Miniature devices have many applications and researchers are constantly finding new uses for them. One such use, which we’ve blogged about before, is a microfluidic device that could let patients conduct immune detection tests by themselves. But to work in the microscale, devices like this one, of course, rely on even smaller components such as micropumps.
Let’s turn to a tutorial model of a valveless micropump mechanism that was created by Veryst Engineering, LLC using COMSOL Multiphysics version 5.1.
The micropump in the tutorial model creates an oscillatory fluid flow by repeating an upstroke and downstroke motion. The fluid flow enters a horizontal channel containing two tilted microflaps, which are located on either side of the micropump. The microflaps passively bend in reaction to the motion of the fluid and help to generate a net flow that moves in one direction. Through this process, the micropump mechanism is able to create fluid flow without the need for valves.
The geometry of the micropump mechanism tutorial.
Please note that the straight lines above the microflaps are there to help the meshing algorithm. Check out the tutorial model document if you’d like to learn how this model was created.
The tutorial calculates the micropump mechanism’s net flow rate over a time period of two seconds — the amount of time it takes for two full pumping cycles. The Reynolds number is set to 16 for this simulation so that we can evaluate the valveless micropump mechanism’s performance at low Reynolds numbers. The Fluid-Structure Interaction interface in COMSOL Multiphysics is instrumental in taking into account the flaps’ effects on the overall flow, as well as making it an easy model to set up.
Left: At a time of 0.26 seconds, the fluid is pushed down and most of it flows to the outlet on the right. Right: At a time of 0.76 seconds, the fluid is pulled up and most of it flows from the inlet on the left.
The simulation starts with the micropump’s downstroke, which is when the micropump pushes fluid down into the horizontal channel. This action causes the microflap on the right to bend down and the microflap on the left to curve up. In this position, the left-side microflap is obstructing the flow to the left and the flow channel on the right is widened. This naturally causes the majority of the fluid to flow to the right, since it is the path of least resistance.
During the following pumping upstroke, fluid is pumped up into the vertical chamber. Here, the flow causes the microflaps to bend in opposite directions from the previous case. This shift doesn’t change the direction of the net flow, because now the majority of the fluid is drawn into the flow channel from the inlet on the left.
Due to the natural deformation of the microflaps caused by the moving fluid, both of these stages created a left-to-right net flow rate. But how well did the micropump mechanism do at maintaining this flow over the entire simulation time period?
The net fluid volume that is pumped from left to right.
During the two-second test, the net volume pumped from left to right was continually increased, with a higher net flow rate during peaks of the stroke speed. This valveless micropump mechanism can function even at a lower Reynolds number.
The valveless micropump mechanism could have many future applications, one of which is to work as a fluid delivery system. In such a scenario, a micropump mechanism could take fluid from a droplet reservoir on its left and move it through a microfluidic channel to an outlet on its right. In this post we have shown just one set of simulation results. By experimenting with the tutorial model set up by Veryst Engineering, you can visualize how a valveless micropump may work in different situations and use this information to discover new uses for micropump mechanisms.
Marangoni convection — also called thermocapillary convection — is important in a number of processes, including welding, crystal growth, and electron beam melting. Due to the types of metals used and the extremely high temperatures involved, performing experiments to analyze Marangoni convection often proves to be rather challenging. The impact of gravity, which mixes up this convective effect with the Marangoni effect, also adds to the difficulty of studying this phenomenon.
At NASA, researchers analyzed Marangoni convection to see how mass and heat move within a fluid under microgravity conditions. Conducting the experiment in microgravity enabled the research team to create silicone oil columns much larger than those that could be studied on Earth, offering a more detailed look at the flow and instability within them. Additionally, suppressing the influence of gravity helped eliminate the possibility of gravity-induced deformation, thus enhancing the accuracy of their results.
With numerical experiments, it is very easy to separate effects that are simply impossible to remove in an experiment on Earth. Our Marangoni Effect tutorial uses a transparent liquid at ambient temperatures to find the velocity field induced through the Marangoni effect in a fluid with known thermo-physical properties. The transparency of the silicone oil makes it easy to implement and compare our simulation results with the microgravity experimental findings.
To begin, we must solve the Navier-Stokes equations to model the velocity field and pressure distribution in the fluid. Keep in mind that variations in temperature affect the velocity and cause a buoyancy force that needs to be represented in the equations. This can be done by using the Boussinesq approximation in the Navier-Stokes equations.
With the Laminar Flow interface, we can solve the momentum balance equations. To solve for heat transfer, we use the Heat Transfer in Fluids interface. Finally, we use the Non-Isothermal Flow multiphysics coupling to set the convective term in the heat equation and the Marangoni Effect multiphysics coupling to impose that the shear stress is proportional to the temperature gradient.
The setup of the tutorial model. The Multiphysics node contains both the nonisothermal coupling and the Marangoni effect.
This simulation presents three multiphysics couplings that must be solved using the nonlinear solver:
In our simulations, we analyze a gradual increase in temperature difference between vertical walls. For an almost unnoticeable temperature increase of 1 mK, the temperature field and velocity field have only a slight relation, and the decrease appears linear from left to right.
The results of a Marangoni effect simulation after only a small change in temperature. The background color represents the temperature field and the red arrows indicate the velocity field. The black lines are isotherms.
With an increase of 50 mK, Marangoni convection increases the fluid flow and temperature distribution. The temperature decrease is no longer linear across the plot.
The results of the simulation after a temperature increase of 50 mK.
Finally, we test a temperature difference of 2 K. The temperature and velocity fields are distinctly coupled and the fluid accelerates at the surface where the temperature gradient is highest.
The results of the simulation when the temperature difference is raised to 2 K.
As indicated by the simulation results, the Marangoni effect becomes predominant as the difference in temperature increases.
For the same temperature difference of 2 K, we can easily remove the gravity contribution and keep the Marangoni effect. With the same objective of understanding how buoyancy forces compare with the Marangoni effect, we can simply disable the Marangoni contribution at the surface, leaving the surface free of stress. The results show that the Marangoni effect is predominant versus buoyancy forces. The shape of the curve shows a peak close to the cold right wall, which is characteristic of the fluid behavior of high Prandtl numbers.
The results of the horizontal velocity at the surface versus the horizontal coordinate (m) for a temperature difference of 2 K. Blue represents both the Marangoni effect and the buoyancy effect; green represents only the Marangoni effect; and red represents only the buoyancy effect.
