Inkjet printers are widely used to provide highresolution 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 addon 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 singlephase fluid flow. When the liquid moves through the air, the flow becomes a twophase flow.
We won’t go into the details of how to build this model here, because you can download the stepbystep 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 builtin 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.
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 NavierStokes 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 singleuse autoinjector pen, two key design considerations are:
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 needlebased drug delivery design.
First principles for this problem is achieved by describing the physical parameters of this application through the HagenPoiseuille 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 NavierStokes 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 BYSA 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 BYSA 3.0, via Wikimedia Commons.
Absolute pressure and relative pressure are related according to:
p_{A} = p + p_{ref}
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 noslip 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 noslip 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 yaxis 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, lowerpressure 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 builtin 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 buoyancydriven 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)
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)
If we neglect the dynamic effects of the density changes, we get:
(3)
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)
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 zdirection. This force looks as follows:
(5)
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.
Buoyancydriven 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)
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 zdirection. The small change in density is obtained by linearizing the fluid’s density at a reference temperature. The zcomponent of the volume force becomes as follows:
(7)
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 ΔT is the temperature difference measured against the reference temperature.
The advantage of using the Boussinesq approximation for buoyancydriven 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 timedependent 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 buoyancydriven 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 wellposed problem, even when we do not prescribe pressure at a boundary.
For weakly compressible and incompressible flows, the timedependent 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 socalled 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 NavierStokes 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.
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 BYSA 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 merrygoround 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 dishshaped or flat bottom.
Side views of a flatbottom mixer (above) and a dishedbottom 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 sixblade Rushton disc turbine, which is a radial impeller used for highshear mixing, and a more generalpurpose pitchedblade impeller, which is an axial impeller.
A Rushton disc turbine with six blades (left) and a pitchedblade 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 flatbottom 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 Viscoelastic Flows with Reference to Laminar Stirred Vessels by J. Hall (see Ref. 2 in the model documentation).
Two mixer geometries, one combining a baffled flatbottom mixer and a Rushton turbine (left) and one with a baffled dishedbottom mixer and a fourblade pitched impeller (right).
Moving on, our next two examples deal with the turbulent mixing of water within a baffled dishedbottom mixer. This mixer contains a pitched fourblade 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 kepsilon (kε) model, which has a quick convergence rate, to a komega (kω) model, which works better for flows with recirculation regions.
Let’s begin by looking at the velocity magnitude and inplane 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 (xzplane) and inplane velocity vectors (yzplane) for the laminar mixing example.
On the other hand, the velocity magnitude and vector projection for the turbulent flow kε model indicate that the fluid is expelled axially and radially by the pitchedblade 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 (xzplane) and inplane velocity vectors (yzplane) for the kε turbulence model example.
The third study reveals that the turbulent flow kω model has a large zonal vortex, similar to the kε 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 kε model. While the kω 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 kε model. Comparing simulation results to experiments is, therefore, a necessary next step.
The velocity magnitude and inplane velocity vectors for the kω 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 timedependent 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.
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 MaxwellStefan equations and the mixtureaverage 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)
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)
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 userfriendly 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 MaxwellStefan equations to the mixtureaverage 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.
In order to transport food and other perishables, refrigerated trucks are designed to maintain a cold temperature. If these products are not kept properly cooled, they can become heated above an acceptable temperature and thus harmed while intransit. This is especially true in cases where products are exposed to heat for a prolonged period of time.
A refrigerated truck.
While it may seem straightforward to keep a truck properly cooled, there are a few important elements to consider. For one, the cooling system’s design must be optimized in order to preserve the correct temperature. Additionally, the truck’s walls need to be sufficiently insulated to help maintain this desired temperature, which requires choosing the right materials for the job.
Now consider when goods are loaded or unloaded from the truck. The open and closeddoor cycles that refrigerated trucks undergo makes addressing the above elements even more challenging. It is, however, important to address these periods in order to obtain realistic results for the vehicle’s cooling efficiency.
Let’s see how SIMTEC, a COMSOL Certified Consultant, helped Air Liquide, a leader in technologies, gases, and services for industries and health, model this process.
When it comes to choosing the best thermal insulation materials for an efficient aircooling system, it is important to have a thorough understanding of the aerothermal configuration inside a truck’s refrigerated box. As we’ll demonstrate here, this can be achieved by coupling CFD and heat transfer in COMSOL Multiphysics.
For their simulation studies, the team at SIMTEC created an aerothermal model to simulate heat transfer during the normal operation of a refrigerated truck. Their goal: Predict the temperature and air velocity distribution inside the truck and find out what happens upon closing and opening the rear door.
