriate heat flux boundary condition at the walls. In doing so, the borehole becomes a local heat sink and heat will travel towards it. If more than one BHE is installed at a local site, the heat exchangers might start to interact after some time. In particular, if groundwater flow in an aquifer is present, the boreholes will connect thermally. This thermal interaction can cause significant efficiency losses of the whole geothermal system. On the other hand, groundwater flow also increases the thermal recovery rate. Reliable predictions can only be made if the geological data set is accurate enough.
The heat transfer through the BHEs over a period of one year. The top surface temperature changes seasonally between about 2°C and 22°C according to meteorological weather data from Berlin, Germany. Notice how the heat pattern elongates in the direction of flow in the region of the aquifer.
Today, we have introduced how you can use multiphysics simulation in geothermal energy applications. Next up in this blog series we will present a more advanced application, involving coupling heat transfer in heat pipe arrays with subsurface porous media flow. Stay tuned!