In this blog post, we have demonstrated how to set up a model representing an experiment combining gravity and Marangoni effects. Separating these two effects is challenging in an experimental setting. In numerical simulations, this process is straightforward, facilitating an understanding of each effect.
You can reproduce the results shown here by downloading the Marangoni Effect tutorial from our Application Gallery. This example uses the Non-Isothermal Flow and Marangoni Effect multiphysics couplings available in the Heat Transfer Module.
While we have focused our attention here on single-phase flows, it is worth mentioning that the Marangoni effect is also handled in the two-phase flow interfaces, which are available in the CFD Module and the Microfluidics Module.
]]>Since we released version 5.0 of the COMSOL Multiphysics® software, you have the ability to create simulation apps — either starting from scratch or with a demo app from the Application Library. Today, we’ll introduce you to an app that can be used for understanding and optimizing mixer design and operation for a given fluid. The exemplified application models and simulates stirred tank mixers, which are used for reactors in the fine chemical, pharmaceutical, food, and consumer products industries.
In addition to the aforementioned industries, stirred batch reactors are also frequently used for lab-scale kinetic studies and when developing new processes and synthesis. In all of these processes, it’s required to obtain a relatively uniform reactor solution composition and temperature. Achieving this would allow for a reproducible and uniform product quality.
By creating an app, you can provide a user-friendly simulation environment where scientists, process designers, and process engineers can investigate the influence that vessel, impeller, and operational conditions have on the mixing efficiency and the power required to drive the impellers. We created the Mixer app to help you get started building such an app on your own.
A challenge when designing applications is getting an automatic update of geometry, physics, and mesh settings for fully parameterized geometries. It can also be difficult to include completely different geometry objects, depending on the user’s input when running the app. The Mixer application demonstrates the use of geometry parts and cumulative selection for automatic model settings.
In addition, the application also demonstrates how to build an app using COMSOL Multiphysics and how geometry parts and cumulative selections can be used to automatically set domain and boundary settings in models embedded in an application. These settings can be created automatically, even when the choices an app user makes create very diverse geometries.
The demo app can be used as a starting point for more elaborate mixer apps containing more options for the fluid flow. For example, two-phase flow and non-Newtonian flows, as well as for vessel and impeller geometries.
The annotated screenshot of the user interface (UI) below shows the 11 different types of impellers (1) that can be added to a model and the different tank types (2): dished, flat, and cone bottom, with and without baffles. The dimensions of the impellers and vessels can be set for the different types. The fluid properties and rotational speed of the impeller are selected in the Fluid Properties & Operating Conditions section (3). The Home ribbon tab (4) contains the mesh and compute buttons, which generate the numerical model and solve the model equations.
The results show the eddy diffusivity and the velocity field, both in 3D and in a vertical cut plane along the reactor (5). The cut planes can be rotated to show the results at different angles around the shaft (6). The results also give an estimate of the mixing time scale (7).
Annotated screenshot of the Mixer app.
In this specific case, the impeller is equipped with three axial impellers with c-shaped double blade sections, so-called Ekato® Intermig impellers, distributed over the length of the impeller shaft. The cut plane shows the eddy diffusivity, which is a measure of the local mixing, where red is “good” and blue is “bad”.
The 11 different types of impellers give great freedom in the creation of various impeller shapes. For example, by using hydrofoils or constant-pitched impellers in combination with cuts and fillets, you can create different propeller-type impellers.
Different impeller shapes created with the hydrofoil with constant pitch impeller type, which is one of 11 available impeller types in the app.
The pitched impeller with folded blades can create asymmetric blades with rounded folds using fillets.
The model geometry is defined in the embedded model by a number of so-called geometry parts. For example, there are eleven different parts, one for each impeller type.
The parts are called in the main geometry sequence by the part instances (1). The part instances use parameters as input. You may compare the part instance to a function call with the input parameters as arguments. These input parameters (2) control the dimensions and the configuration of the geometry defined by the part. For example, the c-shaped double-blade axial impeller may receive parameters for the radius of the impeller and the number of blades. The geometry defined by the part may also be positioned relative to other objects in the main geometry sequence, such as relative to a work plane in the main sequence (3), for instance.
The output of the part is the geometry object (or several objects) itself together with a number of selections. An example of a selection created in a part is the Impeller Blades selection.
The selection created by the part instance may contribute to selections for domain, boundaries, edge, and point selections. For example, the Arm selection and the C-shaped blades selection both contribute to the boundary selection “Impeller Blades” (4). The “Impeller Blades” selection is referred to as a cumulative selection.
Side note: We have covered cumulative selections in more detail in this blog post.
The part instance in the embedded model takes the output selections from the part and gathers them in cumulative selections according to the settings specified in its settings window.
In the same way, several impellers and impeller types can contribute to the Impeller Blades selection. When setting boundary conditions, the cumulative selections listed in the Contribute to column will be available in the list box for all boundary condition features.
The same methodology is used for the definition of the vessel and the baffles attached to the vessel wall.
By creating selections (in the geometry part) that contribute to other cumulative selections defined in the part instance, we allow for the automatic update of domain, boundary, edge, and point selections in the physics, mesh, and plot settings. A change in the geometry automatically updates the selections in all other settings, see the figure below.
The physics in the model are quite straightforward. The model equations are the Navier-Stokes equations for fluid flow combined with a distributed algebraic equation, defined in every point in space, for the turbulent viscosity. The algebraic equations use the distance from no-slip walls as a variable in the equations. This is computed in the model by solving a wall distance function equation. The flow equations are solved using the frozen rotor approximation for rotating machinery.
The cumulative selections are extensively used in order to automatically update the domain settings and the boundary conditions. For example, the Rotating Interior Wall boundary condition in the Fluid Flow interface gets its selection from the Impeller Blades cumulative selection (see the figure below).
Screenshot detailing that the Rotating Interior Wall receives its selection from the associated “Impeller Blades” cumulative selection.
The embedded model uses a physics-controlled mesh. The mesh, which is produced automatically, includes boundary layer meshing in order to resolve the boundary layers formed along no-slip walls and rotating walls.
The solution is computed in two steps by the solver. The first step computes the wall distance, since this problem is independent of the flow field. The second step computes the solution to the Navier-Stokes equations and the turbulence equation. The turbulence equation uses the wall distance function as input.
Image showing the resulting eddy diffusivity and velocity streamlines in the Mixer app with three c-shape double blades as above but with four baffles added.
The figure below shows the Mixer app in the Application Builder’s Form Editor, where the user interface can be designed using drag-and-drop of different graphics and widgets selected in the ribbon menu.