We can begin by taking a closer look at their model geometry. The engineers modeled the refrigerated area of the truck (the refrigerated box) as a parallelepipedic box with a cooling system. The cooling system is comprised of two circular apertures used for air extraction and a rectangular aperture used to blow refrigerated air into the main box. Since the model geometry features a yz symmetry, it is possible to fully describe the physical phenomena occurring in the entire refrigerated compartment by using only a model of half of the box.
Geometry of the refrigerated truck. The entire refrigerated box geometry, the halfbox geometry, and a closeup view of the ventilation and cooling system are shown. Image by Alexandre Oury, Patrick Namy, and Mohammed YoubiIdrissi and taken from their COMSOL Conference 2015 Grenoble presentation.
For their analyses, the team investigated two stages of a refrigerated truck’s operating cycle, using two different computational methods to predict temperature and air flow distribution. In the first stage, which lasts for about three hours, the truck’s rear door is closed and the cooling system is turned on. Both the fans and refrigerating unit operate in order to cool the refrigerated box. The air from the refrigerating system is initially at a temperature of 27°C and later heats up when coming into contact with the warmer box walls. During this cooling period, the engineers partially decoupled their simulations to minimize the system’s degrees of freedom.
Cool air moving through a refrigerated truck with its rear door closed. Image by Alexandre Oury, Patrick Namy, and Mohammed YoubiIdrissi and taken from their COMSOL Conference 2015 Grenoble presentation.
In the second stage, which lasts around ten minutes, the truck’s rear door is opened after the cooling period, and both the refrigerating unit and ventilation system are switched off. Unlike the first stage, this step fully couples CFD and heat transfer to solve the laminar CFD and heat transfer equations. The team then used these equations to analyze the airflow into the box as well as the temperature change during this time period.
To answer this question, let’s begin by looking at the simulation findings from the first stage, when the truck’s rear door is closed. The figure below illustrates an air flow streamline of the local velocity after 10,000 seconds (approximately 2 hours and 45 minutes). At this point in time, the air reaches its maximum velocity at the roof of the box facing the inlet and along the door wall. The velocity decreases rapidly as the air flows through the rest of the box.
Streamlines showing the air’s local velocity after 10,000 seconds in the closeddoor stage. Image by Alexandre Oury, Patrick Namy, and Mohammed YoubiIdrissi and taken from their COMSOL Conference 2015 Grenoble paper.
The temperature field has a very similar distribution to the air velocity, with the coldest areas corresponding to highvelocity regions and vice versa. The plot below indicates that the warmest region is the recirculation zone located at the bottom of the box, where the air is greater than 0°C.
Streamlines showing the air flow’s local temperature after 10,000 seconds in the closeddoor stage. Image by Alexandre Oury, Patrick Namy, and Mohammed YoubiIdrissi and taken from their COMSOL Conference 2015 Grenoble paper.
The engineers also calculated the heat losses within the refrigerated box. By doing so, they found that air cools down inside the truck and global heat losses increase over time. Most of the heat is lost through the lateral and rear walls, both of which feature similar loss profiles. Because the floor is composed of the thickest materials and is the most insulated surface, a very limited amount of energy is lost through its surface.
Averaged heat loss for each of the five walls during the closeddoor cooling period. Image by Alexandre Oury, Patrick Namy, and Mohammed YoubiIdrissi and taken from their COMSOL Conference 2015 Grenoble paper.
Shifting gears to the second stage… With the cooling and ventilation system both off, the engineers stressed that the only driving force for air is the natural convection caused by the difference between outer and inner air temperatures. Since the temperature within the box is much cooler, warmer air flows into the box.
As illustrated in the following simulations, hot air rapidly enters the box at first, but after 50 seconds, the average air velocity drops below 10 cm/s. At 500 seconds, the average air velocity is as low as 2 cm/s, which may be due to the fact the temperature difference between the box and the outside environment is greatly reduced.
The average air velocity in the truck box at 2, 10, 50, and 500 seconds after opening the rear door. Images by Alexandre Oury, Patrick Namy, and Mohammed YoubiIdrissi and taken from their COMSOL Conference 2015 Grenoble paper.
As for the temperature, about 10 seconds after the truck’s rear door opens, the temperature for most of the box matches the outside temperature (around 25°C). One exception is the area around the walls, where a thermal inertia helps the surrounding air stay cool.
The temperature in the truck box at 2, 10, 50, and 500 seconds after opening the rear door. Images by Alexandre Oury, Patrick Namy, and Mohammed YoubiIdrissi and taken from their COMSOL Conference 2015 Grenoble paper.
