One way to simplify the designs of key process equipment, such as rotating packed bed reactors and centrifugal disc atomizers, is to replace conventional centrifugal pumps with a less complicated alternative: rotating cone micropumps.

*A conventional centrifugal pump (left) and rotating cone micropump (right). Images by A. Uchagawkar, M. Vasilev, and P. Mills and taken from their COMSOL Conference 2015 Boston presentation.*

Before they can be used as replacements, rotating cone micropumps need further analysis. In particular, research on the velocity profiles over the rotating conical surfaces is quite important in process manufacturing industries. Such studies show that fluid velocity distributions greatly affect the efficiency of process equipment.

To fill this research gap, a team from Texas A&M University used COMSOL Multiphysics CFD simulations to accurately investigate the pump performance of a rotating cone micropump — a more efficient approach than trial-and-error empirical studies.

The researchers used COMSOL Multiphysics to develop a realistic fluid dynamics model of a rotating cone micropump, which analyzes both laminar and turbulent flow regimes using the 3D transient Navier-Stokes equations.

The researchers looked at how the flow and pressure fields are affected by changing the micropump’s geometrical and operational parameters, including:

- Cone height and semiangle
- Ratio of outer to inner radius
- Angular rotational speed

As seen in the following schematic, the model geometry is comprised of a vertical and rotating inner cone, rotating inner solid cone, and stationary outer cone. The fluid in this model is water.

*Rotating cone micropump geometry. Here,* H *is the height of the cone,* α *is the cone semiangle, and* R_{0} *is the upper radius. Image by A. Uchagawkar, M. Vasilev, and P. Mills and taken from their COMSOL Conference 2015 Boston presentation.*

The research team used an unstructured mesh with tetrahedral elements and tested different element sizes to see how the number of elements (mesh resolution) affects the computation results. The results show that the computed pressure head varies to a very small extent with the number of elements (48,000; 92,000; and 124,000), indicating that the results are within the required accuracy for the lowest number of elements (48,000).

*Tetrahedral mesh with 48,000 elements. Image by A. Uchagawkar, M. Vasilev, and P. Mills and taken from their COMSOL Conference 2015 Boston presentation.*

To expand the reach of this study, the researchers created a dedicated user interface for running the model (an app) using the Application Builder in COMSOL Multiphysics. For more information about the researchers’ model and simulations, check out the full paper.

Let’s go over a few of the research team’s results now, beginning with the fluid velocity and pressure profiles for rotating cone micropumps with two different cone semiangles. The plots below show that the velocity magnitude remains below 0.1, which for this system corresponds to a Reynolds number of around 1.5. Within this range of Reynolds numbers, viscous forces are the main driver of flow for micropumps. As for the velocity patterns, these switch from the axial direction at the inlet to the angular direction at the angled region. Upon leaving the cone’s angled region, the velocity transitions into a combination of angular and axial behavior.

*Velocity profiles for a rotating cone micropump with a semiangle of 12° (left) and 45° (right). Both cones give a volumetric flow rate of 1 ml/s. Images by A. Uchagawkar, M. Vasilev, and P. Mills and taken from their COMSOL Conference 2015 Boston paper.*

Moving on to the corresponding fluid pressure profiles, these indicate that the micropump’s hydraulic head is a weak function of both the cone semiangle and rotating cone’s height. While the micropump head is greatly affected by the frequency of rotation, the hydrodynamic head remains below 135 Pa, even at the maximum rotational speed of 12,000 RPM.

*Fluid pressure profiles for a rotating cone micropump with a semiangle of 12° (left) and 45° (right). Both cones have a volumetric flow rate of 1 ml/s. Images by A. Uchagawkar, M. Vasilev, and P. Mills and taken from their COMSOL Conference 2015 Boston paper.*

The researchers also predicted the head curve for various angular rotational speeds by repeatedly running the model for each volumetric flow rate and angular speed pair as well as calculating outlet pressure. The outlet pressure shows a near linear decrease with an increasing volumetric flow rate. The micropump head is also almost proportional to the square of the rotational speed.

*Comparison of the micropump head with different RPM values. Image by A. Uchagawkar, M. Vasilev, and P. Mills and taken from their COMSOL Conference 2015 Boston paper.*

Other factors, including the fluid’s viscosity and density, can also affect the pressure head that the rotating cone micropump produces. To demonstrate, two head curves for liquids with different viscosities and densities and the same operating conditions are compared below. Water, the more viscous and dense fluid, generates a larger pressure difference throughout the range of flow rates than the other fluid, diethyl ether. For conventional centrifugal pumps, the opposite is true. However, the pressure head behavior trends for water and diethyl ether are similar and agree with those for centrifugal pumps.

*Comparison of pressure head curves for water and diethyl ether at 12,000 RPM. Image by A. Uchagawkar, M. Vasilev, and P. Mills and taken from their COMSOL Conference 2015 Boston paper.*

Overall, rotating cone micropumps can create comparatively large throughputs, but the pressure heads that they create don’t compare to those of conventional centrifugal pumps. Therefore, this rotating cone micropump design is best suited for applications requiring small pressure heads like microprocess systems. In these cases, rotating cone micropumps are a good choice due to their simplicity and performance.

Moving forward, the team notes that they can optimize their rotating cone micropump design. By using CFD simulations, they can evaluate the effects of adding modifications to the cone head surface, such as spiral fins. These results can then be compared with empirical data.

- Get more details about the research: “Hydrodynamic Modeling of a Rotating Cone Pump Using COMSOL Multiphysics® Software“
- Watch an 18-minute archived webinar on CFD simulation
- Browse additional CFD blog posts:

Spending time in a hot building in the summer isn’t pleasant, and depending on the temperature, it can even be dangerous to your health. This is especially true in hot climates like the Mediterranean, where large amounts of solar radiation generate hot temperatures indoors and outside. As a result, recent European environmental policies state that maximizing indoor thermal comfort while minimizing cooling consumption is a key goal.

*Sun hitting a roof. Image courtesy of ICP.*

While there are many strategies for enhancing the thermal performance of buildings, we’ll focus on optimizing roofs. To understand why roofs are important, picture a house on a sunny day. As solar radiation beams down from above, it hits the roof on top of the building. The heat transfers through the roof’s surface, causing the house to heat up.

This effect is amplified because a roof has a larger surface area than other parts of a building. Also, its angle of incidence is conducive to being struck directly by the sun’s rays, thus increasing its impact on the heat management of the entire house. Due to these factors, researchers are looking to optimize roof designs to reduce the energy required for space cooling and air conditioning.

*A ventilated, pitched roof made of Marsigliese tiles. Image courtesy of UNIFE.*

One way to reduce energy costs is to use a tilted or pitched roof along with a system of battens and counterbattens to create a ventilation layer beneath overlapping roof tiles. The ventilation layers allow air to flow from the eaves section (an intake vent) to the ridge. This promotes insulation to reduce heat transfer between the roof tiles and the house’s structure. It also increases heat dissipation via the airflow. Using this strategy, called above-sheathing ventilation (ASV), technicians reduce the energy required to cool a house. Further, the air permeability of the overlapping tiles also functions as another intake/exhaust air-vent system.

ASV can be improved by increasing the air permeability, and thus airflow rate, between tiles. This is the goal of the European Life HEROTILE project, which seeks to enhance the energy efficiency of buildings by designing superior roof tiles.

*Logo for the Life HEROTILE project.*

While this idea came from the Department of Architecture at the University of Ferrara (UNIFE), it has expanded into the Life HEROTILE project. The project spans multiple countries and partners, such as the Industrie Cotto Possagno, Monier Redland Ltd, Terreal SAS, ACER, and ANDIL.

Today, we’ll focus on a preliminary study by researchers from the University of Ferrara that analyzes the performance of novel tile shapes.

