With the use of COMSOL Multiphysics, Tingcheng Wu, Guillaume Escamez, Clement Lorin, and Philippe J. Mason from the Department of Mechanical Engineering at the University of Houston were able to perform simulations to analyze how individual components of the structure impacted its overall performance. By applying various parameters to the structural design, they were able to conclude which factors had the greatest effect on the machine, both structurally and thermally. Thus, they could determine how to achieve a balance between the two.

The team of researchers modeled the structure as a rotor shaft separated into five individual parts connected by bolts. As a means to provide thermal insulation, different materials were used on particular areas of the shaft, as you can see below.

*Bolts were used to connect stainless steel with G10, a glass fiber material characterized by low thermal conductivity and a high yield stress. Image by T. Wu, G. Escamez, C. Lorin, and P. Mason, and taken from their poster submission.*

Seeking to analyze the issues of heat transfer and solid mechanics in these machines, the researchers used the Heat Transfer Module and Structural Mechanics Module to create their simulations. The figure below highlights the team’s findings regarding temperature, depicting that the G10 components take on the greatest temperature gradient.

In the next figure, the connection bolts underwent the most stress within the structure, a factor that was found to decrease as the cross-section area was decreased.

*Simulations highlight the thermal and structural pressure that the shaft endured, especially along its connection bolts. Image by T. Wu, G. Escamez, C. Lorin, and P. Mason, and taken from their presentation.*

With the continued funding and efforts of NASA and other research teams, progress continues to be made in the design of aircraft. As torque transfer components within fully superconducting rotating machines continue to be optimized, researchers gain momentum in their quest for developing structures with greater power densities and the potential for electric propulsion. In addition to making air travel a quieter and more energy efficient process, implementing this technology paves the way for its potential use within modes of ground transportation as well.

- Access the paper, presentation, and poster: “FEA Mechanical Modeling of Torque Transfer Components for Fully Superconducting Rotating Machines“

GREENKITCHEN project, a European initiative that supports the development of energy-efficient home appliances with reduced environmental impact, was founded to help develop and share knowledge about home appliance manufacturing in order to advance eco-friendly strategies for improving energy efficiency in households across Europe.

*COMSOL Multiphysics model of Whirlpool’s Minerva oven showing the predicted temperature distribution of the oven’s surfaces.*

One of the main goals of the project is to improve heat exchange in domestic ovens. Since ovens are one of the least energy-efficient appliances in a household ( only about 10-12% of the input power is used to heat the food being prepared ), they offer one of the best areas for improvement in regards to energy efficiency.

Using Whirlpool’s Minerva oven as a model, researchers at Whirlpool R&D working on the GREENKITCHEN project modeled the heat transfer processes of conduction, convection, and radiation in the oven. “Our goal for the project is to reduce energy consumption of Whirlpool’s ovens by 20 percent,” described Nelson Garcia-Polanco, Research and Thermal Engineer at Whirlpool, in an article from the 2014 edition of * Multiphysics Simulation*. This improvement in efficiency would result in a reduction of 50 million tons in CO

The first step in the project was to create a COMSOL Multiphysics model of the oven and verify its accuracy by performing experimental tests. One of the tests used by the team was the “chilled wet brick test” or “brick test”, in which a wet brick representing a food matrix is heated from 5°C to 60°C over a set period of time. The temperature distribution and evaporation of the water from the brick is then measured to determine the oven’s performance and energy usage. Below are images showing both the model of the oven and brick (right), as well as a photo of the oven during experimental testing (left).

Using the model, Garcia-Polanco measured the temperature profile and water concentration in the brick over the course of a 50-minute heating cycle at 200°C. This information was then compared to the results from the experimental test in order to verify that the oven simulation could reasonably predict the temperature changes throughout the test’s duration.

Water evaporation from the brick was also taken into account. The model predicted that an average of 166 grams of water would have evaporated after the 50-minute heating cycle, while the actual value was found to be only slightly higher, at 171 grams. With this information about water concentration and heating profiles, Garcia-Polanco and the GREENKITCHEN team will be able to optimize the oven to achieve a high energy efficiency, while being able to maintain the quality of the final product — ensuring that the energy-efficient oven distributes heat throughout the oven and provides optimal performance in the cooking process.

Currently, Garcia-Polanco and the team are using the validated model to test design ideas for both efficiency and performance.

By the way, Garcia-Polanco will discuss their project and designs further in his upcoming keynote speech on “Multiphysics Approach to Improving the Performance of Domestic Ovens” at the COMSOL Conference 2014 in Cambridge, UK.

To learn more about the research by Nelson Garcia-Polanco and the team at GREENKITCHEN, check out:

- The full-length article on page 24 of the
*IEEE Spectrum*insert,*Multiphysics Simulation* - Read the Garcia-Polanco et. al paper from the COMSOL Conference 2013, “Multiphysics Approach of the Performance of a Domestic Oven“

To optimize the Passive Vaccine Storage Device (PVSD), engineers at Intellectual Ventures, as part of the Global Good Program, turned to thermal and vacuum system modeling with COMSOL Multiphysics together with experimentation.

In the early development stages, they began with a design similar to a cryogenic dewar — a specialized vacuum container commonly used in the field. Typical dewars are able to store ice for a few days before it melts, which is not nearly enough time for long trips to remote destinations. Traveling from a source point to areas where people need vaccinations could take weeks depending on their locations. Long travel times in combination with extreme climates present major challenges for experts working in the medical community. The PVSDs need high-performance insulation to create the temperature-controlled environment required for vaccine storage. Each layer of the device can impact overall performance and is designed to add to its insulative strength.

*The Passive Vaccine Storage Device can hold vaccines in a temperature-controlled, easy-to-transport compartment for longer durations than ever before.*

The shell of the PVSD is made of multilayer insulation, which is similar to the materials used for temperature regulation in spacecraft. This design is especially necessary for areas of the world that get incredibly hot because the vaccines need to stay in a cool and narrow temperature range (between 0°C and 10°C). The multilayer insulation consists of several layers of reflective aluminum, a low conductivity spacer, and nonconducting vacuum space.

*When modeling the PVSD, the Intellectual Ventures team considered physics phenomena and design variables including heat transfer, outgassing, and hold time.*

Vaccines require cold chain storage, which entails proper handling from the moment they are manufactured up until they are administered to a patient. Live virus vaccines can quickly deteriorate as soon as they leave their temperature-controlled space, and inactivated vaccines can lose potency from very short temperature fluctuations. Each year, countless doses of vaccines are thrown away or rendered useless because they were not stored and handled correctly.

For experimental tests, the researchers used an environmental chamber to recreate extreme outdoor conditions. In addition to experimental evaluation, multiphysics models were implemented using the Molecular Flow Module and Heat Transfer Module with COMSOL Multiphysics to optimize the PVSD design with regard to thermal performance and hold time. The outside of the device, composed of metal, prevents air inflow and helps maintain the cool temperature within. Added rubber absorbs shock to protect the contents during bumpy travels. Inside the PVSD is a small insulating shell (pictured below) that contains several compartments where the life-saving vaccines are stored.