The application UI consists of two areas:
The application UI during construction in the Application Builder.
The Application Builder screenshot above hints at the structure of the application’s UI. The form denoted main (1) is linked to the Main Window node (2) using the form reference in the Main Window settings (2). The main form contains two form collections with tabs.
The first form collection contains the general (3) and impeller_settings forms (4), with the titles General and Impeller shown in their respective tabs (5). The general and impeller_settings forms are in turn section form collections including all other forms that begin with general_ and impeller_ in the Application Builder tree. Section form collections group their member forms in sections, for example the general_tank form (6) that has Tank Type & Dimensions as the section title (6).
The second form collection in the main form contains the graphics forms (7). The graphics forms are also grouped in tabs and are the forms that begin with graphics_ in the Application Builder tree. For example, the graphics_geometry form (8) is the form with the title Geometry (9) and is shown at the top in the screenshot above.
Almost any widget contained in the forms can be linked to a command in the embedded model. For example, the Tank Type list box (10) is linked to a method that selects the tank type that will be built when the application user updates the geometry. The Update button widget (11) in the ribbon is in turn linked to the method that updates the geometry in the embedded model.
Despite its apparent complexity, the Mixer application is quite straightforward to create once the embedded model is available. The design of the geometry parts and the cumulative selections also make the embedded model relatively easy to parameterize and automate.
Future versions of the application will contain two-phase flow as well as data output for required impeller torque, flow number, and pumping capacity.
Ekato® is a registered trademark of Ekato Holding GmBH.
]]>The semiconductor industry uses ion implanters to implant dopants into wafers. To optimize the design of these devices, engineers need to quickly and easily test a wide range of parameters. Simulation apps help streamline the design process of ion implanters by sharing the capabilities of a simple and fully customizable interface with colleagues who don’t have a simulation background. Here, we introduce you to our Ion Implanter Evaluator demo app.
Known as doping in semiconductor device fabrication, ion implantation involves using an electric field to accelerate ions generated within an ion source to achieve a desired energy for successful implantation. For ions of a certain charge-to-mass ratio to reach the wafer, a selection magnet chooses ions of the correct charge state and bends the ion beam. In addition to ensuring that only certain ions reach the wafer, only specific parts of the wafer should be implanted. An organic photoresist is used to mask parts of the wafer and create the desired pattern.
One common issue with ion implanters is that when struck by the ion beam, the photoresist emits gas molecules of its own. When these outgassing molecules interact with the ion beam, it results in undesirable charge-to-mass ratios at different points along the path of the beam. If the molecules reach the wafer, it degrades the uniformity of the implant and affects the accuracy of the measurements for implant doses. To avoid this issue, the number density of the outgassing molecules should remain as low as possible within the ion beam line.
By transforming this model into an app, you can evaluate key parameters to ensure that these undesirable effects are avoided. The simple, intuitive interface of an app provides colleagues with the ability to make their own changes to your ion implanter simulation.
The Ion Implanter Evaluator simulation app is based on our Ion Implanter Evaluator tutorial model, created with the Molecular Flow Module. The ion implanter includes a wafer held on a carrier plate that can rotate its center axis to create different angles for implantation. The plate is positioned in a chamber with three large cryopumps placed on cylindrical vacuum ports. The device also features a vacuum path through the magnetic field that enters the main chamber opposite the wafer. There’s an additional cryopump at the beginning of the beam path and an aperture at the entrance of the chamber that reduces flux in the connector.
Model geometry of an ion implanter.
The user interface for the ion implanter demo app contains a range of simple tools designed to be used by simulation engineers and non-simulation experts alike. The ribbon contains different buttons that you can use to solve the app, such as:
There are also fully customizable forms, shown on the left side bar, which allow you to:
The simple user interface for the Ion Implanter Evaluator demo app.
The parameters for the model above can be easily and repeatedly changed by both you and your colleagues. You are able to test the angle of the wafer to find optimal placement to keep the number density of the outgassing molecules low, and therefore preserve the uniformity of the wafer. Other possible parameter adjustments include:
By creating your own ion implanter app, you can include all or some of these parameters, as well as a range of other possibilities based on your own simulation needs. You can easily visualize the results of your parameter inputs on the right side of the screen. In our demo app, we are able to view the number density, pressure, molecular flux, and the average number density along the beam line to start evaluating and optimizing the ion implanter.
Ensuring that you are able to model an ion implanter with optimal performance will aid in more efficient and reliable semiconductors.
When fluid flow passes a body, it will exert a force on the surface. As shown in the figure below, the force component that is perpendicular to the flow direction is called lift. The force component that is parallel to the flow direction is called drag. For simplicity, let’s assume that the flow direction is aligned with the coordinate system of the model. Later on, we will show you how to compute the lift and drag forces in a direction that is not aligned with the model coordinate system.
Schematic of lift and drag components when fluid flow passes a body.
There are two distinct contributors to lift and drag forces — pressure force and viscous force. The pressure force, often referred to as pressure-gradient force, is the force due to the pressure difference across the surface. The viscous force is the force due to friction that acts in the opposite direction of the flow. The magnitudes of pressure force and viscous force can vary significantly, depending on the type of flow. The flow around a moving car, for instance, is often dominated by the pressure force.
COMSOL Multiphysics offers complete access to all of the internal variables and makes it very easy to compute surface forces via integration on a boundary. Here, we will demonstrate how to compute the drag forces on an Ahmed body. You can download this model from our Application Gallery.
Simulation of airflow over an Ahmed body. The surface plot shows the pressure distribution, and the streamlines are colored by the velocity magnitude. The arrow surface behind the Ahmed body shows the circulation in the wake zone.
There are several ways to compute drag depending on the physics. The most straightforward way is to integrate the total stress — which includes contributions from the pressure force and the viscous force — in each direction. To do so, we first need to define a surface integration operator under the Derived Values node, as illustrated below.
TIP: Alternatively, you can also use a boundary probe or integration operator in the component coupling to define such integration. The difference is that the operations defined in the physics setting can be used during the simulation — for example, drag force computed with a boundary probe as an objective or a constraint in an optimization study.
Next, we can select the boundaries to perform the integration. In this example, we chose all of the boundaries on the body. Drag in this model is in the y-direction. We can type in the expression: spf.T_stressy
, which represents the total stress in the y-direction.