In the preliminary stages of this project, researchers investigated two popular roof tile shapes, Portoghese and Marsigliese, and focused on optimizing their shape. Using these two known shapes, the team created over 20 new designs and performed CFD analyses comparing the air permeability of the new and standard tiles. UNIFE’s team noted that “COMSOL [Multiphysics® software] was fundamental in the initial phases of [the] HEROTILE project.”

*A Portoghese roof tile prototype. Image courtesy of Monier Technical Centre GmbH.*

The team verified the model and calibrated its parameters using experimental measurements from a highly controlled test rig. The geometry was created using a CAD package and then imported into COMSOL Multiphysics using the CAD Import Module. The three-dimensional CFD model was defined using incompressible turbulent flow, which describes the airflow from the wind through the tiles.

*Left: The modeling domain in the study consists of four unsealed tiles in the test rig and their equivalent borders. Right: Mesh details of the modeling domain. Images courtesy of UNIFE.*

Let’s focus on one design in particular: a novel Marsigliese roof tile featuring a higher sidelock and new headlock pattern. Using the CFD simulation, the team ran parametric studies to analyze the airflow variations passing through the tile at various incident wind speeds and directions.

*Marsigliese roof tile models. Green indicates the design variations. Images courtesy of UNIFE.*

To evaluate the performance of the tile designs, the researchers analyzed factors including:

- Wind speed
- Roof slope
- Horizontal wind direction

Together, they studied a total of 30 different wind condition scenarios. They also compared how the tiles reacted to changes in air pressure drops and volumetric flow rates through the tiles.

*The researchers studied a variety of roof slopes, wind intensities, and wind directions. Image by the research team and taken from their COMSOL Conference 2016 Munich presentation.*

The team noted that when horizontal wind angles changed from blowing directly up the roof slope to blowing sideways, the airflow rate increased for the novel design and decreased for the standard Marsigliese design. By investigating the pressure difference between an external and internal point probe, the UNIFE team also found that the airflow rate of the standard tile design diverged clockwise, while the new shape diverged counterclockwise.

*Velocity magnitude surrounding the roof. Images taken from Bortoloni et al., Summer Performance of Ventilated Roofs With Tiled Coverings, 34 ^{th} UIT-Heat Transfer Conference, Ferrara, 2016.*

Further results confirmed that the new Marsigliese tile shape generates higher airflow rates than the standard shape due to the new sidelock geometry. This form acts like an intake vent, increasing airflow through the tile’s side.

*The airflow streamlines on the roof tiles. Image courtesy of UNIFE.*

As for air permeability, when compared to the standard shape, the novel shape and headlock pattern increase air permeability by up to 100%. As such, the researchers achieved their goal of designing an innovative tile shape that offers improved air permeability.

Novel roof tile designs can help reduce the significant environmental and fiscal costs that come with indoor air conditioning. The UNIFE team considers “the first phase of the project a success” and that the “results of [the] COMSOL [Multiphysics] simulations had [a] strong effect” on the project.

For instance, using this research, the Life HEROTILE project began real-scale testing of their designs as part of their Action C.3 phase. In this phase, the team plans to evaluate the energy and waterproofing performance of their novel tile designs and compare them to standard tiles and other types of waterproof materials.

*The roof tile testing site in Ferrara, Italy. Image courtesy of the University of Ferrara.*

The team is collecting data from two monitoring stations at climate-appropriate mock-ups in Ferrara, Italy and Yeruham, Israel. Each of these mock-ups consists of two buildings, one with a pitched roof and the other with a flat roof. The pitched roof is divided into sections with different coverings. In the Ferrara mock-up, there is a section for the standard Marsigliese tiles as well as the new HEROTILE Marsigliese tiles discussed in this blog post. This mock-up is also equipped with the standard Portoghese tiles and the Portoghese version of the new HEROTILE tiles.

At this stage, the team hopes to create a comprehensive database of experimental tests to match their simulation data.

- Read the full paper from the COMSOL Conference 2016 Munich: “The Design of a Novel Roof Tile Shape Using CFD Analysis“
- Take a look at other applications of heat transfer analysis on the COMSOL Blog:

Bubble entrapment is a common problem in microfluidic devices, as bubbles often become stuck in microchannels. This interrupts the path of the fluid, creating disruptions in the flow and negatively affecting the performance of the device. For example, the presence of bubbles can result in incorrect readings in microchannel sensors or block the formation of jets in inkjet printers. Bubbles can’t always be avoided or removed by design, so the solution is to prevent bubbles from getting stuck in the microchannels of microfluidic devices.

*A simple microfluidic device.*

Preventing bubble entrapment starts with the design of the microfluidic device. There are several factors influencing bubble movement through the microchannel, including the geometry of the channel, surface properties of the walls, and fluid flow characteristics. To better understand the effect of these aspects, researchers can use simulation to find the optimal conditions for ensuring that bubbles successfully pass through the microchannel. They can then design microfluidic devices that have a reduced risk of bubble entrapment.

To study these factors, Veryst Engineering — a COMSOL Certified Consultant — modeled a bubble moving through a microchannel with COMSOL Multiphysics.

For their analyses, Veryst Engineering created a model using the level set method in the CFD Module, an add-on to COMSOL Multiphysics, and the *Multiphase Flow* interface. They chose to model a 0.3-mm bubble in a 1-by-2-mm microchannel that is mildly restricted in the middle. The bubble’s initial position is near the bottom of the channel and it moves upward with the fluid flow. The average fluid speed in the microchannel is ramped up from 0 to 50 mm/s in the first 10 microseconds of the simulation.

*The bubble in the microchannel geometry. Image courtesy of Veryst Engineering.*

For their model, the engineers first took the surface tension between the bubble and surrounding fluid into account. They also accounted for the surface properties of the microchannel walls, such as whether they were hydrophilic, neutral, or hydrophobic. The engineers assumed that the walls have a neutral contact angle with the bubble, except for in the constrained section of the microchannel.

Note that contact angle hysteresis — the difference between the advancing and receding contact angles — increases the chance of the bubble getting stuck. This effect was not taken into account in the model.

The team from Veryst then modeled the bubble at two different contact angles. The first simulation shows the bubble at a 22.5º contact angle with the microchannel wall. As seen below, the bubble becomes stuck near the end of the constriction. This forces the fluid to move around the obstacle, resulting in a nonsymmetric velocity field.

*The bubble at a 22.5º contact angle with the channel wall. Toward the end of the constriction, the bubble becomes stuck. Animation courtesy of Veryst Engineering.*

In the second simulation, the bubble is at a 90º contact angle. It now moves smoothly through the constriction and continues moving through the microchannel.

*The bubble at a 90º contact angle with the wall. The bubble successfully passes through the constriction. Animation courtesy of Veryst Engineering.*

The bubble’s average speed in both simulations is compared below. As it moves through the constriction, the bubble’s speed quickly increases due to the reduction of the channel area, as shown between 0.05 and 0.1 seconds.

*Comparison of the bubble’s average velocity for two different contact angles. Image courtesy of Veryst Engineering.*

The Veryst engineers also compared the model predictions to analytical estimates. They showed that for this simple channel geometry and bubble size, a 22.5° contact angle at the constriction leads to bubble entrapment, while a 90° contact angle does not.

After finishing these simulations, the Veryst engineers applied this approach to a more realistic microchannel geometry. Their new results provided insight into various geometries and contact angles that result in bubble entrapment.

Knowing more about these factors can help to optimize the design of microfluidic devices that experience bubble entrapment. This in turn makes it easier for those who are working with these devices. With less chance of bubble entrapment, a microfluidic device’s performance is more reliable.