The team later compared their thermal simulation results to physical experiments where the rear door of the truck was opened and closed multiple times. They found that the model predictions were in reasonable agreement with the experimental temperatures. The simulation results, however, do show oscillations occurring during the opendoor periods that were not observed in the physical experiments. An explanation suggested by the engineers for this is that the temperature was calculated in a different location in the model and physical sensor. Another possible cause is the sensor’s intrinsic inertia, which may have a small leveling effect on the temperature. The model itself shows an instantaneous temperature of air.
A plot comparing simulation results with experimental data. Image by Alexandre Oury, Patrick Namy, and Mohammed YoubiIdrissi and taken from their COMSOL Conference 2015 Grenoble presentation.
With COMSOL Multiphysics, the team at SIMTEC found it easy to couple turbulent CFD and heat transfer in order to perform aerothermal simulations of a refrigerated truck model. These findings have the potential to serve as a powerful tool for designing the next generation of refrigerated trucks, identifying the optimal materials for their walls and indicating ways to enhance the power specifications and locations of their cooling systems.
To learn more about our COMSOL Certified Consultant SIMTEC and their services, visit their website.
When it comes to describing the velocity and pressure fields inside the system you are analyzing, there are many equations that could be appropriate. You could, for example, adequately describe a fluid slowly moving in a porous bed with Darcy’s law. But if the fluid moves rapidly, you may need to use the Brinkman equation. While there are many options available, today we will focus on the NavierStokes equations, as they are the most common in fluid flow analysis. Note that most of the explanations and practices highlighted here will also apply to the equations referenced above.
The first step is to characterize the type of flow that you are modeling based on fluid density. All fluids are compressible, that is, their density depends on absolute pressure and temperature through a thermodynamic relation, . However, from a practical point of view, most liquids can be safely described as having a density that depends uniquely on temperature, . Density is, of course, a function of an alternate element in some cases — for example, salt concentration in the Elder problem.
In the SinglePhase Flow interface available in COMSOL Multiphysics, there are three possible formulations for momentum and mass conservation equations: Compressible flow (Ma < 0.3), Weakly compressible flow, and Incompressible flow. You can easily select from these compressibility options within the Laminar Flow settings, as highlighted below.
Choosing a compressibility option in COMSOL Multiphysics.
In general, the various properties of a fluid are not constant and may depend on a number of quantities. Whether it is necessary to account for such dependencies in your modeling processes is up to you. Since the focus is on mass, momentum, and energy conservation equations in this blog post, we will review how COMSOL Multiphysics deals with viscosity , density , thermal conductivity , and heat capacity for the different compressibility options.
We’ll begin our discussion with isothermal flow simulations and later turn our attention to nonisothermal cases.
Pumps, mixers, airfoils, and multiphase systems… These are just some of the devices that are often modeled as isothermal. Isothermal flow simulations assume that , , , and are not dependent on temperature. If the properties are defined as a function of , they are evaluated at the reference value. This means that the energy equation is weakly coupled with the other two equations through the convective term, and it can be computed at a later time, if needed. It is, however, not always possible to make such an approximation.
Given the freedom that the user interface (UI) of COMSOL Multiphysics grants us, we can first study and solve a single physics problem and then build a multiphysics problem on top of the initial solution. Keep in mind that neglecting the energy conservation equation, even if you’re not directly interested in the temperature field, is valid only below a certain Mach number ().
Let’s take a look at how to select the appropriate compressibility option for your modeling case.
Compressible flow (Ma < 0.3), the most general case, makes no assumptions for the system that is being solved. COMSOL Multiphysics takes into account any dependency that the fluid properties may have on the variables. In isothermal flows, temperature is typically uniform and fluid properties (density and viscosity) remain constant, respectively, evaluated at the reference value. Even so, the properties can still vary with pressure or other quantities, such as concentration. With this formulation, which is the most computationally expensive, we can model any kind of flow and also describe incompressible situations. Our Flow Around an Inclined NACA 0012 Airfoil tutorial model provides one example of how to use the compressible flow formulation.
For the Weakly Compressible flow option, new with COMSOL Multiphysics® version 5.2a, the equations look the same as they do for the Compressible flow (Ma < 0.3) option. The only difference is that if the density is pressure dependent, the density will be evaluated at the reference absolute pressure. In this case, all the other dependencies of the density, like species concentration, are accounted for, so we can still use this formulation to account for volume forces given by concentration gradients.