*The inner shell of the PVSD holds individual vaccine vials that aid workers can easily access without disrupting the vacuum space or controlled environment.
*

By breaking down geographical barriers for aid workers, many lives will be impacted by the PVSDs and their important cargo.

- Molecular Flow Module
- Heat Transfer Module
- Read the full-length article “Innovative Thermal Insulation Techniques Bring Vaccines to the Developing World” in the 2014 edition of
*COMSOL News*

Our test set-up consists of a copper coil wound around a laminated iron core with some plastic and aluminum parts for stability. A conventional computer fan is placed one meter away from it. The occurring electromagnetic losses have to be calculated as well as the turbulent non-isothermal fluid flow around the device. The iron core has an air gap, which is intentionally included in order to analyze the influence it has on currents inside the coil and aluminum parts.

*The inductor device.*

*Schematic of the test set-up.*

Engineers — especially those working within project deadlines — are always looking for the right balance between computational (and modeling) efforts and accuracy. Therefore, it is a good idea to start with thinking of a suitable simplification, since the aspect ratio of the model geometry is quite challenging.

The distance between the fan and device is roughly one meter, while the interior gaps between the copper winding are about 0.1 millimeters, resulting in an aspect ratio of 10,000. In order to keep the processing time as low as possible, we choose a submodeling approach. A first model with a simplified transformer geometry is used to calculate the large-scale flow field around the device. Due to symmetry, only half of the geometry is modeled. The results of this model are exported and used as an inlet condition for the following step.

*Streamline plot of the velocity field. This field was used as an inlet boundary condition in the detailed model (at the position of the slice plot).*

The geometry of the detailed electrical device is built in SolidWorks® software and imported into COMSOL Multiphysics® via the CAD Import Module. Only a small part is used for the non-isothermal flow calculation in the detailed submodel (about 400 mm by 900 mm). The electromagnetic part needs to be solved for an even smaller domain (200 mm by 200 mm).

The iron core is laminated in order to reduce eddy currents. We’ll use the same approach as described by TU Dresden & ABB. The material is homogenized and defined with an orthotropic electrical conductivity. This allows us to keep a single domain and a coarser mesh rather than resolving the lamination geometrically with all small plates.

Due to the alternating current at 500 Hz, inductive effects in the coil (skin and proximity effect) have to be resolved. Additionally, eddy currents in the aluminum plates and iron core will heat up the device.

Due to hysteresis, there are also some magnetization losses. These are quite small in comparison to the eddy current losses and are not explicitly solved for. The table below shows the magnetization losses as functions of the magnetic flux density Q_{mag} = f(B). We could use an interpolation function instead of solving hysteresis time-dependently.

Part |
Electromagnetic losses |
---|---|

Copper coil |
37.2 W |

Aluminum, eddy currents |
36.2 W |

Laminated core, eddy currents |
0.02 W |

Laminated core, magnetic losses |
0.004 W |

The device reaches a maximum temperature of 125°C on the backside of the coil.

Today, our task was to find the best solution for computing thermal designs of transformers. In the case of BLOCK Transformatoren, they decided that COMSOL Multiphysics was the most suitable for their application after comparing the handling and results of several simulation tools.

In the end, this model involved simultaneous solving for a maximum of 8 million degrees of freedom (DOFs), using a robust combination of direct and iterative solvers. Memory (RAM) usage peaked at 89 GB of memory.

In order to be able to solve highly complex models, they chose the Ready-to-Go+ (RTG+) package with a benchmarked cluster for optimal performance. With everything being set for advanced simulations at BLOCK, we can expect their products to be pushed even closer to the limit in the future.

*SolidWorks is a registered trademark of Dassault Systèmes SolidWorks Corp.*

Power transmission systems need to deliver a required level of current without overheating the cables. Cable structure, internal electric losses, installation system geometry, and environmental conditions — such as the ambient temperature or the properties of the surrounding material — all contribute to system’s thermal response. Even external loads, such as those caused by solar radiation (in the case of an aboveground cable) or in proximity to other systems (e.g., other cables crossing the route), must be accounted for in the design.

While simple mathematical models and calculations suffice for many cases, cable systems are more commonly being installed in environments with many external influences that make it hard to predict performance and side effects.

That’s where multiphysics simulation enters the scene.

*COMSOL Multiphysics simulation showing temperature distribution in the cross section of a double-armored umbilical cable.*

Engineers at Prysmian use the COMSOL software for optimizing the geometry and arrangements of components in their high-tech cable solutions. Such systems may require a careful combination of geometry, materials, and positioning of individual parts. For example, composite cables may contain power conductors, hoses for fluid delivery, and cables for signal transmission.

The team has carried out simulations that analyze the physical effects in individual cables and entire power transmission systems by accounting for the relevant loading and environmental conditions. This allowed them to test, in a virtual sense, their designs before building and running prototypes. In one case, they had to take mechanical loads into account and simulated impact testing in medium-voltage cables. They used their results to optimize the thickness and material choices used in the external layers.

The simulations proved particularly useful for understanding the coupled structural, thermal, and electromagnetic problems that are challenging to solve. Massimo Bechis, a modeling and simulation specialist at Prysmian, comments that they’ve used COMSOL Multiphysics to “do extensive transient analyses to account for daily variations in solar irradiation and ambient temperature conditions […] multiphysics simulation really solves these kinds of problems that were very difficult or even impossible to do before.” They have simulated a range of events and phenomena, including impact testing, temperature distribution in double-armored umbilical cables, and computational fluid dynamics (CFD) analysis in high-voltage systems with limited ventilation.

*Simulation results showing a thermal and fluid flow coupled analysis of a high-voltage cable system installed inside a tunnel with only natural ventilation.*

Bechis explains that simulation has improved the way products are developed, and that using the COMSOL software has helped them take a big step forward in the level of services they are able to offer customers and designers. “COMSOL is able to solve these kinds of problems because we can build a parametric model to optimize the geometry, the laying of the cables, and we can include the physics needed to account for the convection in the air […] we can account for current load changes instead of considering constant operating conditions. This allows us to satisfy requests to consider transient conditions due to load changes.”

The superb results at Prysmian show strong promise for future developments. Simulation has opened up a whole world of possibilities for design, testing, and optimization prior to prototyping. It has reduced their costs, prototyping time, the number of lab tests, and has provided a straightforward way to optimize designs ahead of time, affirming their already-spectacular product record. The team expects that these methods will eventually lead to changes not only in the design phase of development, but also in manufacturing processes.

Simulation has also given them an avenue for communicating their design choices. As Bechis concluded, “Now we are able to optimize, among other things, the structure of our cables and still meet the specifications. We can also explain why we use a certain amount of material in a certain layer and show how we came to our decisions based on the modeling [...] we have improved procedures for designing our cables and power transmission systems. We have an additional and powerful way to respond to requests from clients.”

- Read the full story in
*Multiphysics Simulation*

Before going into the physics, we should briefly recall the system of frames used in COMSOL Multiphysics. When geometric nonlinearities are considered, the *Solid Mechanics* interface makes the distinction between *material* and *spatial* frames. The material frame expresses the physical quantities in the coordinates of the initial state \mathbf{X} = (X, Y, Z), while the spatial frame uses the coordinates \mathbf{x} = (x, y, z) of the current state.