- Learn more about Veryst Engineering
- Explore other blog posts about how Veryst uses multiphysics simulation:
- See other examples of modeling microfluidic applications on the COMSOL Blog

The purpose of fat-washing is to extract flavors contained in fats and oils from meat and vegetables. In the fat-washing process, some of these flavors can be dissolved in ethanol, which has the ability to dissolve both hydrophilic and hydrophobic solutes. For example, water-soluble lemon juice can be mixed with the fatty acids from pecan butter to create new flavors.

The ability of ethanol to dissolve both polar and nonpolar solutes is what gives wine and liquor their flavor complexity. It is also the main reason for the variety of tastes that can be obtained when using ethanol as a solvent. For instance, the use of oak barrels for distilling whiskey and making wine adds complexity to their flavor due to the ethanol that dissolves the solutes contained in the oak. These solutes simply can’t be dissolved by water.

*Fragrances and flavors may be extracted by mixing solvents like ethanol with oils.*

The phenomena and processes involved in making fat-washed liquor are also found in industrial processes for the household product, cosmetic, pharmaceutical, and food industries. These include processes for recovering vitamins, separating flavors and fragrances, decaffeination, and many more applications.

Let’s look at the steps for fat-washing liquor and how they can be scaled up to an industrial process.

The first step of the fat-washing process is to melt or dissolve the meat or vegetable fat. If there is a solid matrix, we also have to extract the fatty flavors from this matrix. Extracting the flavors from an oak barrel is an example of this process, called leaching.

To melt the fat in an industrial process, we can use a mixer equipped with a heating mantle. As for the extraction via leaching, it’s possible to use methods similar to those used when making whiskey, but in most cases, we can simply grind the solids to increase the extraction rate. For example, toasted oak chips are used to make oak-infused vodka.

If needed, the next step is to remove the solids from the molten fat or oil. Bartenders can do this with a strainer — a process that is referred to as filtration. At an industrial scale, we can replace the strainer with a belt filter, which has a filter cake that can be removed. To avoid the oil solidifying into fat, the filtration needs to be done at an elevated temperature.

*The Hawthorne strainer is an important piece of equipment for the unit operations involved in cocktail making.*

After filtering the oil, a bartender pours it into a shaker and mixes it with liquor. To scale this process up, we would have to feed the liquor and oil into an extraction column.

During the extraction of flavors, we have a two-phase system: an oil (molten fat) phase that may dissolve hydrophobic flavors and an ethanol phase that may contain both hydrophobic and hydrophilic flavors. Now, we have to transport the flavors in the oil phase to the ethanol phase, a process referred to as liquid-liquid extraction. During liquid-liquid extraction, we need to maximize the size of the phase boundary per unit volume. We can achieve this by creating the smallest oil droplets possible in the ethanol, hence the importance of mixing during extraction.

A bartender simply shakes the oil and ethanol mixture. This creates a lot of droplets and also enhances the transport of solutes within each phase.

*To get an efficient liquid-liquid extraction of the flavors in the oil, we need to create small droplets of oil in the ethanol solution. Image by JD. Licensed under CC BY 2.0, via Flickr Creative Commons.*

In an industrial process, where larger volumes are used, we can replace the shaking by letting the oil and ethanol flow through a liquid-liquid extraction column, such as a pulsed column. A complication in scaling up a bartender’s process is temperature. In an industrial column, some of the oil has time to solidify into fat. This means that the extraction column must be kept at a temperature high enough to prevent the oil from solidifying.

To break the oil droplets into smaller droplets, we need to apply forces that exceed the surface tension that holds each droplet together. We can achieve this in an extraction column by pulsating the ethanol phase from top to bottom and the oil phase from bottom to top. Just like a bartender’s shaking motion, the pulsating enhances the transport of solutes to and from the droplets’ surfaces.

At the phase boundary between the oil and ethanol, we have a very rapid interchange of solutes. This process is so fast that it is usually assumed to be at equilibrium. The solute concentration in the oil and ethanol phases on each side of the phase boundary is related according to the partition constant:

\[{P_{oil/ethanol}} = \frac{{{c_{oil,eq}}}}{{{c_{ethanol,eq}}}}\]

where *P* denotes the partition coefficient and *c* is the concentration of a solute.

If we examine one droplet at a microscopic scale, the rate of the solute’s transport between phases is determined by the transport from the bulk of the oil droplet to the phase boundary and from the phase boundary to the bulk of the ethanol phase. We need a very large surface area for the phase boundary to also get a large cross-sectional area for the solute flux moving to and from this boundary.

According to the research of Gibbs, we have a surface phase that is a mixture of the oil and ethanol molecules, seen in the figure below. On each side of this surface phase, we have an equilibrium concentration of the solutes in the oil and ethanol phases. The relation between the two equilibrium concentrations for each solute is given by the partition coefficient.

When the extraction starts, we have large gradients in the solute concentration both within the oil droplets and in the ethanol phase around the droplets, shown in the left figure below. When the extraction is complete, the concentration everywhere in the droplets and ethanol phase is equal to the equilibrium concentration according to the partition coefficient, as seen in the right figure below.

*An oil droplet (turquoise) in ethanol (blue) at the beginning of extraction (left) and after the extraction is complete (right). The graph shows the concentration along a line that runs perpendicular to the phase interface and across the interface. The oil and ethanol phases are separated by a surface phase that contains molecules from both phases.*

The animation below shows an oil droplet transferring a solute to the continuous phase (ethanol in our case) as it rises. As the bubble rises, it leaves a trace of the solute along its path. The stationary oil on top also contains the solute, but the transport is very slow until the droplet hits the surface, which produces some mixing at the phase boundary.

*An oil droplet rising in ethanol. As it moves, a solute is extracted from the droplet to the continuous ethanol phase. There is also a layer of oil resting on top of the ethanol.*

Studying one droplet, such as the rising oil droplet above, gives us a better understanding of the processes occurring at a microscopic scale. However, even at the scale of a cocktail, we are dealing with thousands of droplets. In such cases, we need to study the liquid-liquid extraction using dispersed two-phase flow models.

In these models, we don’t deal with the exact shape of the interface between the oil and ethanol. Instead, we use variables for the volume fractions, bubble sizes, and specific interface area between the two phases. From the velocity difference between the phases, velocity gradients within each phase, volume fraction, number of droplets, and size distribution, we can estimate a mass transfer coefficient between the oil and ethanol phases. These types of dispersed models are used in the modeling and simulation of industrial liquid-liquid extraction columns.

*A two-phase flow mixture model of a liquid-liquid extraction column that shows where the oil fraction is high (red) and low (blue). On average, the oil fraction is constant over the column, but it’s concentrated at the center of the oil’s inlet and outlet. The flow is countercurrent, with the ethanol flowing downward while the oil flows upward. There is plenty of time for extraction of flavors between the oil and ethanol.*

The liquid-liquid extraction process is finished when the partition of the species that may dissolve in both phases is at equilibrium. This does not mean that the concentration of the flavors is equal in the two phases, since the hydrophobic species always has a higher concentration in the oil phase and the concentration of the hydrophilic species is higher in the ethanol phase.

After the extraction is complete, the two-phase system must be allowed to rest in order for the oil phase to separate to the top of the ethanol as a continuous phase (instead of being a dispersed phase of droplets within the ethanol). Bartenders can achieve this by letting the shaker rest for a while.

In an industrial column, the oil phase is separated at the top as a liquid phase, while the ethanol can be separated at the bottom of the column. The ethanol may still contain small droplets of dispersed oil, but we can use settling basins to separate out the remaining oil.

Once a bartender has two continuous separated phases in the shaker, he or she can put the two-phase liquid in the freezer. The frozen fat can then be removed as a solid cake. Then, the bartender removes any remaining pieces of fat left in the liquor with a tea strainer, leaving a clear liquor as the base for creating the cocktail.