The Incompressible flow formulation, meanwhile, is valid whenever can be regarded as constant (i.e., when modeling isothermal liquids or gases at low velocities). This option is also used in immiscible two or threephase flow simulations where density is constant. When applying the Incompressible flow formulation, COMSOL Multiphysics automatically uses the reference temperature and pressure to evaluate . In addition, it uses the reference temperature to evaluate . Of course, there are many applications in which the quantities mentioned above are dependent on another variable, like a specy concentration. In these cases, the density must be explicitly evaluated at a reference value for these variables within the interface. To learn more, download our Water Purification Reactor model example.
Density specification in the case of isothermal weakly compressible and incompressible flows.
Also note that the form of the equations, which are shown within the Equation section, change according to the selected option.
The different formulations for compressible and incompressible NavierStokes equations.
Nonisothermal flow simulations typically relate to cooling and heating applications, namely conjugate heat transfer. These simulations can refer to systems that are governed by natural, forced, or mixed convection.
Depending on the type of system that is being analyzed and the hypothesis that is assumed to be true, any of the compressible options can be appropriate for nonisothermal simulations. Since Compressible flow (Ma < 0.3) is the only meaningful formulation for gases subject to high pressure changes, we will focus here on systems that are below the Mach number limit, and those with fluid properties that are uniquely dependent on temperature. (There is a dedicated interface available for modeling high Mach number systems, as highlighted in this Sajben diffuser tutorial model.) The system of equations — mass, momentum, and energy conservation — is completely coupled, as the velocity appears inside the energy equation. Meanwhile, the pressure will appear explicitly in the momentum equation; the temperature will appear explicitly in the energy equation; and both temperature and pressure may be inside the fluid properties in these two equations.
Couplings of momentum and energy equations. In the case of natural convection, a part of the volume force depends on temperature gradients.
For convective heat transfer simulations, the Compressible flow (Ma < 0.3) option can be used to analyze forced and natural convection.
Forced convection refers to when the properties of the fluid vary in a nonnegligible way from pressure and temperature. This is the case for highspeed systems where the pressure changes are nonnegligible in their influence on density. As previously noted, the density of liquids rarely depends on pressure, which makes this exactly the same as the Weakly compressible flow formulation. See our ShellandTube Heat Exchanger tutorial model to learn more.
In natural convection, the driving force is the buoyancy force due to temperature gradients. The Compressible flow (Ma < 0.3) option must be used for gases in closed cavities in order for the system of equations to be consistent. In fact, if the volume cavity and total mass are constant, then the average density needs to be constant. Pressure changes help balance out density variations that are caused by temperature variations. Interested in modeling such a system? Refer to our Free Convection in a Light Bulb tutorial, which exemplifies the set up of transient conjugate heat transfer models with radiation.
The Weakly compressible flow option comes with a simplified formulation that usually leads to increased computational speed. It is the default option when the predefined Nonisothermal Flow or Conjugate Heat Transfer coupling is opened in COMSOL Multiphysics. This formulation, which basically neglects the green arrow coupling shown in the previous image, can be used to analyze forced and natural convection.
In the case of forced convection, the Weakly compressible flow option can be applied to the simulation of water or other fluids and is often valid for modeling gases in open systems (see this heat sink model example). The same considerations are valid for forced convection, as demonstrated in our vacuum flask tutorial.
The Incompressible flow option can be applied to both forced and natural convection as well. The initial case applies when you want to make simulations in cascade. For instance, sometimes it is interesting to compute the flow field at a reference temperature and then compute the temperature field in a second simulation. This can provide a powerful approximation when fluid properties do not vary much within the simulation’s temperature and pressure range. One good candidate for this type of modeling is a heat exchanger that includes liquids. You can also apply this approach to obtain more consistent initial values for highly nonlinear stationary problems. After computing the flow field and temperature field with ‘frozen velocity’, it is possible to gain considerable convergence improvements by using the initial values for the fully coupled simulations.
For the former case, COMSOL Multiphysics implements the Boussinesq approximation. The reference temperature and pressure, specified in the interface, are used to compute density, viscosity, heat capacity, and thermal conductivity. Further, the software automatically computes the coefficient of thermal expansion for the fluid, , taking the derivative of the density around the reference temperature, , and uses it to impose the buoyancy force, , where denotes the gravity vector. You also have the option to enter the coefficient directly, as shown below.
Options for specifying the density.
When utilizing the gravity option, an important concern is the need for consistent boundary conditions and initial values, particularly when dealing with natural convection simulations. This could be a nontrivial task since a domain force, the buoyancy force, is working inside the system, and we need to account for its presence. Consider, for instance, a system like a pipe where a hydrostatic head is present. Here, it is clear that we simply can’t impose a constant pressure as a boundary condition if the boundary itself is not perpendicular to the gravity vector.