The two figures below present the example of a square submitted to compressive strain. The square is ten centimeters long and its bottom-left corner is initially located at (X, Y) = (1~\textrm{cm}, 1~\textrm{cm}). It is then compressed by boundary loads at its left and right sides. This deformation modifies the position of almost all points of the square. For instance, the bottom-left corner moved to a new location (x, y) = (1.54~\textrm{cm}, 0.82~\textrm{cm}).

*A deformed square represented in material coordinates, initial state on the left and final state on the right.*

*A deformed square represented in spatial coordinates, initial state on the left and final state on the right.*

The material coordinates always refer to the same particle in time, which was initially at a given point (X, Y, Z). The momentum equation of Solid Mechanics is formulated in this coordinate system. On the other hand, a point (x, y, z) in spatial coordinates refers to any particle that would be located there at the current state. The heat equation is formulated in this coordinate system.

In these two frames, volume-related physical quantities have different values. For instance, without any mass source, the density in material coordinates remains constant before and after transformation while the density in spatial coordinates changes according to the volume change. Hence, in order to couple an equation formulated on the material frame (structural mechanics) and another equation formulated on the spatial frame (heat transfer), these values need to be properly evaluated on each frame. The following table provides a list of conversions for some thermal physical quantities from spatial to material frame. These conversions involve the deformation gradient \mathbf{F} = {\partial \mathbf{x}} / {\partial \mathbf{X}} and its determinant, J. Both are evaluated using the displacement field computed by the *Solid Mechanics* interface.

Quantity | Material | Spatial |
---|---|---|

Temperature | T | T |

Density | \rho_0 | \rho = J^{-1} \rho_0 |

Thermal conductivity tensor | \bold{k}_0 | \bold{k} = J^{-1}\mathbf{F}^T \bold{k}_0 \mathbf{F} |

Pressure work | W_{\sigma, 0} = \boldsymbol{\alpha} T : \frac{\mathrm{d} \mathbf{S}}{\mathrm{d} t} | W_\sigma = J^{-1} W_{\sigma, 0} |

Heat source | Q_0 | Q = J^{-1} Q_0 |

*Conversion of thermal physical quantities from material to spatial frame.*

These conversions also reflect the fact that stress and strain affect the heat transfer by modifying the geometrical configuration (represented in the spatial frame). For example, a stretched boundary is more likely to receive a higher amount of heat by radiation (Q_\mathrm{r} > Q_{\mathrm{r}, 0}), as shown below.

*Radiative heat flux received at the top surface of a solid, initial state (left) and after stretching the top surface (right).*

Another example, the thermal conductivity expression in the spatial frame, usually using the initial state value \bold{k}_0, involves the quantities \mathbf{F} and J related to solid strain.

*Modification of the thermal conductivity on the spatial frame after deformation of a solid.*

The equations of Solid Mechanics are defined in the material frame. They relate the displacement, \mathbf{u}, the second Piola-Kirchhoff stress tensor, \mathbf{S}, and the elastic strain tensor, \mathbf{E}_\mathrm{el}, by a linear momentum balance equation and a stress-strain relation:

(1)

\rho_0 \frac{\mathrm{d}^2 \mathbf{u}}{\mathrm{d}t^2} = \nabla \cdot (\mathrm{\mathbf{FS}}) + \mathbf{f}_\mathrm{vol}

(2)

\mathbf{S} = \mathbf{C}:\mathbf{E}_\mathrm{el}

Here, \mathbf{C} is the elasticity tensor, which is often defined from the Young’s modulus and the Poisson coefficient. It may depend on the temperature as it is the case for Carbon Steel 1020.

*Young’s modulus of Carbon Steel 1020, depending on the temperature.*

Without any plastic effects, the elastic strain tensor, \mathbf{E}_\mathrm{el}, carries the temperature dependence via the thermal strain tensor, \mathbf{E}_\mathrm{th}, according to:

(3)

\mathbf{E}_\mathrm{el} = \mathbf{E}_\mathrm{tot}-\mathbf{E}_\mathrm{th}

(4)

\mathbf{E}_\mathrm{tot} = \frac{1}{2}(\mathbf{F}\mathbf{F}^\mathrm{T}-\mathbf{I})

(5)

\mathbf{E}_\mathrm{th} = \boldsymbol{\alpha} (T-T_\mathrm{ref})

The coefficient of thermal expansion, \boldsymbol{\alpha}, characterizes the ability of the material to contract and expand because of temperature variations. It is often scalar but may more generally take a tensor form. The table below shows a list of typical values of isotropic \boldsymbol{\alpha}.

Material | Coefficient of Thermal Expansion (10^{-6} K^{-1}) |
---|---|

Acrylic plastic | 70 |

Aluminum | 23 |

Copper | 17 |

Nylon | 280 |

Silica glass | 0.55 |

Structural steel | 12.3 |

*Coefficients of thermal expansion for some materials.*

In addition, \boldsymbol{\alpha} can, itself, depend on the temperature as shown by the example below.

*Coefficient of thermal expansion of Carbon Steel 1020, depending on the temperature.*

As seen in these examples, the values of \boldsymbol{\alpha} are most often of the order of 10^{-5} K^{-1}. Hence, for \mathbf{E}_\mathrm{th} to become significant, a high temperature difference from the reference state is necessary. For instance, aluminum needs to reach about 500 K above the reference temperature to show a thermal elongation of only 1.2%.

*Example of thermal expansion of a constrained aluminum beam heated 500 K, using a deformation scale of 1:1.*

Note that in the formulation of Equations (3)-(5), the thermal strain is subtracted from the total strain. This is an appropriate approximation for small strains, which the thermal strains normally are, due to usually low values of \boldsymbol{\alpha}. The more accurate multiplicative formulation, valid for large thermal strains, is shown below but not discussed further. This formulation is used for the hyperelastic materials in COMSOL Multiphysics.