At an industrial scale, freezing the two-phase liquid just to remove the fat is inefficient. Instead, we can use settling basins, continuously removing the liquor at the basin’s bottom and letting an overflow outlet remove the fat from the top. At steady state, the settling basin has a thin film of oil resting on the ethanol, which also has the positive effect of slowing down ethanol evaporation. If some of the oil solidifies into fat, it can be gently scraped off with a surface scraper without agitating the basin.

I must admit that I have not tried to fat-wash liquor myself. I feel a strong resistance against mixing perfectly good whiskey or bourbon with fat from frying bacon or ham. Perhaps I’ll try it with oak-infused vodka instead. That actually sounds delicious and feels closer to the oak flavors already present in whiskey and wine.

*Oak-infused water of life — or whiskey, as we usually call it.*

Then again, maybe I’ll just take my whiskey with an ice cube, while I study the Karman vortex street from a cigar.

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Supersonic flows, which are faster than the speed of sound and Mach 1, are not something we encounter on a daily basis. However, they are particularly relevant in fields like the aerospace industry. One example is NASA, who studies supersonic flight to advance aeronautics research and design vehicles that perform well at supersonic speeds. (A new supersonic passenger jet is a recent example.)

*A supersonic jet. Image by EyeMindSoul. Licensed under CC BY 2.0, via Flickr Creative Commons.*

Simulation can be a powerful tool for analyzing supersonic flow for these and other purposes. But when you obtain results, you want to be confident in their accuracy. COMSOL Multiphysics includes features and functionality that enable you to generate reliable results for trustworthy analyses. To demonstrate this, we’ll share a benchmark model of an expansion fan, a key element of a supersonic flow.

Expansion waves occur when a supersonic flow encounters a convex corner, also called an expansion corner, and turns away from itself, increasing the area. (The opposite scenario of a supersonic flow turning into itself and causing a reduction in the area creates an oblique shock wave.)

*An oblique shock wave (top) and a Prandtl-Meyer expansion fan (bottom).*

For supersonic expansion fans, otherwise known as Prandtl-Meyer expansion fans, the total conditions don’t change across the wave and entropy is conserved. Further, there is a decrease in static temperature, pressure, and density as well as an increase in the Mach number. Expansion fans are isentropic processes that generate continuous and smooth changes in the flow, causing the flow’s total properties to be conserved. Here, total values are the values that the flow properties produce if brought to stagnation, while static values represent the actual values of the flow properties at a certain speed.

Our 2D example model investigates an expansion fan that has a 15° expansion corner, supersonic inlet flow with Mach number 2.5, and assumed supersonic flow at the outlet. For our analysis, we assume that the flow is inviscid and therefore apply a Slip boundary condition to the walls. The fluid in this model is air.

*The expansion fan geometry.*

To solve this problem, we use the inviscid compressible flow equations along with the *High Mach Number Flow, Laminar* interface in the CFD Module. To prove that the tutorial’s results are accurate, we can compare them to the inviscid compressible flow theory. Let’s jump ahead to these results now.

*The expansion fan’s Mach number (left), pressure contours (middle), and temperature contours (right).*

We see that the supersonic flow past the expansion corner expands in a succession of Mach or infinitesimal waves. These increase the Mach number while simultaneously decreasing the static pressure and temperature — as expected for expansion fans. As for the velocity streamlines, these smoothly vary their direction over the expansion fan, eventually becoming parallel to the surface below.

*The expansion fan’s velocity contours and streamlines.*

Moving on, let’s find out how the model’s results compare to those from the compressible flow theory. In the table below, we see that the Mach number from the simulation agrees well with the inviscid flow theory. We can also confirm that the total properties of the flow are conserved and similar to existing theory.

Mach Number | Total Temperature | Total Pressure | Total Density | |
---|---|---|---|---|

Model | 3.2331 | 305.432 K | 82,213 Pa | 0.93787 kg·s |

Compressible Flow Theory | 3.2372 | 305.432 K | 82,737 Pa | 0.94347 kg·s |

From this, we can conclude that our simulation results correlate well with those gathered from the inviscid compressible flow theory. As such, CFD models like our Expansion Fan tutorial can help to accurately analyze supersonic flow phenomena.

Click the button below to try modeling this benchmark example, which we improved using a few simple tricks:

- To avoid sharp gradients and discontinuities, we rounded the corner in the model
- To refine the mesh at the expansion fan, we used the
*Adaptive Mesh Refinement*feature

- Find other CFD benchmark tutorials:
- Browse through our collection of CFD blog posts

Inkjet printers are widely used to provide high-resolution 2D printouts of digital images and text, where the printhead ejects small droplets of liquid from a nozzle onto a sheet of paper in a specific pattern. In addition to printing images onto paper, the inkjet technique is also common in 3D printing processes. The printhead moves over a certain type of powdered printing material and deposits a liquid through the nozzle onto the powder to effectively bind it into a predetermined 3D shape. (Tip: Check out the video on 3dprinting.com to see this process in action.) Inkjet printheads are also prevalent in life science applications for diagnosis, analysis, and drug discovery. The nozzles are used as part of a larger instrument to deposit microdroplets in a very precise fashion.

*An inkjet nozzle deposits an ink droplet, which travels through the air before reaching its target. The model was created using the COMSOL Multiphysics® software.*

No matter what device or machine relies on the inkjet printhead to deposit material, precision is crucial. Therefore, the quality of the end product hinges on the nozzle design.

The droplet size for an inkjet nozzle is a key design parameter. In order to produce the desired size, you need to optimize the design of the nozzle and the inkjet’s operating conditions. Rather than build nozzle prototypes and test them in a lab, you can use simulation software to understand the physics of the fluid ejection and determine the optimal design. COMSOL Multiphysics® is one such software package.

When you expand COMSOL Multiphysics with either the CFD or Microfluidics add-on module, you can create models that help you understand how the ink properties and nozzle pressure profile affect the droplet velocity and volume as well as the presence of satellite droplets.

*Model created using the level set method to track the interface between air and ink. The color plot around the droplet signifies the velocity magnitude in the air.*

What happens inside the inkjet nozzle when the liquid is emitted? First, the nozzle fills with fluid. Next, as more fluid enters the nozzle, the existing fluid is forced out of the nozzle. Finally, the injection is halted, which ultimately causes a droplet of liquid to “snap off”. Thanks to the force transmitted to the droplet by the fluid in the nozzle, it travels through the air until it reaches its target. In terms of physics, inside the nozzle, there is a single-phase fluid flow. When the liquid moves through the air, the flow becomes a two-phase flow.

We won’t go into the details of how to build this model here, because you can download the step-by-step instructions in the Application Gallery.

As the simulation specialist in your organization, you are a member of a small and rather exclusive group of people tasked with serving a larger pool of colleagues and customers who rely on your models to make important business and design decisions. Wouldn’t it be nice if these stakeholders could take on some of the work that goes into rerunning simulations for different parameter changes?

The COMSOL Multiphysics software comes with the built-in Application Builder, which enables you to wrap your sophisticated models in custom user interfaces. By building your own apps, you can give your colleagues or customers access to certain aspects of your models, while hiding other aspects that may be unnecessary to change and too complicated to expose. For example, suppose that your colleagues in design or manufacturing want to test the performance of an inkjet nozzle for different geometries and liquid properties. Instead of coming back to you each time they want a minor change to the underlying model, they can input different values in simple fields and click on a button to plot new simulation results in the app you provide them. Since they can run their own analyses, your time can be spent on new projects, models, and apps.

To show you what we mean — and to inspire you to make your own apps — we have made a demo app based on our inkjet tutorial model. In this example, the app user can analyze various nozzle designs to see which version produces the ideal droplet size. Contact angle, surface tension, viscosity, and liquid density are all taken into account in the app. As you can see in the screenshot below, an app user can adapt the nozzle shape and operation by changing different input parameters.