Besides enabling you to model all of the above systems and situations, COMSOL Multiphysics helps to address initial values and boundary conditions for each case. To learn more, check out our Gravity and Boundary Conditions tutorial model.
For forced convection, the flow and temperature coupling is taken care of at the Multiphysics node level. Within the NonIsothermal Flow interface, the equations are coupled and fluid flow and heat transfer properties are synchronized (see the screenshot below). Depending on the compressibility option that is chosen, COMSOL Multiphysics will operate in the background to make the appropriate changes for the fluid properties, making them consistent with the selected formulation. Additionally, the NonIsothermal Flow interface takes care of implementing thermal wall functions and computing turbulent heat conduction.
Settings window for the NonIsothermal Flow interface.
If it is necessary to include a buoyancy force due to temperature or concentration gradients, then you must select the Include gravity check box. This will generate values in the Reference Values section that can be used to compute the hydrostatic pressure approximate, alongside reference temperature and pressure. Selecting the Include gravity check box also causes a new subnode to appear: Gravity. Here, you can specify the direction of the acceleration acting on the system. When the Gravity subnode is added, the hydrostatic contribution taken at the reference temperature and pressure is automatically added inside the boundary conditions, if appropriate.
To model natural convection, it is simply necessary to use both the Gravity feature and NonIsothermal Flow interface. Together, they model flow and temperature fields coupled in the presence of gravity acceleration.
Changes prompted by selecting the Include gravity check box.
The following simulation plots correlate with our vacuum flask tutorial model, which evaluates the thermal performance of a bottle holding hot fluid. This system is composed of a gas, air, outside the flask that is flowing in an open system — the flask is leaning on a table in a wide room. Such considerations make this example a helpful resource for understanding the use, assumptions, and results of the various formulations. The Compressible flow (Ma < 0.3) option, for instance, is always applicable. Since air is flowing in an open system, the Weakly compressible flow option is also applicable. And lastly, because the density changes are small, the Incompressible flow option can represent the system appropriately as well.
Plots comparing velocity, temperature, and density fields (respectively) for simulations using the Compressible flow (Ma < 0.3), Weakly compressible flow, and Incompressible flow formulations. We performed these simulations by simply toggling between the three compressibility options.
Graphs comparing velocity, temperature, and density fields (respectively) for simulations using the Compressible flow (Ma < 0.3), Weakly compressible flow, and Incompressible flow formulations. We performed these evaluations at the red dashed line, shown in the plot on the right in the previous set of images, after a simulation time of 10 hours.
Choosing the correct compressibility option is key for solving your system in an accurate and efficient way. COMSOL Multiphysics provides you with functionality that allows you to model both natural and forced convection with ease, while still offering various modeling choices and giving you complete control over the simulations at hand. This results in an optimized approach to the numerical analysis of fluid flow and temperature fields, further advancing your engineering design.
Use the table below as a helpful guide for choosing the compressibility option that is most appropriate for your modeling needs.
Compressibility Option  Isothermal Flow  Nonisothermal Flow 

Compressible flow (Ma < 0.3) 


Weakly compressible flow 


Incompressible flow 


In the various tropical and subtropical regions where it is cultivated and naturalized, the date palm tree is recognized as a valuable resource. The wood, for instance, can be used to construct huts, bridges, and aqueducts. The leaves, when mature, are incorporated into the design of mats, screens, and baskets. But what this type of palm tree is most widely known for is the sweet fruit that it produces: dates.
A date palm tree. Image by Madhif. Licensed under CC BY 3.0, via Wikimedia Commons.
Since ancient times, dates have been cultivated across various regions, from Mesopotamia to Egypt. The ancient Egyptians, for example, used the fruits to make wine and also ate them at harvest. As traders began to bring these fruits to other areas, the popularity of dates and their cultivation extended to Northern Africa, Spain, Mexico, and the United States.
Today, dates are commonly eaten as a snack and are an ingredient in many savory dishes. They are also sometimes used to make vinegar and syrup as well as to form feedstock when mixed with a grain.
In Tunisia, a leading cultivator of dates, these sweet fruits are highly valued as both a food source and financial resource. This is especially true with regards to the Deglet Nour date. Grown in inland oases, Deglet Nour dates are easily recognized by their translucent light color and honeylike taste.
Deglet Nour dates. Image by M. Dhifallah. Licensed under CC BYSA 3.0, via Wikimedia Commons.