Sometimes, engineers can obtain greater insight into designs by examining the pressure force and viscous force separately. COMSOL Multiphysics features a predefined variable, spf.K_stressy
, for viscous stress in the y-direction. We can readily evaluate the viscous force by integrating the viscous stress.
What about the pressure force? Pressure, denoted by the variable p
, is a scalar. To project in the direction of drag, we need to multiply pressure by the y-component of the normal vector on the surface, spf.nymesh
. Therefore, we can evaluate the pressure force by integrating spf.nymesh*p
on the surface.
In some special turbulent flow cases where the wall function is used, it is more accurate to compute the viscous force using the friction velocity, spf.u_tau
. In COMSOL Multiphysics, the k-epsilon and k-omega turbulence models use the wall function.
To learn more about turbulence models in COMSOL Multiphysics, read our blog post “Which Turbulence Model Should I Choose for My CFD Application?“.
We can obtain the local shear stress at the wall by:
Therefore, the local shear stress in the y-direction is:
where u^T is the tangential velocity at the wall. We can further rewrite u^T as u_\tau*u^+, where u^+ is the tangential dimensionless velocity.
Without going into too many details on derivation, we can translate the previous equations into COMSOL variables. We can integrate the local wall shear stress in the direction of drag (the y-direction) with the following expression: spf.rho*spf.u_tau*spf.u_tangy/spf.uPlus
. In this expression, spf.rho
is the density of the fluid, spf.u_tangy
is the velocity in the y-direction at the wall, and spf.uPlus
is the tangential dimensionless velocity.
The table below summarizes the expressions used to compute each force.
Fluid Flow Without Wall Function | Turbulent Flow with Wall Function | |
---|---|---|
Pressure Force | spf.nymesh*p | spf.nymesh*p |
Viscous Force | spf.K_stressy | spf.rho*spf.u_tau*spf.u_tangy/spf.uPlus |
Total Force | spf.T_stressy | spf.nymesh*p + spf.rho*spf.u_tau*spf.u_tangy/spf.uPlus |
Note: In this example, the drag force is in the y-direction. You may need to change the projection direction based on the orientation of your model.
It is common that the geometry may not be aligned perfectly with the flow direction. The angle between the center reference line of the geometry and the incoming flow is called angle of attack (often denoted by the Greek letter \alpha). In aerospace engineering, the angle of attack is frequently used as it is the angle between the chord line of the airfoil and the free-stream direction. The following figure shows the relationship between lift, drag, and angle of attack on an airfoil.
Schematic illustrating lift, drag, and angle of attack on an airfoil.
Using a 2D NACA 0012 model, we will show you how to compute lift with an angle of attack correction. This model is available for download in our Application Gallery.
There are two ways to change the angle of attack of the model. We can either rotate the geometry itself or we can keep the geometry fixed but modify the flow direction at the inlet. Here, we will use the second approach. It is much simpler to adjust the velocity field at the inlet boundary condition as we would not need to remesh the model for each angle of attack. As shown in the figure below, the airfoil is fixed while the streamlines show the flow at an angle of attack due to the adjusted inlet velocity direction.
Simulation of flow passing a NACA 0012 airfoil at a 14-degree angle of attack. The surface plot shows the velocity magnitude along with the streamlines (shown in black).
This example uses the SST turbulence model, which does not use the wall function. Therefore, we will use total stress to compute lift. At a zero angle of attack, the lift is simply spf.T_stressy
. If the angle of attack is nonzero, we can project the force onto the direction of the lift using the following expression: spf.T_stressx*sin(alpha*pi/180)-spf.T_stressy*cos(alpha*pi/180)
. Here, alpha represents the angle of attack in degrees.
You may also be interested in the nondimensionalized forms of lift and drag — the lift coefficient and the drag coefficient. It is often easier to use the coefficients instead of the dimensional forces for the purpose of validating experimental data or comparing different designs. The lift coefficient in 2D is defined as:
Since we have already calculated the dimensional lift, we can simply normalize the lift by the dynamic pressure and the chord length. With the dimensionless lift coefficient, we can compare our simulation results with experimental data (Ref. 1).
Note: In 3D, the lift coefficient is nondimensionalized by area instead of length: C_L = \frac{L}{\frac{1}{2} \rho U_\infty A}
Graph comparing simulation results and experimental data of the lift coefficient on a NACA 0012 airfoil at various angles of attack.
As illustrated in the above graph, no discernible discrepancy between the computational and experimental results occurs within the range of the angle of attack values used in this simulation. The experimental results continue toward a high angle of attack regime where the airfoil stalls.
In this blog post, we have explored ways to compute lift and drag on an Ahmed body and an NACA 0012 airfoil. We have demonstrated how to compute pressure force and viscous force, while also examining the special case where a wall function is used in the turbulence model.
Each of the approaches we have presented here are certainly not limited to these specific simulations. You can compute the body forces on any boundaries or surfaces, thereby gaining insight into designs through multiphysics simulations. With the Optimization Module, you can take this analysis one step further and optimize lift or drag.
In CVD, a substrate is exposed to chemical processes that will react or decompose onto its surface to create a deposit, or thin film. We turn to CVD to create high-quality and strong materials, such as graphene, the unique and powerful material that continues to make headlines in the science world. CVD is also valued for its ability to produce a wide range of versatile products, from carbon nanofibers to synthetic diamonds. In semiconductor applications, CVD is used for epitaxial layer growth of high-purity silicon.
Graphene is one type of material that can be formed through UHV/CVD. Image attributed to AlexanderAlUS. Licensed under Creative Commons Attribution-Share Alike 3.0 Unported via Wikimedia Commons.
UHV/CVD is performed at pressures below 10^{-6} Pa, or about 10^{-8} torr. In this process, molecular flow is used to achieve gas transport. UHV/CVD doesn’t involve any hydrodynamic effects such as boundary layers and due to the low frequency of molecular collisions, there is no gas-phase chemistry. The growth rate of the material on the substrate is determined by the molecular flux of the species arriving on the surface. Graphene can also be grown using UHV/CVD, as my colleague Daniel Smith wrote about in Part 4 of our blog series on “The Graphene Revolution“.
As previously mentioned, UHV/CVD experimentation is costly in both time and money, requires exotic materials and sophisticated thermal management. Fortunately, the Free Molecular Flow interface in the Molecular Flow Module (an add-on to the COMSOL Multiphysics simulation platform) can simulate this chemical process. For this model, wafers are arranged close together on a moveable boat within a quartz tube, surrounded by a furnace. The wafer cassette is positioned within the tube by a delivery rail.
