Many industrial problems require advanced mathematical modeling and sophisticated computing techniques. Our team brings together the broad range of skills required for a holistic treamtent of transport phenomena, i.e., processes in which particles or other quantities of interest are physically displaced from one location to another.  Classically transport phenomena are broadly categorized into 3 types: transport of mass, transport of energy, and transport of momentum. Starting from real laboratory data, we build and analyze abstract mathematical models, generate simulations, predictions, and data ``in silico'', and ultimately return to the laboratory with knowledge to aid in the development and enhancement of tangible technologies. 

Our group is currently has the following  industrial partners, Four Stones, Inc. (based in Edmonton), the Automotive Fuel Cell Corporation, Ballard Power (both based in Vancouver), Simula Research Lab (based in Oslo, Norway), IBM (based in Toronto and New York), and the Environment Canada National Hydrology Lab (based in Saskatoon).

The main themes in this project are: developing models and computational tools for simulation of fuel cells, electrical activity in myocardial tissue, and hydrological flows and design of catalytic converters for carbon sequestration from point-source emitters such as coal-fired electricity-generating stations. Below we provide more details on each of these themes.

Fuel cell research:

Four Stones, Inc., the Automotive Fuel Cell Corporation, and Ballard Power Systems are prominent Canadian leaders in the development of fuel cell technology. Their designs promise cleaner, more efficient power for the automotive industry and for stationary electrical power plants. We are developing analytical and numerical models that describe the reactant gas (hydrogen and oxygen) and water movement in fuel cell stacks. Such models are helping these companies to improve cell efficiency and durability.


Scientific Computing is an extremely important tool for many industrial applications. There are well-developed fluid dynamics codes, for example, that are widely used to optimize designs or investigate the structures of a flow. Similarly commercial codes are available for simulating elastic structures and for investigating fatigue and plastic failure. It is possible to use these simulation tools to make improvements to product design where physical design and testing is too expensive or not possible. For instance, Ford has invested more than 20,000 person-hours in the computer modeling of P2000 (their new all aluminum engine).

However, many problems of industrial interest involve interrelated physical processes that evolve on widely disparate time scales. Before sophisticated numerical methods can be employed, it is often wise for the models and their discretizations to be analysed to develop a reduced set of equations that describe the phenomena of interest but which are amenable to analytical and scientific computational methodologies. Our group has the expertise in physical modeling and the associated computational tools to work with its industrial partners to develop such reduced models.

We have expertise in four overlapping areas: industrial liaison, mathematical modeling, asymptotic analysis, and scientific computation. An overlapping of skills is essential for the development of effective industrial collaborations. Good modeling is also a fundamental aspect of computing, and many times a "pre-analysis" can isolate the really interesting phenomena that can then be computed with more ease and accuracy. Similarly, computations can often reveal the physical phenomena of interest in poorly understood models and in turn lead to better models.

Project Details

The hydrogen fuel cells that we study combine hydrogen and oxygen to produce electricity with a pollution-free end product of water (if pure hydrogen is used as fuel). The reaction is catalyzed by a thin layer of Platinum, to which humidified reactant gases are delivered via a series of pressurized flow channels on either side of a Nafion membrane. The membrane is permeable to only water and protons, and it is supported by a gas diffusion electrode (GDE) that is currently made from a porous teflonated carbon fiber paper. The GDE allows reactant gases to reach the active catalyst sites on the fuel cell membrane and carries current away from the sites. We are developing analytical and numerical models of problems arising in the durability and efficiency of fuel cell stacks. There is a broad family of modeling and design issues that we have addressed over the lifetime of the project:

Transport processes in fuel cells: There are several important transport processes in fuel cells. The most straightforward perhaps is mass and heat transfer through the GDE (from the flow channels to the catalyst layer and membrane). Of interest is the effect that the GDE geometry and material parameters have on cell performance and longevity. Basic questions must be answered concerning gas, liquid, and heat convection and diffusion within the GDE. Another fundamental issue is water management. The membrane must be kept fully saturated with water to function optimally and avoid deterioration; however water must be removed quickly from the cathode (oxygen) side GDE to prevent pore blockage that inhibits oxygen from reaching the catalyst sites. The gas flow modeling in the porous GDE involves Darcy's Law with the diffusion of multi-component gas relative to mole-averaged velocities given by the Maxwell-Stefan equations. Thermal convection and diffusion, including heat of reaction and transfer to the graphite plate, reactant gas flow, and coolant, must also be accounted for. Even in the relatively simple structure of the GDE, there are some tough questions concerning the nature of two-phase flow and also questions about two-phase flow in the gas channels (see below). Pore network simulations and detailed volume of fluid calculations have been done by the group to gain some understanding of the various phenomena.