*An example of what an inkjet printhead design app might look like. In this demo app, users can modify liquid properties, the model geometry, and simulation time intervals.*

When you build apps, you can empower other stakeholders to make better decisions faster without actually giving them access to your full underlying model. The model simply powers the app and you, as the app designer, decide what inputs the users can modify. Your original model file stays safely untouched in your care, but a variety of results are accessible by those who rely on them most.

Get started by downloading the .mph file and accompanying documentation for the tutorial model and demo app from the Application Gallery.

All you need to download the documentation is a COMSOL Access account. To get the .mph file, you will also need a valid COMSOL Multiphysics® software license or trial. Note that you can access these files directly within the product as well, via the Application Libraries.

- Watch a keynote video on industrial inkjet printheads from the COMSOL Conference 2014
- Get an introduction to modeling separated three-phase flow with COMSOL Multiphysics
- Learn about modeling piezoelectric actuators

Eli Lilly and Company leverages first principles — the fundamental rules governing the behavior of a system or process — in the design and development of their pharmaceutical products. First principles thinking enables the company’s design engineers to predict and explain why, for a specific set of circumstances, they get one type of behavior and not another.

The approach, as Bernard McGarvey explains, involves identifying a specific design decision and applying a first principle to a generic model of the design. After applying computational methods, such as a simulation in the COMSOL Multiphysics® software, the generic solution that is found is translated into a more specific solution to the original design challenge.

*Eli Lilly’s first principles approach to designing pharmaceutical products. Reproduction based on Bernard McGarvey’s keynote talk from the COMSOL Conference 2016 Boston.*

Using first principles when designing a product makes the entire process more efficient and effective. This is because there is no need to justify the first principles involved — such as the ideal gas law or the Navier-Stokes equations — as they are already proven. McGarvey summarizes the benefits of this process, saying: “You’re taking a specific situation and, by taking advantage of its generic principles, you make it efficient.”

In his keynote talk, Bernard McGarvey discusses a simple example of an insulin pen, one focus of Eli Lilly’s pharmaceutical development work. For this specific single-use autoinjector pen, two key design considerations are:

- Keeping the needle gauge small enough to deliver a large dose with minimal backpressure
- Keeping the needle gauge large enough to maintain patient comfort and reduce injection pain

The need for both patient comfort and device efficiency creates a natural opposition, which leads to a challenging problem for the engineers working on the design of these systems.

*From the video: Bernard McGarvey of Eli Lilly and Company discusses the needle-based drug delivery design.*

First principles for this problem is achieved by describing the physical parameters of this application through the Hagen-Poiseuille equation — what McGarvey refers to as “a design engineer’s worst nightmare.” He says this because one variable in the equation is particularly hard to control. This accounts for the delivery of a certain volume of a substance in a certain amount of time and, for the case of the needle design, any change in the needle’s inner diameter (ID) greatly affects the backpressure.

To address this challenge, a needle vendor presented them with an elegant solution: a tapered needle that might cut the required delivery force significantly, while not increasing the pain from the injection for the patient. Eli Lilly worked with the vendor to evaluate a tapered needle design instead of a straight cannula needle. This design reduces backpressure while maintaining the comfort level of the patient during injection. The company used COMSOL Multiphysics to investigate how much of a reduction in backpressure can be expected with the tapered design. They found a 40–50% decrease in backpressure is possible as compared to a straight cannula needle. This is a significant reduction in backpressure and provides Eli Lilly with a design option for future systems.

Want the full story of how Eli Lilly uses modeling and first principles thinking for product design and development? Watch the video of the keynote presentation at the top of this post.

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In fluid mechanics, pressure represents the force per unit area applied to a surface by a fluid. Using COMSOL Multiphysics, we can solve the governing equations for fluid flow, the Navier-Stokes equations, to determine the velocity and pressure fields that describe the flow.

There are two main ways that we can talk about pressure for CFD problems: absolute pressure and relative pressure.

Absolute pressure is the direct measurement of a fluid’s pressure against vacuum. For instance, if we measure the pressure outside on a typical day with a barometer, we see an absolute pressure reading of about 1 atm or 101.325 kPa, which is the atmospheric pressure at sea level. An absolute pressure of zero corresponds to vacuum.

*This barometer measures the outdoor pressure from 950 to 1050 mbar (1 mbar = 100 Pa). Image by Langspeed. Licensed under CC BY-SA 3.0, via Wikimedia Commons.*

Relative pressure refers to a fluid’s pressure with respect to a reference pressure level. Gauge pressure is the pressure measured relative to ambient pressure; i.e., the relative pressure using ambient pressure as a reference. Typically, relative pressure is used to characterize the pressure levels in closed systems. It can be measured using a manometer, which relates the internal pressure to the surrounding pressure.

*Manometers measuring the relative pressure in a pressure control station. Notice how the dials start at zero, which represents the system pressure equaling the reference pressure level. Image by Holmium — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.*

Absolute pressure and relative pressure are related according to:

*p _{A}* =

If we use vacuum as the reference pressure, then the absolute pressure and the relative pressure are equal. In most cases, the reference pressure is set to the atmospheric pressure, which usually is the ambient pressure.

Let’s connect these definitions of pressure to what we see in COMSOL Multiphysics. When we compute the solution to a fluid flow problem, the COMSOL® software solves for the components of velocity (u,v,w) and the relative pressure (p). As we explain later in this blog post, by using the relative pressure instead of the absolute pressure as the dependent variable, we can improve the accuracy of the description of pressure in our simulation. We can then use the values of relative pressure in the initial values and boundary conditions in the model, which we will see in the following example.

Let’s take a look at an example that illustrates how to use the variables for relative pressure and absolute pressure appropriately in a COMSOL Multiphysics model. To demonstrate these concepts, we use a simple model of air flowing into a channel with an inlet velocity of 1 m/s and exiting to an absolute outlet pressure of 1 atmosphere. The top and bottom boundaries represent the no-slip channel walls, except for two short inlet sections where we assume symmetry. The inlet sections are there to avoid inconsistent boundary conditions, which would be the case if we were to define a straight inlet velocity profile adjacent to a no-slip boundary.

*A schematic of a channel with air flowing through it.*

In this model, the variable name for relative pressure is p and the variable name for absolute pressure is spf.pA. In the settings for the *Laminar Flow* interface, we see that the dependent variables to be solved are the components of velocity (u,v,w) and the relative pressure (p).

*Settings window for the dependent variables.*

We can see in the figure below that the reference pressure level is set to 1[atm] by default. This reference pressure level is used to calculate the absolute pressure: spf.pA = p + spf.pref.

We also set the compressibility to *Weakly compressible flow*, which means that the density of air depends on temperature and reference pressure. To learn more about different compressibility settings, take a look at this previous blog post.

*The compressibility and reference pressure settings.*

Now we can specify our boundary conditions. At the inlet, we assign the normal velocity to be 1 m/s. For the initial conditions and outlet boundary condition, we need to enter the relative pressure, since we are using the default setting; i.e., to use a reference pressure. When we add the outlet condition, we see that the default value for the relative pressure is p = 0, which is equal to 1 atm in absolute pressure for the default reference pressure.

*Settings window for the boundary conditions, showing the relative pressure (left) and absolute pressure (right).*

You might be wondering, then, why the COMSOL® software calculates the variable for absolute pressure, spf.pA. The absolute pressure is used when calculating the density of a compressible fluid. For instance, if we navigate to the material properties for the air in our channel, we will see that the density is defined using the ideal gas law, where pA is the absolute pressure and T is the temperature. Since the ideal gas law is calculated based on the absolute pressure, we have to add the reference pressure to the relative pressure (p) to calculate the density. However, in this case, the relative pressure is such a small fraction of the total pressure (0.00025%; see below) that we may as well use the reference pressure to calculate density, which is what we get when using the *Weakly compressible flow* option. In systems with larger pressure variations, we can select the *Compressible flow* option.