A reactant gas, with a ballast gas, enters the reaction chamber at one end through a load lock. A turbo pump is located at the other end of the chamber.
The model geometry of a reaction chamber for a UHV/CVD process.
For this simulation, the reactant gas, silane, enters the inlet of the system with the ballast gas, hydrogen, at a standard mass flow rate of 1 SCCM combined in a ratio of 20% silane and 80% hydrogen. An outgassing wall boundary condition is set for the inlet of the chamber. On the other end of the chamber, a vacuum pump is positioned at a cylindrical port.
Simulation allows us to analyze the process from a range of pumping curves. Three different pumping curves are analyzed for both silane and hydrogen. We enter these curves into the COMSOL Multiphysics software as interpolation functions. Then, we can use a parametric sweep to investigate how the different gases are transported for different pumping curves.
In the surface plot below, we can see the silicon growth on the wafer when using one of the pumping curve options.
The molecular flux fraction of silane at the wafer cassette, which controls the growth on the wafer.
The molecular flux fraction of silane at the surface of the wafer (0.04) is significantly lower than that at the inlet (0.2). This is because hydrogen is much more difficult to pump than silane, due to its lower molecular weight. Since this measurement directly controls the amount of growth on the wafer, the choice of pump for the silane and hydrogen is important to the amount of material yielded through UHV/CVD. As the molecular flux fraction of each material is hard to measure in a physical experiment, simulation helps to analyze and optimize the UHV/CVD process.
In the field of MEMS and lab-on-a-chip devices, microfluidic biochips don’t require a lot of power or sample volume to operate. They are also relatively cheap, offer a fast response time, and are overall highly efficient. Another great advantage is their ability to integrate operations such as detection, sample pre-treatment, and sample preparation onto a single chip. Microfluidic biochips are used in a wide range of applications, including inkjet printer heads, micro drug delivery systems, and DNA and clinical pathology devices.
There are two kinds of microfluidic biochips:
Generally, microfluidic samples are continuous flow, such as drops of blood or saliva for diagnostic tests. This leads to two problems: You cannot control the droplet volume and the excess liquid can’t be removed in the case of overpressure on the device. In cases like these, for analog microfluids to be processed by a DMFB, an analog-to-digital converter (ADC) is needed on the device.
ADCs are microfluidic dispensers that can dispense manipulated droplets from continuous flow or a reservoir of the sample. These devices are fundamental for integrating analog and digital microfluidic biochips into a mixed-signal microfluidic device. Surprisingly, there is not a great amount of research on integrating an ACD into a microfluidic biochip design. Answering the call, a team of researchers from the University of Bridgeport in Connecticut set out to design and simulate a high-throughput microfluidic droplet dispenser as an analog-to-digital microfluidic converter for use in lab-on-a-chip (LOC) applications.
A schematic of a microfluidic droplet dispenser. Image credit: C. Jin, X. Xiong, P. Patra, R. Zhu, J. Hu, University of Bridgeport, Bridgeport, Connecticut. Taken from their COMSOL Conference 2014 Boston paper submission.
The research team aimed to understand the mechanism of a high-throughput droplet dispenser as well as verify its function as an interface between analog and digital microfluidic biochips on an LOC device.
First, they used the Laminar, Two-Phase Flow interface with the level set method in COMSOL Multiphysics to simulate the droplets as they move and split in the dispenser by simulating the electrowetting process. This was done to better understand microfluidic behavior in general and analyze the time settings needed for the next phase of the simulation. The researchers were also able to calculate the force needed to move the droplets, evaluate the droplet shape and movement, as well as analyze the voltage needed to cause their movement.
Simulating microfluidic droplets in a dispenser as they move and split. Image credit: C. Jin, X. Xiong, P. Patra, R. Zhu, J. Hu, University of Bridgeport, Bridgeport, Connecticut. Taken from their COMSOL Conference 2014 Boston paper submission.
Next, the team simulated the droplet dispenser as an interface between analog and digital microfluidic flow. To reduce simulation time, only two digital output ports were included on the model. With COMSOL Multiphysics, the team easily selected mesh elements to create a finer meshed model of the droplet dispenser.
A meshed model of the digital droplet dispenser. Image credit: C. Jin, X. Xiong, P. Patra, R. Zhu, J. Hu, University of Bridgeport, Bridgeport, Connecticut. Taken from their COMSOL Conference 2014 Boston paper submission.
The behavior of the droplets in the output ports was simulated in both alternate and parallel modes to analyze the efficiency of each method.
Simulating the parallel (left) and alternate (right) modes of a droplet dispenser allows researchers to analyze the behavior of the device. Image credit: C. Jin, X. Xiong, P. Patra, R. Zhu, J. Hu, University of Bridgeport, Bridgeport, Connecticut. Taken from their COMSOL Conference 2014 Boston paper submission.
After simulating the droplet dispenser, the researchers verified its function and further concluded that dispensing the droplets in both alternate and parallel modes works effectively for integrating analog and digital microfluidics on a single lab-on-a-chip device. Hopefully, this research will inspire more improvements to these devices, advance the field of microfluidics, and in turn improve clinical diagnostics and other applications.
To learn more about the digital droplet dispenser, download the paper and presentation from the COMSOL Conference 2014 Boston: “Design and Simulation of High-Throughput Microfluidic Droplet Dispenser for Lab-on-a-Chip Applications“.
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Introduced by L.V. King in 1914, the hot-wire anemometer represented the first of its kind: A device that could measure fluid flow using thermal sensing techniques. In a hot-wire anemometer, a thin wire is electrically heated to a temperature greater than its surroundings. The surrounding flow cools down the device up to a given temperature. Because of the known relationship between the electrical resistance of the wire and its temperature, the fluid velocity can be obtained from the knowledge of the wire’s resistance.
Because of their delicate nature, hot-wire anemometers are often not suitable for industrial use as many applications include dirt, which can cause damage to these fragile devices. Hot-wire anemometers are also referred to as intrusive devices, because the sensors do not only measure the flow properties, but they also disrupt the flow. A more viable solution in such cases would be to use a thermal mass flow meter. The thermal mass flow meter is a non-intrusive device, i.e., it leaves the flow path unobstructed. While applying the same concepts as their predecessor, these instruments feature casing around the wires, enhancing their durability as well as their accuracy in measuring fluid flow.
A schematic of a thermal mass flow meter.
Thermal mass flow meters have been used extensively for measurement in gas flow applications, from thermal transfer to chemical reactions. These instruments are particularly favored in the industry for their simplistic design, as there are no moving parts included in the device.
