2. Methods
An integrated parallel plate microreactor configuration is simulated.
The oxidation and reforming reactions are carried out in alternate
channels separated by walls. The stoichiometric methanol-air flow in the
oxidation channel is co-current with respect to the methanol-steam flow
in the reforming channel. Both the fluid streams enter the device at
room temperature and exit at atmospheric pressure. Using the inherent
symmetry of the geometry, only one-half of each channel and the
connecting wall are simulated. The fluid flow as well as the heat and
mass transfer equations are solved in the fluid phase and the energy
balance is explicitly accounted for in the solid wall. A steady-state
solution of the problem is sought using computational fluid dynamics.
More specifically, computational fluid dynamics is used to solve the
steady-state continuity, momentum, energy, and species conservation
equations in the fluid phase and the heat equation in the solid phase
using a finite volume approach. An adaptive meshing scheme is used for
the discretization of the differential equations. The computational mesh
is initialized with 200 axial nodes, 200 radial nodes for the oxidation
channel and the wall sections, and 200 radial nodes for the reforming
channel. This discretization translates into a total of about 80000
nodes. This initial mesh is adapted and refined during a calculation to
increase the accuracy of the solution in regions of high gradients.
Specifically, additional nodes are introduced to refine the mesh using
the tools built in the computational software. Specifically, mesh
refinement before achieving complete convergence reduces the
computational effort, but a too early refinement, namely in a few
iterations, may lead to refinement in wrong regions. Such an adaptive
meshing strategy, starting with a relatively coarse initial mesh
followed by refinement in regions of large gradients, achieves an
adequate balance between accuracy and computational effort.
The fluid density is calculated using the ideal gas law. The individual
properties of various gaseous species, such as thermal conductivity, are
calculated using the kinetic theory of gases, whereas the specific heats
are determined as a function of temperature using polynomial fits from
the thermodynamic database. Mixture properties, such as specific heat
and thermal conductivity, are calculated from pure component values
based on the mass-fraction weighted mixing law. Binary species
diffusivities are determined and then are used to calculate the
multicomponent mixture diffusivities. For the solid wall, a constant
specific heat and an isotropic thermal conductivity are specified. Given
that material conductivity varies with temperature and more importantly
with the material chosen, simulations are carried out over a wide range
of conductivities. To permit modeling of fluid flow and related
transport phenomena in industrial equipment and processes, various
useful features are provided. These include porous media, lumped
parameter streamwise-periodic flow and heat transfer, swirl, and moving
reference frame models. The moving reference frame family of models
includes the ability to model single or multiple reference frames. A
time-accurate sliding mesh method, useful for modeling multiple stages
in turbomachinery applications, for example, is also provided, along
with the mixing plane model for computing time-averaged flow fields.
Another very useful group of models is the set of free surface and
multiphase flow models. These can be used for analysis of gas-liquid,
gas-solid, liquid-solid, and gas-liquid-solid flows. For all flows,
conservation equations for mass and momentum are solved. For flows
involving heat transfer or compressibility, an additional equation for
energy conservation is solved. For flows involving species mixing or
reactions, a species conservation equation is solved or, if the
non-premixed combustion model is used, conservation equations for the
mixture fraction and its variance are solved. Additional transport
equations are also solved when the flow is turbulent.
The flow of thermal energy from matter occupying one region in space to
matter occupying a different region in space is known as heat transfer
[29, 30]. Heat transfer can occur by three main methods: conduction,
convection, and radiation. The net transport of energy at inlets
consists of both the convection and diffusion components. The convection
component is fixed by the inlet temperature specified. The diffusion
component, however, depends on the gradient of the computed temperature
field [31, 32]. Consequently, the diffusion component and therefore
the net inlet transport is not specified a priori. When heat is added to
a fluid and the fluid density varies with temperature, a flow can be
induced due to the force of gravity acting on the density variations.
Such buoyancy-driven flows are termed natural-convection or
mixed-convection flows. Multiple simultaneous chemical reactions can be
modeled, with reactions occurring in the bulk phase and on wall
surfaces, and in the porous region [33, 34]. For gas-phase
reactions, the reaction rate is defined on a volumetric basis and the
rate of creation and destruction of chemical species becomes a source
term in the species conservation equations. For surface reactions, the
rate of adsorption and desorption is governed by both chemical kinetics
and diffusion to and from the surface [35, 36]. Wall surface
reactions consequently create sources and sinks of chemical species in
the gas phase, as well as on the reacting surface. Reactions at surfaces
change gas-phase, surface-adsorbed and bulk species. On reacting
surfaces, the mass flux of each gas specie due to diffusion and
convection to and from the surface is balanced with its rate of
consumption and production on the surface. The model boundary conditions
are described next. Boundary conditions consist of flow inlets and exit
boundaries and wall boundaries. Symmetry boundary conditions are used
since the physical geometry of interest, and the expected pattern of the
flow and thermal solution, have mirror symmetry. The axis boundary type
is used as the centerline of an axisymmetric geometry. Both gases enter
the channels at room temperature with a uniform, flat flow velocity. The
reactor exits are held at a fixed pressure and the normal gradients of
species and temperature, with respect to the direction of the flow, are
set to zero. Symmetry boundary condition is applied at the centerline of
both channels, implying a zero normal velocity and zero normal gradients
of all variables. No-slip boundary condition is applied at each
wall-fluid interface. Overall, the device is adiabatic, namely no heat
losses occur through the side walls. Continuity in temperature and heat
flux is applied at the fluid-solid interfaces. Neither heat-transfer nor
mass-transfer correlations are employed since detailed transport within
the solid and fluid phases is explicitly accounted for.
The pressure-based solver traditionally has been used for incompressible
and mildly compressible flows [37, 38]. The density-based approach,
on the other hand, is originally designed for high-speed compressible
flows [39, 40]. Both approaches are now applicable to a broad range
of flows, from incompressible to highly compressible, but the origins of
the density-based formulation may give it an accuracy, namely shock
resolution, advantage over the pressure-based solver for high-speed
compressible flows [41, 42]. Two formulations exist under the
density-based solver: implicit and explicit. The density-based explicit
and implicit formulations solve the equations for additional scalars,
for example, turbulence or radiation quantities, sequentially [43,
44]. The implicit and explicit density-based formulations differ in
the way that they linearize the coupled equations. In both methods the
velocity field is obtained from the momentum equations [45, 46]. In
the density-based approach, the continuity equation is used to obtain
the density field while the pressure field is determined from the
equation of state. On the other hand, in the pressure-based approach,
the pressure field is extracted by solving a pressure or pressure
correction equation which is obtained by manipulating continuity and
momentum equations. In both cases a control-volume-based technique is
used that consists of division of the domain into discrete control
volumes using a computational grid, integration of the governing
equations on the individual control volumes to construct algebraic
equations for the discrete dependent variables, such as velocities,
pressure, temperature, and conserved scalars, and linearization of the
discretized equations and solution of the resultant linear equation
system to yield updated values of the dependent variables. The two
numerical methods employ a similar discretization process, but the
approach used to linearize and solve the discretized equations is
different. The full computational fluid dynamics problem is solved via a
segregated solver using an under-relaxation method. Convergence of the
solution is monitored through the residuals of the governing equations
and the norm of successive iterations of the solution. The coupling of
the heat equation in the wall and the reacting flow equations makes the
problem stiff due to the disparity in thermal conductivity between the
gases and the wall. Typical computational fluid dynamics simulation
times varies from about several hours to a few days on a single
processor depending on the stiffness of the problem. Natural parameter
continuation is employed to study the effect of various operating
parameters.