Agronomic practices have a strong impact on the dates’ chemical composition — specifically the moisture content — and thus the overall quality of the fruit. In recent years, an increased proportion of dry dates in Tunisia has prompted the use of thermal processing to generate softer fruits that possess a more suitable appearance and characteristics. For this process to be effective, it is important to control the key unit of operation: hydration. For example, if the hydration time is too long, the shelf stability of the fruit could decrease. On the other hand, if the hydration time is too short, the final product quality might be unacceptable.
In an effort to address this concern and optimize the hydration process, one team of researchers turned to experimental tests and simulation analyses. We’ll explore their findings in the next section.
For their experiments, the researchers selected a sample of Deglet Nour dates that were harvested in 2014 and stored at a temperature of 4°C and a relative humidity of 65%. They used these dates to conduct hydration experiments at the laboratory scale. To approach industrial conditions, the dates were placed in a closed environment at atmospheric pressure, with temperatures ranging between 50°C and 65°C.
Such an environment was achieved by placing the dates in the head space of a metallic enclosure, which was filled with water and heated via a temperaturecontrolled hot plate. As they were housed in the head space, the dates themselves did not have any contact with the water. Additionally, as a means to prevent overpressure and maintain atmospheric pressure, no insulation was included on the cover of the enclosure.
Schematic depicting the experimental setup. Image by S. Curet, A. Lakoud, and M. Hassouna and taken from their COMSOL Conference 2015 Grenoble paper.
During the hydration process, the dates were weighed at regular intervals. The researchers determined the average moisture content for the flesh of the dates by drying 3 grams of dates in an oven at a temperature of 105°C for a minimum of 18 hours. By monitoring the dates’ temperature for the various hydration times, they found that the temperature could be considered homogenous.
After conducting their experiments, the group shifted gears to performing simulation analyses of the hydration of dates. The image below depicts the 2D geometry that they designed. The geometry includes the date flesh, date pit, and the saturated air that allows the date to be hydrated.
2D model geometry. Image by S. Curet, A. Lakoud, and M. Hassouna and taken from their COMSOL Conference 2015 Grenoble paper.
Using this model, the researchers computed moisture distribution in the date’s flesh during hydration and then calculated the average moisture concentration as a function of time. Note that only mass transfer phenomena was taken into account, as the temperature was considered to be homogenous.
While experiments were conducted for several types of dates, the researchers focused on using the results from two samples of one type of Deglet Nour date, which were slightly harder and drier than the others. The following series of plots compare the average moisture concentration of Date 1 and Date 2. The simulation curves are found to fit quite well with the experimental data. From these plots, we can also see that there is no decrease in the rate of moisture uptake. One possible reason for this behavior is the short length of the hydration times as compared to the maximum industry processing times.
Plots comparing the average moisture concentration from experiments and simulations in Date 1 (left) and Date 2 (right). Images by S. Curet, A. Lakoud, and M. Hassouna and taken from their COMSOL Conference 2015 Grenoble paper.
The model below shows the moisture concentration distribution within a date’s flesh after 14,640 seconds (around 4 hours) of hydration. As the plot illustrates, the gradient of moisture concentration is higher when closer to the surface in contact with the air. From there, the concentration gradient decreases to zero toward the center of the date’s flesh, where the moisture concentration stays at its initial value. This behavior indicates that diffusion occurs mainly at the outer surface of the date for the range of hydration times considered.
Simulation plot of moisture distribution concentration in a date after 4 hours of hydration. Image by S. Curet, A. Lakoud, and M. Hassouna and taken from their COMSOL Conference 2015 Grenoble paper.
For any food processing technique, the goal is to balance efficiency with overall quality. In the thermal processing of dates, optimizing the hydration method is a key step for realizing this balance, saving energy while ensuring a highquality product.
The simulationbased approach highlighted here offers valuable insight into the mass transfer phenomena that takes place in the hydration operation. As such, the theoretical model underlying this approach can be used as a resource to optimize the hydration of dates by predicting the necessary times to achieve a desired water content, as well as to reduce the overall processing time.
At NASA’s Marshall Space Flight Center, researchers are reaching for the stars, quite literally. As participants in the Advanced Exploration Systems (AES) program, they seek to advance new technologies that could allow for future space missions that extend past Earth’s orbit.
The organization participates in the Life Support Systems Project (LSSP), which is part of NASA’s AES program. The goals of this project are to extend the duration of crew missions, improve reliability, reduce risks, and address gaps in technology with regards to life support systems. Both the LSSP and AES are directly derived from the architecture of the International Space Station (ISS).
The International Space Station. Image by the NASA Goddard Space Flight Center. Licensed under CC BY 2.0, via Flickr Creative Commons.