(6)

\mathbf{E}_\mathrm{el} = \frac{1}{2}(\mathbf{F}_\mathrm{el}\mathbf{F}_\mathrm{el}^\mathrm{T}-\mathbf{I})

(7)

\mathbf{F}_\mathrm{el} = J_\mathrm{th}^{-1/3} \mathbf{F}

(8)

J_\mathrm{th} = \left( 1 + \alpha (T-T_\mathrm{ref}) \right)^3

The heat equation is an energy balance equation deduced from the First Law of Thermodynamics. For solids, it takes the following form when formulated on the spatial frame:

(9)

\rho C_p \left( \frac{\partial T}{\partial t} + \mathbf{u} \cdot \nabla T \right) = \nabla \cdot (k \nabla T) + W_\sigma + Q

The coupling term W_\sigma is the heat source due to compression or expansion of the solid and is defined by:

(10)

W_\sigma = \mathrm{det}(\mathbf{F})^{-1} T \frac{\partial \mathbf{E}_\mathrm{tot}}{\partial T} : \frac{\mathrm{d}\mathbf{S}}{\mathrm{d}t}

which, in the case of \boldsymbol{\alpha} being independent from temperature, reduces to:

(11)

W_\sigma = \mathrm{det}(\mathbf{F})^{-1} \boldsymbol{\alpha} T : \frac{\mathrm{d}\mathbf{S}}{\mathrm{d}t}

Here, \boldsymbol{\alpha} is the same coefficient of thermal expansion as in \mathbf{E}_\mathrm{th}. The low value of \boldsymbol{\alpha}, as seen in the table above, has to be compensated for by high enough values of T {\mathrm{d} \mathbf{S}} / {\mathrm{d} t} to make W_\sigma a significant heat source, that is:

- by a high temperature
- by rapid and high variations of stress

We have now described four key contributions to the multiphysics coupling between Heat Transfer and Solid Mechanics:

- The influence of strain and stress on thermal quantities and boundary heat fluxes in the material or spatial frames
- The temperature dependence of the elasticity matrix
- The temperature dependence of the elastic strain tensor via the thermal strain tensor
- The heat source, W_\sigma, corresponding to pressure work in the solid

Next, we will illustrate the last two coupling contributions and show how to handle them in COMSOL Multiphysics with a couple of modeling examples.

My colleague Nicolas previously described in more detail how to model thermal stress in a turbine stator blade. Here, we display only the results in order to show the effects of J_\mathrm{th}. Because this is a steady-state model, the pressure work, W_\sigma, can be ignored.

*Temperature field on the blade surface, representation in the material frame.*

Due to a hot environment, the temperature field shows values between 870 K and 1100 K compared to the reference temperature of 300 K that the shape of the stator blade is initially. Such high temperatures make the material more prone to thermal deformations. The average coefficient of thermal expansion and temperature being around 1.2·10^{-5} K^{-1} and 1070 K, \mathbf{E}_\mathrm{th} is around 0.9%.

The volume expansion, due to thermal effects, for large deformations is \Delta V/V_0 = J_\mathrm{th}-1 (where J_\mathrm{th} was introduced in Equation (8)). It is still a good approximation for a small strain, giving an expansion of around 2.80%. In postprocessing, the actual volume expansion is found to be 2.76%.

*Temperature field and deformation of the stator blade, exaggerated plot with a scale factor of 3 for more visibility.*

The Bracket — Transient Analysis model is available both in the Structural Mechanics Module Model Library and the Model Gallery. In this model, the arms of the brackets move according to rapid time-dependent loads. Consequently, small variations of temperature should occur.

The existing model neglects these thermal effects, so we need to add a new *Heat Transfer in Solids* interface.

Then, we add the two multiphysics features below to couple the *Heat Transfer in Solids* and *Solid Mechanics* interfaces:

- Thermal Expansion
- This modifies the thermal strain tensor, \mathbf{E}_\mathrm{th}, applied on the whole bracket domain

- Temperature Coupling
- This couples the temperature variable computed by the
*Heat Transfer in Solids*interface with the*Solids Mechanics*interface

- This couples the temperature variable computed by the

Finally, we add the *Pressure Work* subfeature to handle the thermoelastic heat source, W_\sigma.

The study can also be extended to 30 milliseconds to observe more load periods.

Starting from an isothermal profile of 20°C everywhere, the small temperature variations lead to a negligible thermal strain tensor. The main contribution to thermal effects is now the thermoelastic heat source due to rapid stress variations.

*Temperature profile of the bracket over time, exaggerated plot with a scale factor of 10 for more visibility.*

Differences of about 0.8 K can be observed between the extreme temperatures in the bracket. The heating and cooling process is, as expected, located at corners where the stress is more important and its variations stronger.

The heat transfer in a deformed solid is numerically computed by solving the heat equation and the momentum balance equation. For practical reasons, we made the distinction between two systems of coordinates:

- The material frame where the equation of motion is formulated
- The spatial frame for the heat equation

Volume-related quantities in both frames have different values and need a conversion from each other, in particular for specific energies and density.

The two governing equations each contain coupling terms that makes the solid motion dependent on the temperature and the heat transfer dependent on the solid deformation. As shown in the previous two examples, COMSOL Multiphysics provides appropriate functionalities to conveniently account for them.

When temperatures remain near the reference state and without too rapid stress variations, these coupling effects are negligible. Otherwise, they shall be added to the formulation on the model.

To delve deeper into this topic, you can download the files related to the models mentioned here and read a couple of related blog posts via the links in the section below.

- Model downloads:
- Previous blog posts:

There are two different ways in which hyperthermia can be used to treat cancer. The first is called *local hyperthermia* or *thermal ablation*, and it is used to treat a small area of cells, such as a tumor. This treatment is more successful when used to treat tumors that are located on or near the skin’s surface. In these treatments, the area is generally heated to temperatures between 40 and 45°C (104 to 113°F). The second type is called *regional hyperthermia*, and it uses low heat to elevate the temperature of a region of the body, such as a limb or organ. Generally, the temperature isn’t hot enough to destroy tissues, but it is instead combined with chemotherapy or radiation to make these treatments more effective.

Currently, hyperthermic oncology treatments are still undergoing extensive experimental testing. The procedure is only offered in a few treatment centers around the world. Because measuring the temperature inside a tumor is difficult, it is hard to ensure that the area being treated is kept within the exact temperature range needed over a precise period of time, without affecting surrounding tissues. This is where simulation is proving to be a powerful tool. It can be used to study the responses of different tissue types to hyperthermia, to determine the temperatures reached within a tumor, and to help decide the optimal design and placement of temperature sensors.

Let’s take a look at a model of a local hypthermia treatment method that uses microwave coagulation to heat a tumor by inserting an antenna into it. This method is most commonly used to treat hepatocellular carcinoma (small size liver tumors). The microwaves heat up the tumor and create a coagulated area where the cancer cells are destroyed.

The model example takes into account the temperature distribution, radiation, and specific absorption rate (SAR) in liver tissue. The model geometry, which was based off of the paper by Saito et al., is shown below. The model is composed of liver tissue, the antenna from which the microwave coagulation is delivered, and a ring shaped slot that serves to increase the size of the coagulated region in the liver tissue.

*Antenna geometry for microwave coagulation therapy. A coaxial cable with a ring-shaped slot cut on the outer conductor is short-circuited at the tip. A plastic catheter surrounds the antenna.*

Since the model contains rotational symmetry, the computational domain can be represented in 2D using cylindrical coordinates. Because of the computation time saved by modeling in 2D, a fine mesh can be selected, which will provide very accurate results. In the model, the metallic parts of the antenna and slot have been modeled as boundaries. Heat transfer has been modeled in the liver domain and insulation is used where the domain ends. In addition to analyzing heat transfer, the simulation also gives an idea about the degree of tissue injury during the procedure, as computed by the Arrhenius equation. The model assumes that blood enters the liver at 37°C.