The membrane is also a key element in the performance of a fuel cell. Its task is to separate the fuel components (hydrogen and oxygen) and to transport the protons from the anode to the cathode while being an electronic insulator. To provide a high protonic conductivity in membranes that have been used to date (such as Nafion), the membrane must attain a high water content. This is usually measured in numbers of water molecules per fixed ion group (sulphonic acid group) of the membrane. A dry membrane not only performs quite poorly, its life time also decreases dramatically. There is a lack of microscopic understanding of the important processes of water and proton movement in these membranes. It is known that the negatively charged sulphonic acid groups (SO3-) form clusters in which most of the water is present. These so-called "micelles" are of order 50 Angstroms in diameter and connected by even smaller channels. As the water content decreases, so does the channel width and, hence, the proton conductivity of the material. It is an important concept that there are no clearly defined phases (liquid, vapour) of water in the membrane. As the water content is increased from zero, the first molecules in a pore will interact with the sulphonic acid groups to form some sort of bound state. It is only for a critical level of water content that some molecules begin to move freely and the effect of the SO3-groups decreases. In addition, there are also water molecules that are bound to the free flowing protons. It becomes clear that both the proton conductivity and the water diffusivity are increasing functions of the water content. Membrane models based directly on first principles are deemed to be impractical for use in larger simulations. Instead, models of a reasonable structure that are empirically fit to experimental data are used. These models have been shown to match data over a wide range of conditions. Membrane models that include the effects of swelling (with water uptake) against compressive force are currently in development.

The catalyst layer is perhaps the most difficult structures to model in fuel cells. This is the region between the GDE and the membrane. It contains carbon-supported platinum particles, upon which the reaction occurs. The region is also impregnated with membrane material that allows protons to reach reaction sites. Gas pores allow reactant gas to reach reaction sites. Electric current is carried in the carbon grains to the carbon fibres in the GDE, through the graphite plates into which the gas channels are carved, and out to the external circuit. Our group has developed state-of-the-art agglomerate models (in which the microstructure is captured in a limited way) of the catalyst layer as well as considering details of reactions near triple points of pore, membrane, and catalyst.

As we move to model complete 3-D devices, the size and scope of the numerical computations increase dramatically. Certain elements of the models we consider can be done using standard CFD software, but other aspects (the electrochemistry, for example) have to be input as user-defined subroutines. In another approach, reduced-dimensional models that capture sub-layer effects in an average way have also been considered.

Unit Cell Model (1+1D): Hydrogen fuel cells with any appreciable power output are constructed of several unit cells. In each of these cells, a single Membrane Electrode Assembly is sandwiched between two graphite plates into which flow channels are etched. We were asked by our industrial partners to develop a simple computational model that could reproduce a large experimental data set, to which we were given access and in which local current densities of a single unit cell were measured at several locations from inlet to outlet for a wide variety of operating conditions. There were several constraints on the model complexity: the final version would need to include many effects, but it would also need to be run by design engineers on PCs. This was a wide-ranging effort, involving some of our previous insights into the GDE flow, our most advanced understanding of the membrane, standard electrochemical models for the catalyst process, and the use of some reliable literature parameters and others fit to the data. The resulting model is a coupled system of ODEs for anode and cathode channel average molar gas fluxes per unit cell width down the length of the channel. The processes of proton and water transport through the membrane are modelled as one-dimensional and can almost be solved analytically in terms of the channel fluxes (a nested sequence of scalar inverse problems results). Often in this particular data set, the until cell was run in counter-flow mode, that is the flow direction of the cathode gases was opposite to that of the anode. In the model, this manifested itself as a situation where data for different solution components are given at different locations. Numerical solutions are obtained using a technique known as forward-backward shooting. The resulting code can accurately (to within about 10%) predict cell voltage and local current density over a wide range of conditions.

Unit Cell Models (3D): Full 3-D unit cell models have also been developed using commercial CFD code (CFD-Ace) with several additional features added via user-defined subroutines. This has been a joint project of the software company, Ballard, and members of our group. On an on-going basis, we have compared the results of this full code and the reduced-dimensional models above to identify where the 3D models are really needed to give accurate performance predicitions. The development of a hybrid technique, using 3D computations only where needed, is underway.