When it comes to manned vessels, one element that is required for successful space travel is efficient and reliable carbon dioxide removal assembly (CDRA) systems. Since these systems directly affect the health and wellbeing of a vessel’s crew, optimizing their design is of great importance. That, however, is easier said than done. Physical tests require a great deal of time and energy, and simulation studies can hit snags due to the inherently complex nature of CDRA systems. Such challenges have prompted engineers to explore new approaches to system development.
To supplement testing and avoid overcomplicated simulations, researchers at NASA’s Marshall Space Flight Center used COMSOL Multiphysics to create a 1D model of a 4bed molecular sieve (4BMS), a component of the ISS CDRA system. Their goal: Identify problem areas, with the hopes of eventually optimizing a CDRA 4BMS system.
Let’s begin by taking a closer look at the CDRA 4BMS system analyzed in this study. The main method for gas separation used by atmosphere revitalization systems is absorption in packed beds of pelletized sorbents. As illustrated in the schematic below, this system operates by first sending cabin air through a desiccant bed that absorbs water vapor from the air. The cooler and blower then precondition this dry air before passing it through a sorbent bed that removes CO_{2}. When the air stream enters a second desorbing or desiccant bed, the water vapor is added back and the air is returned to the cabin.
Schematic depicting a CDRA 4BMS system. Image by R. Coker and J. Knox and taken from their COMSOL Conference presentation.
As the process above occurs, there is also another process taking place in the 4BMS. The second sorbent bed is closed at one end and heated on the other, which releases CO_{2} from the bed. This is followed by a tenminute air save mode that helps to recover most of the air trapped in the sorbent bed. After this, the bed is vented into space.
The entire sequence highlighted in this section is called a halfcycle and lasts for approximately 155 minutes. In the next halfcycle, the two absorbing beds transform into desorbing beds, and vice versa.
The CDRA 4BMS, as you can see, is an intricate system. Yet here, predictive 1D models are accurate enough to help design the system’s desiccant beds. Although the sorbent beds of this system aren’t cylindrical, and the heaters create the potential for a complex multidimensional flow path, the researchers observed that air flows relatively uniformly through the channels. This prompted them to use a 1D approximation to study the beds, creating a fully coupled model to solve for concentrations, pressures, and temperatures.
For their study, the researchers modeled the transport of two concentrated sorbate species, carbon dioxide and water, in air. This mixture flows through four linked beds of sorbent pellets. Calculated effluent mass fractions of CO_{2} and H_{2}O from the upstream beds were applied as inputs for the next downstream bed. The carbon dioxide beds utilized a heaterassisted vacuum desorption model, with heat transfer involving the gas, porous media, solid housing, and can insulation modeled as well. Further, the researchers used distributed PDEs and Toth isotherms to determine the absorption rates and pellet loading.
The figure below shows the idealized schematic for the 4BMS model. Here, we can see that the team only modeled the glass beads and parts of the beds containing sorbents. In their model, the researchers handled the glass bead layers the same way as the sorbent layers, aside from the H_{2}O and CO_{2} having zero adsorption and desorption capacity.
The idealized 4BMS model. Images by R. Coker and J. Knox and taken from their COMSOL Conference paper submission.
To validate their model, the team used an ISS CDRA 4BMS ground test. It is important to note that some of the inputs, including total sorbent mass, degree of thermal insulation, and pressure drops across the beds, are unknown. Further, in order to achieve faster results and increase numerical stability, the initial bed loading conditions were set close to the expected final results.
The following series of graphs compare the simulation results with experimental data for sorbent bed temperatures and carbon dioxide partial pressures. Overall, the experimental system converged fairly quickly, with the data in the graphs generated from the test’s fourth halfcycle.
When plotting the temperature at the thermocouple (TC) locations for both the baseline data and the COMSOL Multiphysics model (shown below), the researchers noted that although their model’s cooling rate during adsorption was slightly fast, it matched the data fairly well during desorption. This could be due to the fact that the 1D model has a large, simple geometry and uses an ad hoc method to include fins.
Plot comparing sorbent temperature and time. Image by R. Coker and J. Knox and taken from their COMSOL Conference paper submission.
The next step involved comparing the partial pressure of carbon dioxide at the desiccant bed influent and effluent. As the graph below indicates, both simulation and experimental results feature a spike at the beginning of the halfcycle. The plot further shows a rise at the end of the halfcycle, indicating that the sorbent bed is fully breaking through. This is fine, however, since a full sorbent bed maximizes CO_{2} removal efficiency during desorption.
Plot comparing CO_{2} partial pressure and time. Image by R. Coker and J. Knox and taken from their COMSOL Conference paper submission.