A team from the University of Cambridge used COMSOL Multiphysics software to develop a 3D model of thermal flow sensors and analyze the dynamics behind the components’ operation. Let’s explore how simulation enabled this team to describe the behavior of this instrument under every physical aspect.
The design of the model used in the study was based on a silicon on insulator (SOI), complementary metal-oxide semiconductor (CMOS) MEMS thermal flow sensor — or SOI CMOS MEMS thermal flow sensor, for short. The model features a validation chip that includes five parallel metal strips. The strip in the center is used to increase the temperature of the device to 300°C. All of the strips can be used to sense the temperature through the relationship between the metal resistivity and the absolute temperature. A four-wire measurement is used to obtain the resistance value.
When developing this validation chip, deep reactive ion etching was implemented at the back surface to remove the silicon substrate from beneath the sensing elements. This postprocessing step drastically reduced the thermal conductivity observed by the heating element and, consequently, lowered the power required to increase the temperature at the desired value.
The geometry of the thermal flow sensor. The image on the left shows the cross-sectional view of the validation device, while the image on the right shows the top view. Image by C. Falco, A. De Luca, S. Sarfraz, and F. Udrea, and taken from their COMSOL Conference 2014 Cambridge paper submission.
In their analysis, the research team coupled three different physical domains — electric current, heat transfer in solids, and laminar flow — to create a multiphysics model. That is, the bias current is used to locally heat up the component, through the Joule heating effect, and conductive and convective heat transfer dissipate the excess heat.
Initially, the flow sensor was validated in still air. The plot below represents a validation of the relationship between dissipated power in the heater and the temperature sensed by alternate resistors without flow above the surface. The values from the simulation and the experiment are shown to mirror one another.
Comparing the temperature in all the resistors within the simulation and experimental data. Image by C. Falco, A. De Luca, S. Sarfraz, and F. Udrea, and taken from their poster submission.
The figure below provides a complete temperature profile for the current value of 10 mA.
Temperature profile in the area of interest. Image by C. Falco, A. De Luca, S. Sarfraz, and F. Udrea, and taken from their poster submission.
The wall shear stress, defined as the stress that a viscous fluid exerts on a wall, is chosen to characterize the fluid properties. In the next step, the research team calibrated the sensor, with the air movement above the chip included in the analysis, for varying values of wall shear stress. Comparing three values for the biasing current (6, 8.5, and 10 mA), the results showed good agreement between the temperature within the resistors and the experimental data for wall shear stress.
Plots depicting the sensor output as a function of wall shear stress. The graph on the top (a) shows the calorimetric approach and the graph on the bottom (b) represents the anemometric approach. An anemometric approach involves the measurement of the changes in the voltage across the heater; the calorimetric approach senses the variance in voltage between resistors that are placed symmetrically on opposing sides of the heater. Image by C. Falco, A. De Luca, S. Sarfraz, and F. Udrea, and taken from their paper submission.
Here, we have introduced research designed to investigate the behavior of thermal flow sensors. Coupling heat transfer, electric current, and laminar flow, the simulation provides accurate predictions of the sensor’s behavior and, by modifying the model’s geometry and material properties, can be applied to various applications of this technology. With its high precision and range of applications, the thermal flow sensor model serves as a powerful resource in optimizing the design of thermal flow sensors and developing prototypes more efficiently.
For our example, we will use a model that couples the Navier-Stokes equations and the heat transfer equations to model natural convection in a square cavity with a heated wall. The temperature on the left and right walls is 293 K and 294 K, respectively. The top and bottom walls are insulated. The fluid is air and the length of the side is 10 cm.
We will use this model to compare the computational cost of three different modeling approaches:
Each of these three approaches and their variables are defined here.
In COMSOL Multiphysics, the model is solved with a stationary study using the Laminar Flow, and Heat Transfer in Fluids interfaces, and the Non-Isothermal Flow multiphysics coupling:
While setting up the model, it is important to check whether the flow is laminar or turbulent. For a natural convection problem, this is done by calculating the Grashof number, Gr. For an ideal gas, it is defined as
The Grashof number is the ratio of buoyancy to viscous forces. A value below 10^8 indicates that the flow is laminar, while a value above 10^9 indicates that the flow is turbulent. In this case, the Grashof number is around 1.5 \times \hspace{1pt} 10^5, meaning that the flow is laminar.
When using the full Navier-Stokes equation, we set the buoyancy force to \rho \mathbf{g}:
The buoyancy term is added using a volume force feature. The terms nitf1.rho and g_const represent the temperature- and pressure-dependent density, \rho, and the gravitational acceleration, \mathbf{g}, respectively.
When using the Navier-Stokes equations with pressure shift, we have to make three changes.
First, we need to change the definition of the volume force to (\rho-\rho_0)\mathbf{g}, as such:
The term rho0 refers to the reference density \rho_0.
Next, we evaluate the reference density \rho_0 and the reference viscosity \mu from the material properties in a table of variables:
Here, pA and T0 represent the reference temperature and pressure.
The air viscosity is set to the constant \mu_{0}:
Finally, when using the Boussinesq approximation, we need to set the buoyancy force to -\rho_0\frac{T-T_0}{T_0}\,\mathbf{g}:
As with Approach 2, we also evaluate the reference density and viscosity from the material properties. A third and final step with Approach 3 is to set the fluid density to the constant reference density \rho_{0} (the Boussinesq approximation states that the density is constant except in the buoyancy term).
Note: If your model includes a pressure boundary condition (open domain), set the pressure to the hydrostatic pressure -rho0*g_const*y for Approach 1 or to 0[Pa] for Approach 2 and Approach 3. The boundary conditions for models including gravitational forces are also discussed here.
The mesh is made of 15,000 triangular elements and 1,200 boundary layer elements. These are first-order elements.
The resulting velocity magnitude and streamlines are nearly identical for all three approaches. The maximum temperature difference between Approach 1 and 2 is less than 2 \times \hspace{1pt} 10^{-6} K and the maximum temperature difference between Approach 1 and 3 is around 5 \times \hspace{1pt} 10^{-4} K. The only thing that differs is the simulation time.
Velocity magnitude and streamlines.
Because of the short running time of this 2D simulation (around 30 seconds), we look at the computational load by comparing the number of iterations it takes the solver to converge to the steady-state solution. The number of iterations, in this case, is nearly proportional to the CPU time.
The table below compares the number of iterations across all three approaches.