Stack Model: Several unit cells are often placed in series into fuel cell stacks to produce a higher voltage, and hence more power, in a compact unit. Due to the computational complexity of stack modelling, we use copies of the reduced dimensional (1+1D) unit cell models discussed above. These are system-level models, with the reactant flow network through headers and cells is considered as a nonlinear resistive network. The main objective of this study is to predict the variation in flow rate through the unit cells (and so the variation in the voltage produced, using the unit cell models developed above). In addition, cells interact thermally and electrically with each other. These effects have been included in the latest generation of our reduced-dimensional stack simulation tools.

Additional Reactions: Unit fuel cells are placed in series in fuel cells stacks as discussed above. Each unit cell has the same total current. All the models described above deal with fuel cells operating under "reasonable" conditions. For example, it is assumed that the amount of reactant gases supplied to the cell is sufficient to generate the specified total current. When this is not true, some of the total current must come from other reactions, powered by the sum of the voltages in the healthy cells in the stack. In some situations, this current comes from carbon oxidation of the catalyst support. This is a major source of performance degradation in fuel cell systems. Carbon oxidation can also occur during start-up transients. Unit cell models able to predict carbon oxidation in fuel-starved situations have been developed.

Condensation Front Modeling: Comparing the local temperature and vapour pressure found in the GDE to saturation curves, it is possible to predict likely condensation regions. To proceed and predict the motion of the liquid water leads to interesting modeling, analysis, and computational issues. Standard capillary pressure models (Leverett's empirical curve for water in sand, for example) do not apply to our case: the teflonation of the carbon fiber paper, which is essential for water removal, also keeps the water in the non-wetting phase. Experimental work is clearly needed to develop accurate models of capillary pressure. The models of multi-phase motion naturally lead to several possible zones in the GDE, each separated by a free boundary: water only, gas only, and two-phase zones where temperature and vapour pressure obey a saturation relation. Within this project, we have turned to simpler condensation problems in a system with only water and water vapour and simple boundary conditions, for which there is experimental work for comparison. In the Engineering literature, two-phase zones are often modeled as being at constant temperature, and it is an interesting asymptotics problem to determine under what conditions this is valid (the GDE is quite thin, and it is unclear whether this assumption will be valid under this scaling, for example). The simple water-vapour system is also a good forum for developing computational methods for the two-phase free boundary problem. This can be considered as a  generalized steady state Hele-Shaw problem. We are considering shape optimization techniques and level set methods as candidates for these computations. Such a study requires significant computational resources, even in two dimensions.

Liquid Water Flow in Channels: Our group is also considering the movement of liquid water in the channels. Liquid water in small rivulets or droplets is blown out of the cell by the shear forces of the channel gas flow. Experimental estimates of the water flux in each channel can be obtained easily from the current density and geometry of the fuel cell plates, corrected for the amount that remains in the vapour state. If the water is in a rivulet form, it is a relatively straightforward process to calculate the size of the rivulet (indicating what percentage of the channel it will block) and how that will affect the pressure drop needed to maintain the channel flow. Models of droplet motion are less straightforward. The physics at the contact line is poorly understood. We have proceeded in two ways. First, like many others, we have applied slip boundary conditions at the solid boundary. We are considering some novel approaches using immersed boundary and shape optimization type techniques. Second, we have made some attempts to understand the contact line dynamics in a more fundamental way.

Heart simulation:

Mathematical models and computer simulation are becoming important tools in cardiovascular research.  Mathematical models can simulate heart conditions and the effects of certain drugs designed to treat them.  Presently, the development of a drug often costs hundreds of millions of dollars. One aim of computer simulation is to reduce this cost, e.g., by reducing the number of physical experiments needed in designing a drug. The other is to improve our understanding of heart function and its pathologies non-invasively or in ways experiments cannot provide insight.

Electrophysiological models of the heart describe how electricity flows through the heart, controlling its contraction. The flow of ionic currents at the myocardial cell level are governed by systems of ordinary differential equations (ODEs). Cardiac electrophysiological models are often based on the Nobel prize-winning work of Hodgkin and Huxley in the 1950s that modelled neural tissue mathematically as a circuit.  Modern cardiac electrophysiological models adapt this work to describe electrical activity in the heart and include data gathered from experiments to form models with increasing physiological accuracy. 

