Research
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.
Methodology
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.).