*Defining density using the ideal gas law in the settings window.*

Now that we have defined the boundary conditions for our problem, we can compute the solution and visualize the velocity profile with streamlines.

*Velocity profile with streamlines and a vector plot of flow through a channel.*

We can also look at the pressure profile along the inlet (along the *y*-axis at the left vertical boundary). We can see in the plot below that the pressure variation along the inlet is around one tenth of a Pascal compared to the reference pressure, which is of the order of magnitude of 1·10^{5} Pa. This means that the reference pressure is about one million times larger than the variations in the inlet pressure!

*Relative pressure along the vertical inlet boundary.*

The default way to solve fluid flow problems in COMSOL Multiphysics is by using the relative pressure as a dependent variable and adding the reference pressure when an absolute pressure is required; for example, to compute the density of the fluid. This improves the accuracy of the description of the fluctuation of the pressure field around the reference pressure and the description of the gradients of the pressure field.

Let’s return to our channel example and calculate the pressure drop. If we use the *Line Average* feature to evaluate the relative pressure at the inlet, we will determine the pressure to be about p_{inlet} = 0.26 Pa.

Now, imagine that we solved our problem using absolute pressure instead. The absolute inlet pressure would be 101,325.26 Pa and the absolute outlet pressure would be 101,325.00 Pa. The relative change of the pressure field between the inlet and the outlet is 0.000253814%. As shown in the inlet pressure plot, the variations at the inlet are even smaller: one millionth of the absolute pressure. This is a very small relative change to look for when we solve the equations.

Since we are solving this problem numerically, we are approximating the real pressure field. This is defined at every point, with a numerical approximation defined at a relatively few number of points. We introduce a numerical error due to truncation and interpolation errors. In addition, the numerical equations can only be solved to a given tolerance. This boils down to a relative error in the computed numerical approximation of the pressure field that would disturb the relatively small fluctuations that we are looking for. By using a reference pressure, we can better resolve the gradients of pressure and the fluctuations around atmospheric pressure at viable values for the relative numerical error in the pressure field, compared to the case with the absolute pressure.

Now that we understand why the COMSOL Multiphysics software uses the relative pressure to solve fluid flow problems, we can also appreciate the importance of specifying an accurate reference pressure level. Obviously, a reference pressure level of 1 atm is appropriate for systems working around atmospheric pressure. For very high or low pressure systems, we should use a reference pressure level that matches the pressures expected in the flow.

For instance, in a traditional incandescent light bulb, lower-pressure argon is housed in a glass bulb to prevent oxidation of the filament. In the Application Gallery tutorial model, we see that the reference pressure level changes to match the pressure of this gas (50 kPa). In the *Initial Values* section, the relative pressure is set to p = 0, which corresponds to an absolute pressure of 50 kPa due to the updated reference pressure level.

*A simulation of the free convection of argon within a light bulb.*

For very low pressure systems, it’s important to check that fluid can still be considered a continuum. You can calculate the Knudsen number, which is the ratio of the mean free path to the length scale of the device, to determine if the flow is best solved using molecular flow physics.

In today’s blog post, we explained how the absolute pressure of a system is a direct measurement of pressure, while the relative pressure describes the pressure with respect to a reference pressure level.

COMSOL Multiphysics solves CFD problems using relative pressure to improve the numerical accuracy of the pressure field. This means that the initial conditions and boundary condition should be defined using relative pressure values. However, when calculating the density of a gas, the absolute pressure is used and the reference pressure is added to the relative pressure automatically. For high or low pressure systems, the reference pressure level should be changed to match the average pressure in the system.

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Natural convection is a type of transport that is induced by buoyancy in a fluid. This buoyancy is in turn caused by the fluid’s variations in density with temperature or composition.

You may be familiar with the concept of natural convection in indoor climate systems. In this scenario, hot air rises to the ceiling close to heat sources and cool air sinks to the floor close to cold surfaces, such as the windows during winter.

Electronics cooling is another type of process that often depends on natural convection in order to work. For example, we do not want to use noisy fans to cool the amplifiers and TVs in home cinema systems. Electronic devices that need to operate in quiet environments often rely on natural convection to circulate air over their built-in heat sinks.

*Free convection around a splayed pin fin heat sink that is heated from below. The animation shows the value of the velocity in the air around the heat sink.*

Less obvious natural convection problems are found in industries such as chemical and food processing. Environmental sciences and meteorology also involve natural convection problems, as scientists and engineers try to predict and understand transport in air and water.

In all of the cases mentioned above, it is important for engineers and scientists to understand and design systems to control natural convection. In this context, mathematical modeling is the perfect tool. In the latest version of COMSOL Multiphysics, it is easier to define and solve problems involving natural convection. We have introduced a number of new capabilities for this purpose.

The *Weakly compressible flow* option for the fluid flow interfaces neglects the influence of pressure waves, which are seldom important in natural convection. It allows for larger time steps and shorter solution times for natural convection problems.

The *Incompressible flow* option with the Boussinesq approximation for buoyancy-driven flow linearizes density using a coefficient of thermal expansion. This option includes the density variation only as a volume force in the momentum equations. This implies an even larger simplification compared to the *Weakly compressible flow* option, but it still gives an excellent and efficient description for systems with small density variations. This simplification is almost always valid for free convection in water subjected to small temperature differences.

The *Gravity* feature makes it easy to define a reference point for hydrostatic pressure and also automatically accounts for hydrostatic pressure variations at vertical boundaries.

Let’s learn more about these new features and how you can apply them in your natural convection modeling problems.

The *Nonisothermal Flow* interface includes the *Weakly compressible flow* option, which simplifies flow problems by neglecting density variations with respect to pressure. This option also eliminates the description of pressure waves, which requires a dense mesh and small time steps to resolve, thus also a relatively long computation time. In natural convection, there is usually very little influence of pressure waves, which means that we lose very little fidelity in the model’s description of reality by making this simplification.

The continuity equation for a compressible fluid looks as follows:

(1)

\[\frac{{\partial \rho }}{{\partial t}} + \nabla \cdot \left( {\rho {\mathbf{u}}} \right) = 0\]

where *ρ* denotes density and **u** is the velocity vector.

For a gas, density is proportional to pressure and temperature. For example, for an ideal gas, this gives:

(2)

\[\frac{{\partial \rho }}{{\partial t}} = \frac{{\partial p}}{{\partial t}}\frac{M}{{RT}} - \frac{{\partial T}}{{\partial t}}\frac{{pM}}{{R{T^2}}}\]

If we neglect the dynamic effects of the density changes, we get:

(3)

\[\frac{{\partial \rho }}{{\partial t}} = - \frac{{\partial T}}{{\partial t}}\frac{{pM}}{{R{T^2}}}\]

If we use the expression for the density of an ideal gas and neglect the influence of pressure on density, we obtain the following continuity equation:

(4)

\[\frac{{\partial T}}{{\partial t}} + \nabla T \cdot {\mathbf{u}} - T\nabla \cdot {\mathbf{u}} = 0\]

This means that variations of density are taken into account only in terms of temperature variations. The variations in density may cause an expansion of the fluid, but the direct dynamic effects of those expansions on the pressure field are neglected when using the *Weakly compressible flow* settings.

In addition to the density expression in the continuity equation, selecting the gravity check box in the settings for the fluid flow interface adds a volume force in the momentum equation in the direction of gravity. By default, this is the negative *z*-direction. This force looks as follows:

(5)

\[{F_z} = -g\rho \left( T \right)\]

where density, ρ, is a function of temperature.

For an ideal gas, density is inversely proportional to temperature.