Because of their intricacy, obtaining physiologically accurate mathematical models is a difficult task. A further challenge to obtaining physiological accuracy is that of performing the simulation efficiently.  To move effectively beyond models for one cell, enough cells must be included in the model to realistically approximate the geometry and physiology of the heart.  Because the heart has approximately 10 billion cells, any realistic simulation will have enough cells (or clusters of cells) to dramatically magnify any inefficiencies in the numerical method.  This has forced some researchers to reduce the physiological accuracy of their models to make the simulations feasible.  In tissue-scale models such as the monodomain or bidomain models, the ODEs for myocardial cell models are coupled with partial differential equations (PDEs) describing the flow of electricity through myocardial tissue.  The models are numerically stiff, and so standard (explicit) numerical methods are often unable to provide efficient simulations. If the efficiency of the simulation process can be significantly improved, then greater physiological accuracy and subsequently obtain more useful data can be obtained.

The Simula Research Lab is one of the world's leading research groups on heart simulation. We are working closely with them to develop and validate their models, but our greatest contributions are in developing efficient time-integration methods for the differential equations describing electrical activity in myocardial tissue. Our recent results for individual cell models indicate that variable-stepsize implementations of low-order implicit-explicit Runge-Kutta time-integration methods that take advantage of problem structure outperform all other methods used in practice. We are presently working on discovering ways to apply these results to tissue-scale models such as the monodomain or bidomain models.

Catalytic converters for carbon sequestration:

The abundance of man-made greenhouse gases in the atmosphere is believed by many to be inexorably raising temperatures worldwide. The thinking among environmental scientists is that if unchecked, the toll exacted by this global warming will be massive.  For example, with the melting of the polar ice caps, many coastal cities will be flooded.  These coastal cities represent a large fraction of the population of all nations with coastlines of significant length. Even more alarmingly, recent studies show that the amount of greenhouse gas emissions is actually accelerating at a rate that outstrips even the most pessimistic predictions to date.

The solution to the problem of global warming will undoubtedly arise through a combination of efforts, from energy conservation to the development of renewable (or “green”) energy sources. An important component to the solution will be the reduction in emission of greenhouse gases (mainly carbon dioxide and methane) into the atmosphere, especially until new “greener” ways of consuming and producing energy become the norm. In light of this need for reduced emissions, we are partnering with IBM to establish a computation-based investigation into strategies for the removal of carbon dioxide directly from the point of emission, a process known as carbon sequestration. We aim to target the large point source emitters, such as coal-fired electricity generating plants, which are still one of the main sources of the man-made greenhouse gases in the atmosphere.

The goal of carbon sequestration strategies is to react the carbon dioxide with something to form environmentally benign materials. Many of the carbon sequestration strategies fall into the category of “capture and storage” methods.  These methods seek to exploit the chemical properties of carbon dioxide, such as its ability to adhere to metals, to then safely dispose of it by, e.g., storing it underground or further reacting it with other metals, such as calcium, to form stable carbonate salts. Some carbon sequestration strategies seek to form as the end product specialty chemicals, such as formaldehyde, acetic acid (vinegar), methanol, etc. Simulations of these reactions are reported to take copious amounts of computer time, e.g., hours for each micro-second of simulated time.

Specific objectives are to (i) construct a model of the relevant reactions in a typical (or specific) coal-fired electricity generating plant; (ii) apply and/or develop novel numerical methods for the simulation of the model; (iii) optimize the distribution of reaction products over parameters such as operating temperature, pressure, feed gas composition, and catalytic converter composition. It is important to have numerical methods that allow long time integrations in order to study the aging of the catalyst. This research is of particular interest to our industrial partner IBM, specifically their Big Green division, which is interested in a broad range of research to benefit the environment.

Hydrological flows:

Simulation of hydrological flows is a computationally demanding task both in terms of the volume of data and the computer processing power required. To meet these demands, researchers in the field of hydrological flows are turning to high-performance computers.  However, as we reach the physical limits on clock speeds of standard CPUs, computers are becoming increasingly parallel in order to continue to provide the potential for performance increases. There are several levels of parallelism within a typical computer cluster: many computers may be connected by a network, each computer may contain multiple processors, processors may have multiple cores, and each processor-core may be capable of executing multiple threads simultaneously. Furthermore, each level of hardware (network, computer, processor, and core) may require a unique parallelization method to produce optimal performance. This makes realizing the full potential of parallel processing one of the most important problems in science today. Our research group has considerable expertise in the numerical solution of differential equations and algorithms and software for high-performance computing. We hope to leverage this expertise in order to assess the current parallel programming strategies used in the various predictive models of Environment Canada and develop new techniques to improve the their performance (accuracy, efficiency, robustness, etc.).