Approach 1 | Approach 2 | Approach 3 | |
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Number of Iterations | 39 | 55 | 55 |
These results are very surprising!
While the Boussinesq approximation is supposed to reduce the nonlinearity of the model and the number of iterations required for convergence, the full Navier-Stokes equations (39 iterations) can be solved faster than the Boussinesq approximation (55 iterations). We also note that the use of Navier-Stokes equations with a pressure shift leads to the same number of iterations as the Boussinesq approximation.
To better understand these results, we can run a second set of simulations after disabling the pseudo time-stepping algorithm. Pseudo time stepping is used for stabilizing the convergence toward steady state in transport problems. The pseudo time stepping relies on an adaptive feedback regulator that controls a Courant–Friedrichs–Lewy (CFL) number. The pseudo time stepping is often necessary to get the model to converge. In this particular case, however, it is not needed .
Here’s a look at the COMSOL Multiphysics settings window for the default solver settings with pseudo time stepping:
The following snapshot shows the updated solver settings without pseudo time stepping. We recommend that you always keep pseudo time stepping switched on, unless you feel comfortable tuning the solver settings.
Note on the solver settings for natural convection:
Due to the very strong coupling between the laminar flow and heat transfer physics in natural convection modeling, always use a fully coupled solver. The COMSOL software automatically switches to a fully coupled solver when a volume force is added in the laminar flow physics, meaning that you are modeling natural convection.
This second table shows the number of iterations without pseudo time stepping:
Approach 1 | Approach 2 | Approach 3 | |
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Number of Iterations | 9 | 7 | 7 |
These results make more sense than the previous ones with pseudo time stepping. This is because Approach 3, the most linear problem, now converges faster than Approach 1. What is surprising is that Approach 2 and Approach 3 converge with the same number of iterations.
Comparing these results with the first set of results, a speed-up of 8 (from 55 to 7 iterations) is observed for the third approach — the Boussinesq approximation. These results also indicate that the number of iterations in the first set of results not only depend on the linearity of the problem, but also on the tuning of the pseudo time-stepping algorithm.
Here, we have discussed the implementation and benefits of the Boussinesq approximation as well as using the pressure shift method. The results show that, for this particular model, there are no real benefits in terms of computational time for using the Boussinesq approximation, regardless of whether or not pseudo time stepping is enabled. This is generally the case since the Boussinesq approximation is only valid when the nonlinearity is small. A much shorter computational time for the Boussinesq approximation with respect to the full Navier-Stokes equations would indicate that the Boussinesq approximation might not be valid.
Because of the small speed-up observed with the Boussinesq approximation and the fact it is not always easy to know a priori if the Boussinesq approximation is valid, we generally recommend solving for the full Navier-Stokes equations. Implementing the pressure shift (Approach 2 and 3), however, does avoid round-off errors and simplifies the implementation of time-dependent problems as well as models with open boundaries. This will be the object of a future blog entry.
Using Approach 3 (Boussinesq approximation with pressure shift) involves more implementation steps and does not reduce the number of iterations as compared with Approach 2 (Navier-Stokes equations with pressure shift). The final simulation time might be slightly shorter for Approach 3, since it does not require the evaluation of the temperature- and pressure-dependent density and the temperature-dependent viscosity, but this speed-up might not be noticeable.
The number of iterations is reduced by a factor 4 to 8, depending on the chosen approach, by disabling the pseudo time-stepping algorithm. Please keep in mind, however, that most problems will not converge without pseudo time stepping or other load ramping or nonlinearity ramping strategies.
You can set up and solve this model using the CFD Module or the Heat Transfer Module. If you have any questions about the models that I’ve presented here, contact our Technical Support team. If you are not yet a COMSOL Multiphysics user and would like to learn more about our software, please contact us via this form — we’d love to connect with you.
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The drag coefficient quantifies the resistance of an object in a fluid environment. It is not an absolute constant for a body’s shape because it varies with the speed and direction of flow, object shape and size, and the density and viscosity of the fluid. The lower the drag coefficient of an object, the less aerodynamic or hydrodynamic drag occurs. In terms of a car, the lower the drag coefficient, the more efficient the car is. As well as affecting the top speed of a vehicle, the drag coefficient also affects the handling. Cars with a low drag coefficient are sought after, but decreasing the drag drastically can reduce the downforce and lead to loss in road traction and a higher chance of car accidents.
Most cars have an average drag coefficient of between .30 and .35. Boxy cars have a higher number, like the HUMMER® H2 vehicle at .57, while more streamlined and agile cars have a lower number, like the Mercedes-Benz® C-Class® vehicle at .24. Again, this is just an average measurement. The exact drag coefficient of a car varies with the Reynolds number and various other factors.
There are certain ways to modify a car to optimize its aerodynamics and decrease the drag coefficient. To streamline the exterior, you can remove certain aftermarket items such as the roof rack, mud flaps, spoilers, and radio antenna. Professional car racers also take off their windshield wipers and side mirrors, but this isn’t recommended for the average driver! You can also add wheel covers, a partial grille block, an under tray, fender skirts, and a modified front bumper to improve the drag coefficient and make your car stand out in the process.
The Ahmed Body was first created by S.R. Ahmed in his research “Some Salient Features of the Time-Averaged Ground Vehicle Wake” in 1984. Since then, it has become a benchmark for aerodynamic simulation tools. The simple geometrical shape has a length of 1.044 meters, height of 0.288 meters, and a width of 0.389 meters. It also has 0.5-meter cylindrical legs attached to the bottom of the body and the rear surface has a slant that falls off at 40 degrees.
The simple geometry of an Ahmed body.
In the Airflow Over an Ahmed Body verification model, our Ahmed body has a 25-degree slant and is placed in the following domain, measuring 8.352-by-2.088-by-2.088 meters, to compute the flow field.
Computational domain and boundary conditions for the fluid flow simulation.
The front of the body is placed at a distance of 2 car lengths (2L) from the flow inlet. To reduce the computational cost, a symmetry plane is introduced to model half of the model.
The flow for this model is turbulent, which is based on the Reynolds number determined by the body length and inlet velocity. The simulation solves for the turbulent kinetic energy in addition to the velocity field. For this simulation, we need a larger mesh size than what is usually common to resolve the turbulent flow. More specifically, we use a finer mesh downstream of the model to capture the wake zone.
The flow for this model is turbulent, which is based on the Reynolds number determined by the body length and inlet velocity. The simulation solves for the turbulent kinetic energy and dissipation in addition to the velocity and pressure fields.