The figure below show the results of the temperature distribution in the liver tissue for a microwave input power of 10 W after 10 minutes. The temperature is highest near the antenna and decreases with distance, reaching 37°C near the outer regions of the domain (top image). It appears that the relatively cool temperature of the blood limits the region of tissue that is heated. The image on the bottom shows the fraction of necrotic tissue.

*Temperature in the liver tissue (top) and the fraction of necrotic tissue (bottom).*

The model can also compute the SAR value within the liver, where the graph below shows it computed along a line parallel to the antenna and at a distance 2.5 mm from the antenna’s axis.

*SAR value in W/kg along a line parallel to the antenna and at a distance 2.5 mm from the antenna axis. The tip of the antenna is located at 70 mm, and the slot is at 65 mm.*

Clinical trials have shown that hyperthermia treatment significantly increases the effectiveness of radiotherapy and chemotherapy. However, there are two major complications that make treatment with hypothermia a challenge: the ability to *precisely* determine the temperature within the tumor, and the ability to ensure that a uniform temperature is achieved within the desired area. However, advances in tools for both heat delivery and temperature monitoring are promising for the future of hyperthermia treatments. Advancements in computational techniques have also played a role in its development. In addition to providing more accurate information on how to design, optimize, and implement hyperthermia treatments, simulation is also being used to developed personalized hyperthermia treatments and planning. Current studies continue to rely on simulation to better understand, improve, and personalize hyperthermic oncology.

Check out these resources to learn more about hyperthermia treatments and microwave coagulation:

- Model download: Microwave Cancer Therapy
- COMSOL webinar: Multiphysics Simulation in Bioheating and BioTechnology
- National Cancer Institute: Hyperthermia in Cancer Treatment
- Estimation of SAR Distribution of a Tip-Split Array Applicator for Microwave Coagulation Therapy Using the Finite Element Method
- Hyperthermia in combined treatment of cancer
- Treating cancer with heat: hyperthermia as promising strategy to enhance apoptosis

Many factors influence the performance of power electronics devices. For example, ideal operating conditions may require a specific temperature range, voltage level, and switching frequency that, if unmaintained, can cause problems such as failure, increased resistance, reduced efficiency, or voltage spikes.

This is why a team at APEI decided to develop a new power module — a package meant to contain power components, cool electronics devices, and connect them to other circuits — with improved thermal management capabilities and performance. Brice McPherson, lead engineer, says that their goal was to create a power package flexible enough to be used for many applications. It needed to be small, easy to configure, and exhibit low inductance with good thermal conductivity.

The figure below shows their new design, which is only a little larger than a U.S. quarter:

*The power package, showing individual components (left) and the entire assembly (right).*

The team began with two materials for comparison: gallium nitride (GaN) and silicon carbide (SiC). The advantage of these materials is that both are wide-bandgap semiconductors that can be safely used at high frequencies and temperatures. This means that they can withstand harsh environments where most electronics have a tendency to fail. For instance, it’s something of a challenge to protect electronics used in drilling equipment, where pressure and temperature increase dramatically very quickly, or create electronics that can survive being sent to the surface of Venus.

McPherson used COMSOL software to aid their design process, analyzing the thermal and electrical response of the new power package when it included each of the chosen materials. He analyzed thermal conduction (aiming to maximize it and reduce thermal resistance), inductance, and device size.

“Designing for low thermal resistance involves selecting materials with high thermal conductivity, reducing the distance heat travels to leave the layers, and optimizing layer thickness to take advantage of thermal spreading,” he explained. “That’s where parametric modeling is your best friend: you can set up parametric sweeps to find out exactly what’s influencing the system the most and get the best compromise among performance, complexity, and cost.”

*Simulation results overlaid on geometry, showing current density in the SiC package (left) and the GaN package (right). The SiC package displays low density (ideal for higher currents), with the highest concentrations in the wire bonds. The GaN shows a higher density, but has more available area for conduction, which is preferred for low inductance.*

McPherson compared his designs to a commonly-used transistor outline package, the TO-254, aiming to improve upon current industry standards. His hope was that APEI’s new design would be able to withstand very high temperatures — above 225°C. The simulation results showed that both the APEI power modules reflected a lower thermal resistance and significantly lower inductance than the TO. The results also indicated that they operated successfully in the specified temperature range.

The team at APEI discovered that optimizing the size of the device and thickness of the base plate were the most influential factors in reducing the inductance of each package. He maximized the cross-sectional area and minimized the current path in an effort to minimize inductance and maximize heat dissipation. The COMSOL Multiphysics results showed that the GaN module exhibited the lowest inductance (7.5 nanohenries). The SiC turned out to be the best choice for cases where high currents and thermal loading were present, processing large amounts of energy in an extremely small area.

*A comparison of thermal resistance among the TO-254, SiC, and GaN power modules based on COMSOL Multiphysics results.*

The result? Two unique and powerful package options that are flexible enough to address multiple customer needs; operate smoothly with clean, rapid switching; and can withstand the high temperatures and frequencies demanded by extreme applications.

APEI’s new modules have far surpassed the current industry transistors, and are paving the road to new capabilities and unforeseen applications in the power electronics field.

Anisotropic properties are found in a wide variety of areas, such as rock formations with anisotropic seismic properties; liquid crystals used in LCD displays; materials for the aerospace industry, which must be lightweight and still withstand high loads; or soft tissues that biomedical replacements should mimic for optimal performance.

In a previous blog post, we saw how to use the *Curvilinear Coordinates* interface and how to apply it to account for anisotropic thermal conductivity.

Let’s revisit this application and consider a carbon-fiber-reinforced polymer. The woven fibers embedded in an epoxy matrix have high thermal conductivity along the fiber axis and low conductivity in the cross section. It is almost impossible to express the anisotropy referring to the well-known Cartesian coordinate system. If we had a coordinate system that follows the fibers, it would be straightforward to specify the anisotropic properties.

*Woven fibers in an epoxy matrix.*

How can such a coordinate system be determined? Physically, there are numerous effects that result in a vector field following the shape of the geometry. For instance, flow through the fibers or heat conduction from one end to the other or even a bundle of current-carrying wires that produce a magnetic field. These are precisely the methods that are used in the COMSOL software to compute the curvilinear system. All methods compute a vector field, \mathbf{v}, which forms the *first basis vector*. Since most applications require a normalized vector field, COMSOL Multiphysics automatically normalizes by dividing with |\mathbf{v}|. A second vector field is specified manually and one of the Cartesian coordinates is often a good choice. Starting from this, the *second basis vector* \mathbf{e}_2 is reconstructed to ensure that it is perpendicular to \mathbf{e}_1 and normalized. The cross product of these two gives the *third base vector* \mathbf{e}_3.

