Always Check Your Results!

By Bruce Finlayson - University of Washington, Seattle, WA

engineering professor teaches modeling analysis via ready-to-use software

During the 21st century, modeling has unquestionably played a major role in engineering. This modeling has transformed the old methods that prototype engineers once used in design, research, and development into modeling based on finite element, finite volume, and finite difference analyses. Once the exclusive domain of skilled researchers in numerical science, these advanced modeling methods are now available for all levels of engineers through ready-to-use software packages. As a result of this software, the focus on modeling has shifted from solving the problems to evaluating the solutions obtained from analysis.

Professor Bruce Finlayson, a pioneer in engineering education at the University of Washington, teaches methods based on partial differential equations (PDEs). The engineering students at the University of Washington are privileged to learn from such an accomplished professor in the area of fluid dynamics and transport phenomena. The most important message that he conveys to his students about modeling can be summarized in one sentence: "Always check your results." Professor Finlayson has achieved remarkable results with this approach.

Undergraduate students calculate pressure losses in fuel cells

Partial differential equations form the theoretical basis for ready-to-use modeling packages in transport phenomena. Understanding the implementation and getting quick results move the PDEs from an abstract and theoretical level to a more concrete and practical level - the equations come to life.

Professor Finlayson has had great experiences using modeling tools with his undergraduate research students. He has experimented with different assignments in the field of fluid dynamics, which were solved using numerical simulations. To accomplish these tasks, Professor Finlayson's students used the finite element package COMSOL Multiphysics, which combines equation-based modeling with ready-to-use applications for transport processes.

Figure 1: Marlina Lukman investigated the laminar flow past a cavity and found a non-linear relationship between the pressure losses and the flow at Reynolds numbers above 100. Marlina's model revealed that a recirculation loop is formed, which results in an increased pressure loss.

The first assignment given to the undergraduates involved simulations of the flow in contractions and bends. This assignment originated from a series of questions from a fuel cell company in Seattle, NEAH-Power, regarding pressure losses in contractions and bends in laminar flow. It turns out that values of pressure losses for these devices were mostly available for turbulent flow. The task for these students was to solve the laminar flow field in a number of standard geometries supplied by NEAH-Power and to tabulate the results including the pressure loss.

The example above shows the flow field in an expansion and contraction section in a channel. Marlina Lukman, a chemical engineering student, performed the study. She showed that the pressure losses in the system increased linearly with the average velocity up to Re numbers of about 100. At these Re numbers, recirculation zones appear in the cavities, which increased the pressure losses. Figure 1 above shows the flow field in the cavity at Re numbers below 100 and above 100. Later, Albert Witarsa expanded the study to pipes and rotational symmetrical structures while Marlina continued with three-dimensional simulations.

The second project dealt with flow, where the viscosity depended on the composition of the fluid. This assignment was "kind of exploratory", according to Professor Finlayson. It treated two-way coupling between the equations of change, momentum balance and continuity, with the balance of mass for a dissolved species, the convection-diffusion equation. The viscosity dependence was given first as a linear function of the concentration and then as a fourth-order polynomial of concentration of the dissolved species. The assignment was carried out quickly and successfully. This was partly due to COMSOL Multiphysics's ability to interpret arbitrary expressions of the modeled variable for the physical properties. The students simply typed the analytical expression for the viscosity into a COMSOL Multiphysics window. This allowed for much freedom in the exploration of different expressions for viscosity. Students could directly study the implications of the results.

The third assignment involved the modeling of reacting flow in a microreactor, supplied by The Dow Chemical Company for investigation of reaction kinetics. Albert Witarsa carried out this assignment by treating the equations of change and mass balance for the reacting species. Albert realized the importance of using correct assumptions. In this case, an oversimplification of the geometry caused inconsistencies in the results. Figure 2 shows the concentration profile in an injection point and the mixing downstream in the reactor. The injection channel causes disturbances in the flow, which gives an improved mixing behind the injection point.

Figure 2: Albert Witarsa studied the flow and concentration profiles around the injection zone in a Dow reactor for kinetic studies. The concentration profile shows that the mixing process is dominated by diffusion and that the injected reactant is forced by the flow towards the upper wall of the reactor. The concentration in vertical cross sections along the length of the reactor is visualized in the one-dimensional plot.

In each assignment presented, the analyses included the influence of the mesh on the results. The students investigated the results for consecutive mesh refinements, which for a trustworthy solution goes toward an asymptotic value. This type of critical self-review goes back to the lesson that Professor Finlayson teaches his students - "Always check your results."