The researchers at NASA’s Marshall Space Flight Center successfully created a fully functional model of a 4BMS that can be used for the predictive modeling of an entire ISS CDRA 4BMS system. Looking ahead, the team notes that their 1D model shows promising capabilities for predicting the behavior of such a system and thus locating any potential problem areas in the 4BMS. They have, for instance, already used it to look for unexpected heat leaks in sorbent beds and to predict breakthrough behavior in desiccant beds.
The plan is to eventually validate the model against other CDRA4EU data sets. After doing so, the researchers can use it as a resource to guide the design process for the next generation of CDRA 4BMS systems, as well as to optimize atmosphere revitalization systems such as the one on the ISS.
The National Aeronautics and Space Administration (NASA) does not endorse the COMSOL Multiphysics® software.
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As laptops are designed to be thinner, faster, and lighter with every generation, the potential for them to overheat rises. Cramming more components into a smaller space means that cooling components must dissipate more heat while using less room. If the laptop produces more heat than the thermal management system can handle, the laptop poses a fire hazard. An efficient cooling system reduces this risk and prevents damage.
A laptop after it overheated and caught fire. Image by PumpkinSky — PumpkinSky Family. Licensed under CC BYSA 3.0, via Wikimedia Commons.
One of the most common ways to alleviate heat is with a heat sink. As mentioned in an earlier blog post, these cooling systems are either active or passive. Active heat sinks incorporate a fan and are smaller than their counterparts. The inclusion of microchannels compensates for their smaller surface area and aids heat dissipation. These traditional microchannel (TMC) heat sinks are effective, yet they undergo large pressure drops and temperature variations.
Adding manifolds to the TMC heat sink overcomes these obstacles. They lie perpendicular to the microchannels, acting as flow dividers for the cooling air and forming numerous inlets and outlets. Manifold microchannel (MMC) heat sinks offer less thermal resistance and a greater surface area to transfer heat to this air. Including manifolds will improve the performance significantly and reduce temperature variations, which makes them more stable when integrated into an electronic device. You can use simulation to determine the optimal number and placement of manifolds.
Although MMC heat sinks dissipate heat efficiently, producing them comes with some challenges. For one, the optimal geometric parameters and flow conditions depend on the fan’s blowing power. The width of the microchannels, inlets, outlets, and manifolds may need to be adjusted so that the heat sink reaches its best performance. Second, contact surface properties affect the thermal resistance of the cooling component. Both increasing surface roughness and low contact pressure results in higher thermal resistance. Since we want the least amount of thermal resistance possible, we need to optimize these properties for an efficient MMC heat sink.
A manifold microchannel heat sink, showing inlet and outlet flow.
Measuring all of these functions in such a small device requires precise calculations and, usually, multiple design iterations. Simulation provides accurate information without the expense of manufacturing a prototype for each design change. COMSOL Multiphysics enables you to easily test the geometry of different heat sink elements so that you can find the design dimensions that provide the best air flow rate and least resistance.
You can take advantage of symmetry with the MMC heat sink and only simulate a section of the device, which consists of the three domains:
We find the temperature field for all three domains and the coupled flow field for the air with the Conjugate Heat Transfer interface.
A simulated section of an MMC heat sink on an electronic component.
Next, we set boundary conditions for air velocity and thermal contact. For this example, say that the laminar inflow velocity is 0.85 m/s and the air temperature is 22°C. Another boundary condition that must be set is the thermal contact between the aluminum heat sink and the ceramic electronic component. The aim is to eliminate as much thermal resistance as possible, so we need to effectively model the contact between the two domains. While the pieces lay flat against each other, there are small surface defects that we need to account for by one of two methods. The first method requires a dense mesh to simulate the geometry of the rough surfaces. Alternatively, describing the thermal contact as nonideal makes more practical sense and accomplishes the same goal.
Left: Simulation results show the air flow patterns and velocity. Right: The microchannel’s resulting temperature.
The plots above show the resulting air flow patterns and velocity in addition to the temperature profile. The air flow speed increases upon exiting the outlet due to the increase in temperature. The temperature jumps by about 0.7 K at the thermal contact point due to light contact pressure. The resulting contact conductance creates a rate of approximately 8900 W/(m^{2}·K).
With heat transfer analysis software, we can evaluate whether the MMC heat sink’s transfer abilities surpass the heat generated by the electronic component. Through the simulation results, we know that this MMC heat sink design is effective because it transfers a large amount of heat away from the device. Heat sinks that prevent devices from overheating not only benefit laptops, but can help to boost the performance of other electronic devices as well.