The total drag coefficient of the Ahmed body is the key measurement for this simulation. It is made up of measurements for the pressure coefficients in the front, slant, and base of the body as well as the body’s skin friction. In the results of our simulation, the total drag is very well predicted but the individual measurements deviate from the experimental results in varying amounts.
A few different factors cause these deviations in data. For the front and roof of the body, wall functions used in the simulation aren’t effective at predicting the flow transitions observed in the experiments.
In terms of deviations in the slant coefficient data, we see the flow along the slant shown by streamlines in the figure below. The thickness of these streamlines is determined by turbulent kinetic energy.
Streamlines behind the Ahmed body have thickness proportional to their turbulent kinetic energy.
In the experimental data, these lines show that the flow along the slant is attached almost everywhere and that there are two small recirculation zones behind the base. In the simulation results, this effect is captured but the extent of the recirculation zones is over-predicted.
Streamlines showing the recirculation zones past the Ahmed body.
The pressure drag for the slant coefficient is very sensitive to the exact shape and location of the recirculation regions, which accounts for the deviation in measurements.
Although the data has quantitative differences, it is qualitatively equal to experimental results because the total drag coefficients are so close. There may be deviating details in the smaller data but the simulation still captures the major features of the flow over an Ahmed body. This simulation is more than adequate for calculating the overall drag coefficient.
HUMMER is a registered trademark or trademark of General Motors LLC.
Mercedes-Benz and C-Class are registered trademarks of Daimler AG Corporation.
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The name syngas gives reference to the role of this fuel gas mixture — comprised mostly of hydrogen, carbon monoxide, and carbon dioxide — as an intermediate in the production process of synthetic natural gas. Syngas, however, is also used to create other products such as methanol, ammonia, and even hydrogen. The idea behind this is a process known as gasification.
In gasification, a solid feedstock is converted to a gas, which can then be used in numerous applications. The gas can be liquefied, for example, by compression. Gasification is particularly valued for its flexibility in the types of feedstock that can be used, with options ranging from coal to biomass. Additionally, this approach simplifies the task of capturing byproducts of the reaction like sulfur or carbon dioxide.
Here, we model syngas combustion in a round-jet burner, comparing our results with experimental data.
In the Syngas Combustion in a Round-Jet Burner model, the burner is comprised of a straight pipe within a slow co-flow consisting of air. A gas made up of carbon monoxide, hydrogen, and nitrogen is fed through the pipe with an inlet velocity of 76 m/s (Ma ≈ 0.25). Meanwhile, the co-flow velocity of the air outside of the pipe is 0.7 m/s.
Upon exiting the pipe, the fuel gas mixes with the co-flow, which generates an unconfined circular jet. The turbulent flow of the jet ensures efficient mixing of the two gases and sustains combustion at the exit of the pipe. This is a non-premixed form of combustion as the fuel and oxidizer come into the reaction zone independently.
A schematic of the round-jet burner.
Within this example, we solve for the mass fraction of six chemical species — the five used in the reactions and the nitrogen initially in the co-flow — to model the mass transport in the reacting jet. In the example, the jet features a Reynolds number of around 16700, meaning that the jet is fully turbulent. Because of this, we can assume that the turbulence of the flow has a significant impact on the jet’s mixing and reaction processes.
The k-\epsilon turbulence model is used to account for this turbulence within the flow field. To model the turbulent reactions, we use the eddy dissipation model, which provides a robust yet simple way of simulating such reactions. Because of heat release in the reactions, there is a significant increase in temperature of the jet — a defining characteristic of combustion. To accurately predict the temperature and composition, we account for the temperature dependence of the species properties as well as the physical properties of the fluid.
The syngas combustion model involves a high degree of coupling, combining turbulent flow with heat and mass transfer. A thorough overview of the solution steps to solve such a non-linear model is shown in the Model Gallery entry.
The first figure below depicts the velocity field within the reacting jet, illustrating the expansion and creation of the hot free jet. Within the outer parts of the jet, turbulent mixing prompts the acceleration of fluid initially from the co-flow and brings it into the jet, a process referred to as entrainment. This transition of fluid is evident in the co-flow streamlines that bend towards the jet downstream of the opening of the pipe.
The velocity flow magnitude and field.
Next, we can analyze the temperature in the jet, using a revolved data set to visualize the model in full 3D. Here, we identify the maximum temperature within the combustion region as about 1960 K.
Jet temperature.
The following figure illustrates the carbon dioxide mass fraction within the reacting jet. CO_{2} is formed in the jet’s outer shear layer, right outside of the pipe. It is in the outer shear layer that the fuel reacts with the oxygen in the co-flow, with turbulent mixing encouraging the reactions. Like the CO_{2} formation, the temperature increase depicted in the previous plot also takes place just outside of the pipe. This suggests no lift-off and the attachment of the flame to the pipe.
Carbon dioxide mass fraction.
Let’s now shift our focus to comparing the simulation results with experimental data. Our analysis begins with the jet temperature profiles along the centerline, as shown in the figure on the left, below. In this graph — and the ones that follow — lines represent the model results and symbols are used to indicate the experimental values. The plot of the centerline shows that the maximum temperature predicted in the model is close to that from the experimental results.
In the model results, you may notice that the temperature profile shifts in the downstream direction. This difference can be attributed to the fact that radiation has been left out of the model. Meanwhile, the figure on the right compares the temperature profiles along a horizontal line at two different positions (20 and 50 pipe diameters) downstream of the pipe exit. Again, there is a good agreement between the values obtained in the simulation and those from the experiment.
Left: A plot comparing jet temperatures along the centerline. Right: Jet temperatures at 20 and 50 pipe diameters downstream of the pipe exit.
When comparing the axial velocity of the jet with experimental data, we can observe that these results are in excellent agreement for both positions (20 and 50 pipe diameters). This is illustrated here:
Axial velocity at the same downstream positions as the previous plot.
Lastly, we evaluate the species concentration along the jet centerline. In the case of the species N_{2} and CO, the axial mass fraction development aligns closely with experimental data. H_{2}O and H_{2} are found to agree fairly well with the experimental values, with a slight shift on H_{2}O. The species CO_{2} and O_{2} feature a similar trend as the experimental results but, just as in the temperature, the profiles are found to shift downstream. Here, the discrepancy can be somewhat attributed to the lack of inclusion of radiation in the model. However, the simplified reaction scheme and the eddy dissipation model are likely to have an influence on the accuracy as well.
Comparing species mass fractions along the jet centerline.