Graduate students validate patents of microfluidic devices

At the graduate level, students in a fluid mechanics class continued the calculation of pressure losses in laminar flow for more difficult cases. The simulations involved the following:

• The concepts of entry length
• A temperature increase caused by viscous dissipation
• The evaluation of patents in the field of microfluidics

In the studies of entry lengths, the students compared the analytical solutions to the results of COMSOL Multiphysics simulations, which showed possible limitations and strengths of using analytical expressions. First, they realized that most of the analytical estimations in fluid dynamics were obtained as series expansions, which might have required the calculation of a large number of the terms to reach a steady solution. Second, they learned that analytical estimations were only valid under certain limiting conditions and the importance of when they can or cannot be used. Part of this assignment was to explain possible deviations between analytical estimations and numerical simulations. Even the simplest case of developing laminar flow between parallel plates requires some in-depth analysis. In the analytical case, the velocity profile is uniform as the flow enters the channel. In a real system, the velocity has already started to develop before the fluid enters the channel, see figure 3. Junior student Trevor Plaisted developed correlations for the approach length - the distance upstream where the velocity has not changed.

In the second assignment, the students coupled the Navier-Stokes equation for fluid mechanics with a heat balance to study the effects of viscous heating. This was introduced as a heat source in the energy balance, which could be taken from the textbook and typed directly into COMSOL Multiphysics's user-friendly graphical user interface.

The third project for graduate students involved the evaluation of different patents in the field of microfluidics. Each student got a different patent. The evaluation included a complete analysis of the patent and detailed studies of the microfluidics in the patented device. This project was highly appreciated by the students because it met their expectations for a real-life project outside of university assignments.

Figure 3: Trevor Plaisted developed correlations for the approach length, which is the distance from the entrance of the channel where the velocity is uniform. The plot on the right side shows the velocity profile in nine cross sections along the height of the channel. The color levels and the cross-section plot show that the flow is influenced already at the level of the first cross section from the bottom.

The in-depth analyses of the patents revealed surprising results. Most of the patented devices could actually perform according to their claims. However, one patent did not hold for a deeper analysis and was more a result of hasty conclusions. In this case, the claimed functionality did not result from the claimed fluid mechanics. This surprising conclusion underlined again the importance of being critical when analyzing published studies and patents - just because it is in print does not mean it is true.

Figure 4 shows results of a patent that really works, according to Finlayson's students. The microfluidic device is supposed to selectively extract a single species from one laminar stream to another. The laminar flow is arranged in an H-shaped cell that puts two streams in contact with a minimum of disturbances. In such case, interchange of mass between the two streams takes place only by diffusion. Using the appropriate dimensions, only the component with the largest diffusion coefficient is transferred from one stream to the other and a selective extraction is achieved.

Figure 4: Diffusion extraction in a microfluidic device. The top graph shows the streamlines and the velocity field in the H-cell while the lower two graphs show the concentration distribution for two species with different diffusion coefficients. In this case, the species with the higher diffusivity is transported into the lower branch of the H-cell while the less mobile species is almost retained in the upper branch.

Apart from the assignments in the graduate courses, Professor Finlayson also has a group of students doing research in the field of ferrofluids. This group studies heat transfer by means of ferrofluid convection, where the temperature field and a magnetic field are used as driving forces for the flow close to a hot metal surface. In principle, the enhancement is obtained by introducing a driving force for the flow close to the surface of the heated walls. The application for this particular system is in the cooling of transformers.

Figure 5: Haijun Sun calculated the temperature and flow distribution in a closed cavity with a cooling and a heating wall (the left and right wall, respectively). The flow is mainly magnetically driven, and the model couples momentum and heat balances with the Maxwell equations for electromagnetic fields.

The problem involves the nonstandard coupling of the Navier-Stokes, heat balance, and the Maxwell equations for electromagnetic fields, which complicates the modeling. The system presents several two-way couplings, since the magnetic field presents a source in the Navier-Stokes equations while the magnetic properties are temperature and flow dependent. Despite this, the problem is easy to model in the COMSOL Multiphysics graphical user interface by just typing in the coupled expressions. The figure above shows the flow induced by the magnetic field and the temperature differences in a closed cavity. The cavity has a cooling wall at the left side and a heating wall at the right side. The driving flow due to temperature differences is introduced using Boussinesq's approximation with an added magnetic force.

Be critical about your own and others results

Professor Finlayson has clearly succeeded in teaching his students to be critical when evaluating the results from numerical simulations. You can always get a colorful plot, but you better be sure you know what it all means. Professor Finlayson feels that it is important to show students traps in the modeling process but also to let them make their own mistakes. With this goal in mind, the students taught themselves to be critical about published results. In the assignments based on numerical models, it was important to produce fairly advanced models in a timely fashion. COMSOL Multiphysics allowed them to type in the expressions for the different couplings in the momentum, heat, and mass balances in a simple and straightforward manner.

Not only did the students gain a valuable lesson, but they also found it very rewarding to explain the results, which were often surprising even to Professor Finlayson. The fact that there was no clear "right answer" to the problems made every assignment more challenging.

This concept of using numerical simulations for teaching has accelerated the learning process in fluid dynamics. It has given Professor Finlayson's students the opportunity to model a large number of diverse systems involving different types of flow. These simulations give the students hands-on experience, which is much more valuable than only an abstract theory from a published textbook.