We can find the settings for the *Weakly compressible flow* option by selecting the *Nonisothermal Flow* interface or the *Conjugate Heat Transfer* interface. Selecting the *Fluid Flow* interface node in the Model Builder shows the settings window below. Selecting the *Weakly compressible flow* option removes the dependency between pressure and density, while selecting gravity automatically adds the volume force of buoyancy in the momentum equation.

*Settings window for the fluid flow interface showing the* Weakly compressible flow *option and gravity feature.*

The figure below shows the flow between two vertically positioned circuit boards. Only the unit cell of one circuit board is shown in the figure. The second circuit board is placed just in front, with its back facing the board that is visible. The flow is completely driven by buoyancy; i.e., there is no fan.

The flow rate at the inlet is around 0.2 m/s and around 0.3 m/s at the outlet. There is no inlet of air from the sides, which means that the difference in flow rate is due to the expansion caused by the increase in temperature along the height of the channel between the circuit boards.

*Buoyancy-driven flow between vertical circuit boards. The expansion is seen in the color legend for the arrows, where the flow velocity is around 0.2 m/s at the inlet and 0.3 m/s at the outlet.*

When the changes in density are negligible in terms of the influence of expansion on the velocity field, we can use the *Incompressible flow* option with the Boussinesq approximation for natural convection. This implies that the continuity equation is simplified even more than with the *Weakly compressible flow* option by treating the fluid as incompressible. In this case, the continuity equation becomes as follows:

(6)

\[\nabla \cdot {\mathbf{u}} = 0\]

Instead, a small change in density is accounted for in a volume force, which is introduced in the momentum equation in the opposite direction of gravity; by default, the *z*-direction. The small change in density is obtained by linearizing the fluid’s density at a reference temperature. The *z*-component of the volume force becomes as follows:

(7)

\[{F_z} = g{\rho _{{\text{ref}}}}\alpha \Delta T\]

Where *g* is the gravity constant, is the density at a given reference temperature, α is the coefficient of thermal expansion of the fluid, and Δ*T* is the temperature difference measured against the reference temperature.

The advantage of using the Boussinesq approximation for buoyancy-driven flow is that the nonlinearities in the fluid flow equations are reduced and the problem becomes easier to solve numerically, requiring less iterations and allowing for larger time steps for time-dependent problems.

A typical example where the Boussinesq approximation can give a realistic description of the flow is for the modeling of liquid water subjected to relatively small temperature differences. The figure below shows natural convection in a glass of water heated from below. Here, we obtain a very complex flow pattern with an upward flow close to the middle and bottom of the glass and with downward flows between the vertical walls and the middle.

*Natural convection in a glass of water. The plot shows the velocity field in the glass and the temperature distribution in the walls of the glass.*

We can obtain the *Incompressible flow* option with the Boussinesq approximation for buoyancy-driven flow by selecting the settings shown in the figure below for the fluid flow interfaces in COMSOL Multiphysics.

*Selecting the* Incompressible flow *option, Gravity feature, and reduced pressure gives the Boussinesq approximation for a natural convection problem.*

When modeling fully compressible flow, the pressure’s time dependency is included in the continuity equation, since density is a function of pressure for compressible fluids. This also means that it is usually sufficient to include an initial condition for the pressure in order to get a well-posed problem, even when we do not prescribe pressure at a boundary.

For weakly compressible and incompressible flows, the time-dependent pressure term in the continuity equation is neglected according to the discussions above. If there are no boundary conditions that set the pressure, the pressure field becomes undetermined, unless we set it in some point in the domain.

In COMSOL Multiphysics, we can use a so-called pressure point constraint in order to avoid an undetermined pressure field. The absence of a reference pressure point is often the source of problems with convergence when solving natural convection problems.

*The settings for the pressure point constraint in the water glass example.*

The equations that describe natural convection usually involve the momentum equation, the continuity equation, and the energy transport or mass transport equation. If buoyancy is driven by temperature differences, then the energy equation is fully coupled with the fluid flow equations (the Navier-Stokes equations). For natural convection, this coupling is fairly tight. This means that the most robust way to solve the equations is to use the fully coupled solver in COMSOL Multiphysics.

*The solver branch in the model tree with the fully coupled solver option.*

For very large problems, a segregated approach may be a preferable option. For example, if there are many chemical species and if buoyancy is caused by variations in density due to chemical composition, then a segregated approach may be the only viable option for getting decent memory consumption in the solution process.

I would like to end this blog post with one more natural convection problem. I often think about natural convection when I smoke a cigar. Although I do not want to promote smoking, my favorite natural convection problem is the smoke from a cigar on a cold winter day. The figure below shows a lighted cigar resting on an ashtray with the flow distribution caused by the heat from combustion.

*Natural convection (with a small forced component) around a lighted cigar resting on an ashtray.*

Some of the flow caused by the lighted cigar is actually forced convection, since a large part of the tobacco goes to smoke, changing the density from around 500 to 1000 kg/m^{3} down to 1 kg/m^{3}. This can be described as an inlet for the flow at the boundary between the ash and the air surrounding the cigar.

- Learn more about natural convection and fluid flow modeling on the COMSOL Blog:

Industrial mixers are a key element in many fields, from the pharmaceutical and food industries to consumer products and plastics. Further, the purpose of mixers can vary greatly. Mixers are not only used to combine elements and create homogeneous mixtures, but to also reduce the size of particles and generate chemical reactions.

*An industrial mixer. Image by Erikoinentunnus — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.*

Mixers are required for efficient and timely production as well as for producing a uniform product quality within a batch and between batches. In some cases, mixers are required for the safe operation of systems, for example, in exothermic reactions that may create hot spots and runaway reactions (explosions) under poor mixing. With modeling, we can run inexpensive and streamlined experiments with different mixer designs in order to optimize the mixing process, avoid poor product quality, and meet safety requirements.

To resolve these issues, you can turn to COMSOL Multiphysics, which provides you with the tools for testing a wide assortment of mixers. In the next section, we’ll discuss three different mixer design examples that speak to the versatility of COMSOL Multiphysics.

A typical batch mixer generally consists of two main components: a vessel and an impeller, both of which can vary in type and shape. Baffles can also be added to the device to improve the mixing by suppressing the bulk’s main vortex formation.

The importance of the baffles depends on the type of impeller. Radial impellers, for instance, require baffles to work. Otherwise, the solution will rotate like a merry-go-round and mixing will not be achieved. Here, the impellers will only create vertical mixing as the solution hits the walls of the vessel. Axial impellers, on the other hand, create a vertical mixing flow at the impeller, which means they do not require baffles to achieve mixing. However, axial impellers also have a radial component, so baffles can be used to increase radial mixing in axial impellers, if desired.

Let’s take a look at a mixer’s vessel, shown below, which is often modeled as either a vertical cylinder with a dish-shaped or flat bottom.

*Side views of a flat-bottom mixer (above) and a dished-bottom mixer (below).*

Within the vessel, the fluid is mixed by a rotating impeller. The rotation and design of the impeller determines the axial and radial direction in which the liquid is discharged. As such, impellers come in many different designs, enabling them to be used for a variety of different industrial purposes. Here, we will investigate a six-blade Rushton disc turbine, which is a radial impeller used for high-shear mixing, and a more general-purpose pitched-blade impeller, which is an axial impeller.

*A Rushton disc turbine with six blades (left) and a pitched-blade impeller with four blades (right).*

By combining these two common types of vessels with two types of impellers, we create two separate geometries (shown below) and three separate studies. All three studies use the *Frozen Rotor* study type and the *Rotating Machinery, Fluid Flow* interface.

The first study involves the laminar mixing of silicon oil in a baffled flat-bottom mixer that contains a Rushton turbine with six blades rotating at 40 rps. While we focus on the highest of three rotation rates in this example, you can easily adjust the rotation to simulate the slower rotation rates. This first example is based on a PhD thesis by M.J. Rice entitled *High Resolution Simulation of Laminar and Transitional Flows in a Mixing Vessel* (see Ref. 1 in the model documentation) and includes comparisons from the PhD thesis *Study of Viscous and Visco-elastic Flows with Reference to Laminar Stirred Vessels* by J. Hall (see Ref. 2 in the model documentation).