Internally, the software uses the Cartesian coordinate system (\mathbf{e}_x,\ \mathbf{e}_y,\ \mathbf{e}_z) for computation and converts all quantities referring to a different coordinate system (\mathbf{e}_1,\ \mathbf{e}_2,\ \mathbf{e}_3). A direction given by a vector, \mathbf{F}=(F_1,\ F_2,\ F_3), in an arbitrary coordinate system can always be transformed into Cartesian coordinates, as follows:

\mathbf{F}=\left(\begin{matrix} F_x\\F_y\\F_z\end{matrix}\right)=F_1\mathbf{e}_1+F_2\mathbf{e}_2+F_3\mathbf{e}_3=\left(\begin{matrix}

e_{11} & e_{12} & e_{13} \\

e_{21} & e_{22} & e_{23} \\

e_{31} & e_{32} & e_{33}

\end{matrix}\right)\cdot\left(\begin{matrix} F_1\\F_2\\F_3\end{matrix}\right)=\mathbf{M}\left(\begin{matrix} F_1\\F_2\\F_3\end{matrix}\right)

e_{11} & e_{12} & e_{13} \\

e_{21} & e_{22} & e_{23} \\

e_{31} & e_{32} & e_{33}

\end{matrix}\right)\cdot\left(\begin{matrix} F_1\\F_2\\F_3\end{matrix}\right)=\mathbf{M}\left(\begin{matrix} F_1\\F_2\\F_3\end{matrix}\right)

Here, \mathbf{M} is the transformation matrix. For the inverse transformation, simply use the inverse, \mathbf{M}^{-1}, and if (\mathbf{e}_1,\ \mathbf{e}_2,\ \mathbf{e}_3) is orthonormal then \mathbf{M}^{-1}=\mathbf{M}^{T}.

The following are available in COMSOL Multiphysics:

- Diffusion method
- Flow method
- Elasticity method

Next, I will illustrate the different methods available in COMSOL Multiphysics that can be used to calculate a curvilinear coordinate system.

Let’s pick a single fiber and have a closer look.

The diffusion method solves Laplace’s equation: -\Delta U=0. The solution, U, is a scalar potential, and its gradient forms the first base vector. Because you solve for a single scalar potential only, this method is computationally inexpensive compared to the other two. The direction of the vector field is specified with the inlet and outlet boundary conditions. If the geometry is a closed loop, you can set the jump boundary condition on an interior boundary to specify the direction.

The diffusion method is equivalent to solving the stationary heat conduction equation with constant temperatures at the inlet and outlet boundaries. The temperature gradient then forms the first base vector, as illustrated below.

*Curvilinear coordinate system (arrows), temperature gradient (streamlines), and temperature (surface).*

Here, you solve the incompressible Stokes equation for a vector field and a scalar. Thus, this method is the most computationally expensive. The boundary conditions are the same as for the diffusion method. A physical analogy would be incompressible creeping flow with a constant normal velocity at the inlet and a fixed pressure at the outlet. The resulting velocity field gives you the first base vector.

*Curvilinear coordinate system (arrows), velocity field (streamlines), and pressure (surface). *

The elasticity method solves the following eigenvalue equation:

-\nabla\cdot((\nabla\cdot\mathbf{e})\mathbf{I}+(\nabla\mathbf{e}+\nabla\mathbf{e}^{T}))=\lambda\mathbf{e}

where \mathbf{e} is the vector field, \mathbf{I} the identity matrix, and \lambdathe eigenvalue. This method is slightly cheaper in terms of computational costs compared to the flow method because you solve for a vector field only. This difference in performance is more apparent in 2D models. The inlet and outlet boundary conditions are identical, \mathbf{e}\times \mathbf{n}=0. The eigenvalue formulation was devised initially to model multi-turn coils (bundles of wires) in 3D magnetic applications with the AC/DC Module. For Multi-turn coils, the current density should be roughly constant in cross section, since it is assumed that each wire carries the same current and the wires are evenly spaced.

*Curvilinear coordinate system (arrows), coil direction (gray streamlines) and magnetic flux density (red streamlines).*

Apart from these predefined methods, the COMSOL software also provides a user-defined input, as usual. You may encounter other scenarios where you want to implement curvilinear coordinates manually, such as anisotropic hyperelastic material for modeling collagenous soft tissue in arterial walls.

At first glance, all three methods lead to the same coordinate system. Most commonly, you want to choose a method that incurs low computational costs. In that case, the best choice is the diffusion method. That said, some shapes require special attention and the choice of the method can lead to significantly different results when the coordinate system is used by a physical application.

Let’s have a look at a geometry with a sharp bend, plotted below. If we look on the right-hand side of the plot, from top to bottom, we see that with the diffusion method, the streamlines accumulate at the inner radius because they take the shortest path. The elasticity method shows some wobbly curves, while the flow method provides a solution where the streamlines follow the expected path.

*Results of different methods for sharply curved geometries.*

In this case, the elasticity method can fail and you obtain an eigenvalue with eigenvectors that do not produce the required coordinate system. Alternatively, you may have to manually search for the correct eigenvalue. We can see that the streamlines also do not follow the shape perfectly at the upper part of the geometry. Similar behavior is observed with the diffusion method. The flow method provides the best results here, but is most computationally expensive.

*Streamlines along the center plane of the geometry.*

Let’s go back to our model with the fibers that have an anisotropic thermal conductivity of 60\ W/mK in the fiber direction and 4\ W/mK perpendicular to this. If these directions coincide with the axes of the coordinate system, the thermal conductivity — a 2^{nd}-order tensor — has zero off-diagonal elements.

k=\left(\begin{matrix}

k_{xx} & k_{xy} & k_{xz} \\

k_{yx} & k_{yy} & k_{yz} \\

k_{zx} & k_{zy} & k_{zz}

\end{matrix}\right)=\left(\begin{matrix}

60 & 0 & 0 \\

0 & 4 & 0 \\

0 & 0 & 4

\end{matrix}\right)

k_{xx} & k_{xy} & k_{xz} \\

k_{yx} & k_{yy} & k_{yz} \\

k_{zx} & k_{zy} & k_{zz}

\end{matrix}\right)=\left(\begin{matrix}

60 & 0 & 0 \\

0 & 4 & 0 \\

0 & 0 & 4

\end{matrix}\right)

To be able to use this diagonal form, the curvilinear coordinate system for the fibers must be calculated prior to solving for the heat transfer. Because the geometry doesn’t have sharp bends or a changing cross section, the diffusion method gives a fast solution for the curvilinear coordinate system.

After that, you can refer to this coordinate system in the associated Heat Transfer node. The anisotropy of the thermal conductivity can be defined in the Materials node, with the syntax k=\{60, 4, 4\}. Alternatively, you can select the user-defined input in the associated Heat Transfer node.

*Definition of anisotropy in the Heat Transfer node.*

In the model, a boundary heat source in the form of a Gaussian pulse is applied to the center of the geometry and the temperature spreads along the fibers.

*Streamlines indicate the vector field obtained with the Curvilinear Coordinates interface.*

If you want to visualize, for example, the *xx*-component of the thermal conductivity (k_{xx}), keep in mind that you plot the *xx*-component in Cartesian coordinates (\mathbf{e}_x,\ \mathbf{e}_y,\ \mathbf{e}_z). Then the thermal conductivity tensor k for the fibers is of non-diagonal form, according to the transformation described above. The local base vector system (\mathbf{e}_1,\ \mathbf{e}_2,\ \mathbf{e}_3), which was used to define k, is now varying through space and so does k_{xx}. In this model, you can plot the components of the thermal conductivity vector in a slice plot, for example. You can select them from the Expressions menu in the corresponding settings window or simply type `ht.kxx`

(where `ht`

is the tag for the *Heat Transfer in Solids* interface that was used for this model).

This blog has described the different methods for defining the curvilinear coordinate systems that are available in COMSOL as well as when to choose one over another.

**Diffusion Method**- Pros: low computational cost
- Cons: computed vector field tends to take the shortest path in bends

**Elasticity Method**- Pros: computational cost lower than the flow method, better representation of moderate bends than diffusion method
- Cons: often requires manual selection of the eigenvalue and is not robust in all cases

**Flow Method**- Pros: most robust method, supports cross section changes and sharp bends
- Cons: computational cost often greater

You can download the model presented here from our Model Gallery.

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When modeling acoustics phenomena using the *Thermoacoustics* interface, there are several things to be aware of. First off, the physics have to be set up correctly and the mesh has to resolve the viscous and thermal boundary layers. It is also important to note that solving a thermoacoustic model involves solving for the pressure, velocity field (for example, 3 components in 3D), and temperature. This means that the model can become computationally expensive and involve many degrees of freedom (DOFs).

Erroneous specifications of the coefficients of thermal expansion and compressibility is a problem that I often see in support cases. If these coefficients are wrong or even evaluate to zero, the result is a model where acoustic waves (pressure or compressibility waves) propagate at the wrong speed of sound or do not propagate at all. The speed of sound relates to both of these coefficients.

A detailed description of both of these coefficients and on how to define them is given in the Acoustics Module’s *User’s Guide* (under The *Thermoacoustics, Frequency Domain* Interface in the section *Thermoacoustics Model*). The model Vibrating Particle in Water: Correct Thermoacoustic Material Parameters, which can be found in the Model Library, also discusses these issues. A simple check is to plot the parameters `ta.betaT`

(isothermal compressibility) and `ta.alpha0`

(thermal expansion) after solving the model to ensure that they have the correct values.

When meshing a thermoacoustics model, it is important to properly resolve the acoustic boundary layer to capture the physics correctly. In order to do this and avoid too many mesh elements, there are a few tricks you can use:

*Create parameters to control your mesh.*For example, create a parameter for the analysis frequency, say`f0`

, and then also create a parameter for the viscous (or thermal) boundary layer thickness at this frequency. In air, we know that the viscous boundary layer thickness at 100 Hz is 0.22 mm, and, in general, you can write the thickness as`dvisc = 0.22[mm]*sqrt(100[Hz]/f0)`

. If you perform a frequency sweep, you can create parameters for the thickest and thinnest value of the boundary layer. Having these parameters at hand can help you build a good mesh.*Use Boundary Layers.*This will keep the number of mesh elements constant for all studied frequencies. This is especially important in 3D. If you simply prescribe a maximum element size on the walls, the number of mesh elements will explode as the boundary layer thickness decreases.*Use logic expressions when defining the mesh.*For example, use`min(,)`

when defining the maximum element size or the thickness of a boundary layer. In the figure below, an example is given of a circular duct with a diameter 2a = 2 mm. The overall “Maximum element size” is set to a/3. A boundary layer mesh is used with five layers and a thickness of`min(a/30,0.3*dvisc)`

. This ensures a constant mesh thickness up to around 500 Hz (keeping the mesh in the middle of the pipe of good quality) and then the thickness decreases with`dvisc`

as the frequency parameter`f0`

increases.

In general, when solving a model using the Frequency Domain study step, it is not possible to have the mesh depend on the frequency variable `freq`

. This is what you would like for this type of models. However, it is possible to achieve this when performing a parametric sweep. Therefore, one workaround is to use a Parametric Sweep around the Frequency Domain study step. Sweep the parameter over `f0`

and set `f0`

to be the frequency in the Frequency Domain step.

Note that when doing this, the COMSOL software will re-mesh every time a parameter in the mesh changes, which may slow down the computation a bit. On the other hand, you can set up a more intelligent mesh in this way and still save time.

A final option is to prepare several meshes, maybe one mesh for each chunk of 1000 Hz, and then use several studies with these meshes selected for a restricted frequency range.

*Example of a mesh that captures the effects in the acoustic boundary layer, here shown at four different frequencies. The color represents the RMS velocity for a wave traveling in an infinite circular duct with a diameter of 2 mm.*

In that it is computationally expensive to solve thermoacoustics models, it is often advantageous to do so only in the parts of your system where thermoacoustics is relevant. These simulations can then be combined with simulations based on less-complex physics that describe the rest of your system. Here are some ideas on how this can be done:

*Couple the thermoacoustics model to pressure acoustics where relevant.*In models where large differences exist in the geometry scale, only use thermoacoustics in the narrow regions and pressure acoustics in the larger domains. The*Thermoacoustics*interface is a multiphysics interface that has the ability to be automatically coupled to the*Pressure Acoustics*interface. This is exemplified in the Generic 711 Coupler model (located in both the Model Library within the software and the Model Gallery on our website).*Use submodels and lumped models.*For instance, extract a transfer impedance from a detailed thermoacoustic model and use it in a pressure acoustics model. A nice example model of this is seen in the Acoustic Muffler with Thermoacoustic Impedance Lumping model. In this example, the transfer impedance of a perforated plate is analyzed and used in a pressure acoustics model.*As frequency increases, the acoustic boundary layer decreases in size and relevance.*This means that at a certain frequency, the boundary layer losses can be considered to become negligible, and you can switch to solving the modeling as a pressure acoustics problem.- In structures of constant cross section you can use the Narrow Region Acoustics models of the
*Pressure Acoustics*interface. These are homogenized fluid models where the boundary layer losses are smeared over the fluid domain. These models provide a first good approximate response of a system without the cost of solving a full thermoacoustic model.

The documentation for the *Thermoacoustics* interface contains some tips and tricks on how to use different solver approaches if the model becomes very large. See: Acoustics Module User’s Guide > The Thermoacoustics Branch > Theory Background for the Thermoacoustics Branch > Solver Suggestions for Large Thermoacoustic Models.

The most important points when modeling acoustics using the *Thermoacoustics* interface are:

- Solve only for thermoacoustics where and when necessary; investigate if the viscous and/or thermal boundary layer thickness are comparable to the geometrical scale or not (depending on the frequency range and geometry scales).
- Check material parameters to be sure that both compressibility and thermal expansion are non-zero.
- Check the mesh size at boundaries and compare it to the viscous and thermal boundary layer thickness.

Examples of systems where the use of thermoacoustics is important are listed below.