*Two mixer geometries, one combining a baffled flat-bottom mixer and a Rushton turbine (left) and one with a baffled dished-bottom mixer and a four-blade pitched impeller (right).*

Moving on, our next two examples deal with the turbulent mixing of water within a baffled dished-bottom mixer. This mixer contains a pitched four-blade impeller that rotates at 20 rpm. It’s possible to reduce the computational time required to solve these models by using periodicity and only simulating a quarter of the domain.

Our turbulent mixing examples enable you to explore how different models affect your results. Here, we compare a *k-epsilon (k-ε)* model, which has a quick convergence rate, to a *k-omega (k-ω)* model, which works better for flows with recirculation regions.

Let’s begin by looking at the velocity magnitude and in-plane velocity vectors for our three models. These results provide a general view of the circulation patterns in the mixing vessels for all three of our examples.

For our first mixer model, the laminar mixing example, we can see that the fluid is discharged radially outward by the Rushton turbine, creating two zonal vortices. The resulting compartmentalization phenomenon, which is common for radial impellers, is also displayed in our simulation results. This leads to mixing in the top and bottom vortices, albeit less intensely than inside each individual vortex.

*The velocity magnitude (*xz*-plane) and in-plane velocity vectors (*yz*-plane) for the laminar mixing example.*

On the other hand, the velocity magnitude and vector projection for the turbulent flow *k-ε* model indicate that the fluid is expelled axially and radially by the pitched-blade impeller. As a result, a large zonal vortex is generated from the top to the bottom of the vessel. Additionally, a small zonal vortex appears below the impeller, which can aggregate the heavy dispersed particles in this area.

*The velocity magnitude (*xz*-plane) and in-plane velocity vectors (*yz*-plane) for the *k-ε* turbulence model example.*

The third study reveals that the turbulent flow *k-ω* model has a large zonal vortex, similar to the *k-ε* example. However, this time, the core is more vertically stretched. For its part, the smaller zonal vortex located beneath the impeller is stretched in the radial direction. Another difference lies with the torque and power draw values, which are both higher than the *k-ε* model. While the *k-ω* model is a good model to use for these types of flows, we still need to determine if its results are actually more valid than the *k-ε* model. Comparing simulation results to experiments is, therefore, a necessary next step.

*The velocity magnitude and in-plane velocity vectors for the *k-ω* turbulence model example.*

Finally, our simulations reveal that all three examples generate good approximations for at least a few averaged flow quantities. Our results from the frozen rotor simulation for the laminar mixing study can be easily used as initial conditions for a new time-dependent study.

It’s easy to modify the mixer geometries presented here to fit a wide assortment of mixer designs and conditions. Simply change the parameters in the supplied model to alter the types of components and properties of the geometry. For further customization, you can also add your own subsequences into the mix. With this, you can create a customized model to fit your specific application.

For more information on how to improve your mixer simulations, check out the resources in the next section.

- Try out the tutorial featured in this blog post: Modular Mixer-Turbulent Mixing (k-omega)
- Read a few blog posts on simulating mixers in COMSOL Multiphysics:
- Watch this in-depth archived webinar: Simulating Mixers and Non-Ideal Reactors
- Download a related tutorial: Laminar Flow in a Baffled Stirred Mixer

The Chemical Reaction Engineering, CFD, and Plasma modules all include different variations of the equations for the transport of chemical species in a concentrated solution, such as the Maxwell-Stefan equations and the mixture-average model. In a concentrated solution, the model equations have to account for the interactions between all species in a solution, while a model of a dilute solution only includes the interaction between the solute and the solvent. The schematics below illustrates the difference between these two descriptions.

*Dilute solutions (left) and concentrated solutions (right). The interaction within dilute solutions is dominated by the interactions between solvent and solute and solvent with solvent. In a concentrated solution, all species interact with one another.*

Along with these interactions, the velocity field in a concentrated solution is defined as the sum of the flux over every species, *i*:

(1)

\[{\mathbf{v}} = \frac{{\sum\limits_i {{{\mathbf{n}}_i}} }}{{\sum\limits_i {{\rho _i}} }}\]

where **n** denotes the flux in kg/(m^{2}s), and ρ represents the density (kg/m^{3}). For a dilute solution, the velocity field is given by the velocity of the solvent:

(2)

\[{\mathbf{v}} \approx \frac{{{{\mathbf{n}}_{solvent}}}}{{{\rho _{solvent}}}}\]

As we can see from the images above and Eq. (1), the transport of species and fluid flow are tightly coupled for concentrated solutions.

In earlier versions of COMSOL Multiphysics, the *Reacting Flow* interface was a single multiphysics interface with its own domain settings and boundary conditions, specifically designed for coupling flow and chemical species transport and reactions. This approach was user-friendly in nature, as everything was predefined. However, some of the general flexibility in COMSOL Multiphysics was lost in this predefined physics interface. Say you wanted to make larger changes separately to the transport of concentrated species equations and the flow equations. To do so, you would have to define the problem by adding the two types of physics interfaces separately, instead of using the predefined multiphysics interface, and then manually construct the multiphysics coupling.

With the new *Reacting Flow* interface for a concentrated solution, you can handle this tight coupling while maintaining the ability to manipulate the transport equations and the fluid flow settings separately. The coupling itself is defined in the Multiphysics node. With such functionality, you can, for instance, change from laminar flow to turbulent flow or change the transport model from the Maxwell-Stefan equations to the mixture-average model.

Let’s see how this is manifested in the model tree and in the settings for the Multiphysics node. As the following screenshot shows, all of the usual nodes for the constituent physics interfaces can be modified while the coupling is predefined in the Multiphysics node. The predefined coupling controls the mass fluxes and, when summed over all species, satisfies the continuity equation for the flow. As such, the two sets of equations are fully coupled in a bidirectional way.

*The model tree with the* Reacting Flow *multiphysics node selected. Here, we can select which physics interfaces to couple. We can also make changes to the flow model that allow us to include turbulent reacting flow. This is an additional flexibility, with preserved ease of use, as compared to previous multiphysics interfaces.*

Another benefit of the new *Reacting Flow* multiphysics interface is found in the Study node. We have the ability to solve for the fluid flow equations to obtain a decent initial guess for the total flux. In a second step, we can solve only for the transport of species, with the velocity field given by the previous solution of the fluid flow equations.

Now we have a decent solution for the fluid flow and for the composition of the solution, which we can use as an initial guess for the fully coupled problem. Therefore, the last study step (Step 3) involves solving for the fluid flow and the transport of chemical species in a fully coupled scheme. Note though that the fully coupled scheme itself may also be sequential for a large number of species in 3D, but the loop over all species and the fluid flow is performed automatically.

*The three study steps (1, 2, and 3) solve for the fluid flow, the transport of chemical species, and the fully coupled problem, respectively. The automatically generated solver configuration shows the intermediate steps that store the flow field, the concentration field, and the final step that then solves for the fully coupled problem using the stored solutions as an initial guess.*

By utilizing the new *Reacting Flow* multiphysics interface, you have the ability to solve a range of interesting problems, such as the one shown below. In this case, we can see the flow and concentration in a tubular reactor that converts methane into hydrogen. The model combines the transport of species in concentrated solutions, fluid flow in free and porous media, and heat transfer with the endothermic reactions and the heated jacket on the reactor’s cylindrical outer walls.

*Concentration of hydrogen in a reactor that converts methane into hydrogen. The reaction is endothermic and heat is supplied at the cylindrical walls, which yields a higher hydrogen production close to the walls.*

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