Electroacoustic transducers are a good example of true multiphysics models where it is essential to include both thermal and viscous losses:

- Blog post: Thermoacoustics Simulation for More Robust Microphone Analysis
- Model downloads:
- B&K 4134 Condenser Microphone, results compared with measurements
- Tutorial model on a simplified 2D axisymmetric condenser microphone model

- COMSOL News article about the use of COMSOL to model hearing aids, “Simulation-Based Design of New Implantable Hearing Device“

- Blog post about using COMSOL Multiphysics to model MEMS microphones

- An example of a vibrating micromirror, which solves for thermoacoustics in order to model Fluid Structure Interaction (FSI) in the frequency domain

The solution of a Thermoacoustics sub-model to find the transfer impedance of a perforated plate in a muffler system. The impedance is subsequently used as a transfer impedance condition in a Pressure Acoustics model:

- Model download: Acoustic Muffler with Thermoacoustic Impedance Lumping

Modeling the response of an Ear Canal Simulator, the so-called 711 coupler. The model results are compared to IEC standard curves and to a lossless model. The results clearly show the necessity to include thermal and viscous losses.

- Model download: Generic 711 Coupler an Occluded Ear Canal Simulator

Advanced application using the *Thermoacoustic* interface to model photoacoustic applications.

- Model download: Photoacoustic Resonator

- A thermoacoustic tutorial model describing the importance of setting up the compressibility and thermal expansion material parameters correctly

- COMSOL Documentation: Acoustics Module User’s Guide.
- COMSOL Documentation: Acoustics Module User’s Guide > The Thermoacoustics Branch.

The more DNA one has in a sample, the easier it is to identify and diagnose. In many situations, such as old crime scenes or large ecological systems with dilute samples, you want to amplify the amount of available DNA before testing for it. One technique that is most effective in amplifying DNA and making it useful in medical diagnostic, chemical, and biological analysis is *Polymerase Chain Reaction*, commonly known as PCR. It is also referred as a *DNA amplification technique*. Further, there has been great interest in recent years in developing portable PCR-based micro total analysis systems (μTAS; also known as lab-on-a-chip systems) for point-of-care applications.

One strategy that seems very promising is natural convection-based PCR. In this strategy, many copies of a DNA template can be made by cycling between hot and cold regions within a microreactor via a buoyancy-driven flow. The multiphysics nature of this strategy makes it a perfect candidate for simulations with COMSOL software.

A COMSOL model can provide more quantitative understanding of the dynamics and kinetics involved in the process that will be very useful in designing and developing efficient μTAS. Here, I will show you a model that was developed to study the spatio-temporal variation of the concentrations of single-stranded (ssDNA), double-stranded (dsDNA), and primer-annealed (aDNA) DNA components during natural convection-based PCR. The model is available for download from the Model Gallery.

The driving mechanism for the fluid motion is the temperature-induced density differences that results in buoyancy flow. This approach eliminates the need for an external driving mechanism as the difference between the temperatures is sufficient to circulate the PCR mixture and amplify DNA in a closed loop, as shown below.

*Schematic of buoyancy-driven PCR.*

In general, a PCR mixture consists of a DNA template, primers, nucleotides, and the enzymes (Taq polymerase, the most widely used amplifying enzyme). The mixture is repeatedly cycled between three temperature zones:

- Denaturing zone (95°C): Double-stranded DNA template is separated into two single strands.
- Annealing zone (55°C): Primers bind to the ends of the single-stranded DNA.
- Extension zone (72°C): A complement to each of the annealed single strands is created by enzyme resulting in two new copies of double strands DNA template. Repeating the above process results in an exponential increase in DNA concentration.

In the figure above, you can see the 2D model geometry of the buoyancy-driven PCR. A dilute PCR mixture fills the channel with an impermeable wall. The right and left boundaries of the channel are maintained at 95°C and 55°C, respectively. The upper and lower boundaries are insulated. The temperature gradient produces variation in the density of fluid that drives the buoyant flow. The model uses temperature-dependent properties for water. The predefined *Non-Isothermal Flow/Conjugate Heat Transfer* interface available in the CFD Module or the Heat Transfer Module is used to model coupled heat transfer and the fluid flow physics.

The DNA components are generated and consumed by the temperature-dependent chemical reactions as proposed in “Polymerase chain reaction in natural convection systems: A convection–reaction-diffusion model” by E. Yariv, G. Ben-Dov, and K. D. Dorfman. A simplified first-order reaction mechanism (given below) illustrates the PCR process that is used in this model. The rates of change in the concentration of DNA components were determined from the stoichiometric balances.

\mathit{dsDNA}\xrightarrow{k_d}2\mathit{ssDNA} Denaturation (95°C)

\mathit{ssDNA}\xrightarrow{k_a}\mathit{aDNA} Annealing (55°C)

\mathit{aDNA}\xrightarrow{k_e}\mathit{dsDNA} Extension (72°C)

These reaction rates constants were spatially modulated using the below Gaussian mapping function. This is done in order to localize each reaction in its respective temperature zone.

f_i=\exp{\left(-{\frac{{(T-{T_i})^2}}{2{\sigma^2}}}\right)}

Here, \sigma (=1°C) is the standard deviation of the reaction rate with respect to temperature and T_i is the ideal temperature for denaturation (95°C), annealing (55°C), and extension (72°C) reaction (at which the function is maximized). The multicomponent mass transfer of DNA components in the channel is governed by the convection and diffusion equation, and is modeled using the *Transport of Dilute Species* interface from the Chemical Reaction Engineering Module.

A step-wise approach is used to solve the coupled system of heat transfer, fluid flow, and mass transfer. Non-Isothermal Flow physics is solved at steady state, and the velocity field that is obtained is used in the mass transfer equation to obtain time varying concentrations of DNA components.

Below, we can see the steady-state velocity profile and temperature contours across the channel. We can note that a large convective cell occupies the channel and the fluid flows along the boundaries and is faster at the vertical walls where the temperature variations are highest.

*Temperature distribution and velocity magnitude.*

Next, we can look at the reaction rates for denaturation, annealing, and extension reactions after 120 seconds. It is clear that these reactions are localized around 95°C, 55°C, and 72°C respectively. It will be interesting to see the variation of these reaction rates with respect to time.

*Reaction rates for denaturation, annealing, and extension reactions after 120 s.*

Initially only denaturation is taking place in the channel, but after a certain amount of time, annealing and extension effects are also visible.

*Variation of reaction rates for denaturation, annealing, and extension with time.*

Interesting results are obtained when the concentrations of DNA components are monitored. In the plot below, you can see the concentration profile of DNA components for a simulation time of 120s.

*Concentration Profile of DNA Components.*

It is evident that the concentration of the double-strand DNA template (dsDNA) initially decreases due to denaturation, but later on exponentially increases due to the amplification effect in the extension region. It is common to use the doubling time in order to quantitatively characterize PCR efficiency. The doubling time is the time required to double the concentration of initial double-strand DNA template (dsDNA) used in the mixture. From the concentration plot, the doubling time is around 60 seconds, which is of the same order of magnitude as given in the aforementioned literature. In general, the doubling time of a laboratory scale conventional PCR system is around two minutes.

This blog post briefly described a multiphysics model for DNA amplification by PCR in a buoyancy-driven flow. It can be used to understand the amplification rates as a function of the device parameters, which will be very useful in device design and optimization. More realistic reaction data can be brought on-board, and geometric parameters can be varied. In general, it is possible to extend the scope of this model using the powerful and flexible user interfaces of the Heat Transfer, CFD, and Chemical Reaction Engineering modules.